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Design and analysis of computer algorithms (dave mount, 1999)

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Page 1: Design and analysis of computer algorithms (dave mount, 1999)

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; < = > ? @ < A B C D ? @ E < F G > @ D H ? = > I D GJ K L M N O P Q R S T U V W S X Y Y Y Z[ \ ] ^ _ ` a b c d e � � � � f g � h � � � i � g j � � f � � � � � g j d g k h f l � � � � mn � � o � � � � � d � p � i f � c � � l f � q � k i d � � � f d p o � � i c d e � � � � f g � h m

; < = > ? @ < r B s E t u v > D > I = E w G H x ? u u w > I D G EJ K M U O P Q R S T U V y S X Y Y Y Z[ \ ] ^ b h � l f � q c d e � � � � � j � j d g k z f g � h � � � i � g j � � f � � � � � g j d g k h f l � � � � m{ | ] } _ ~ ] ` ] � a � � _ } | � � � � � � � � � k � � g � � d g � � � � � � � � � � � � � d g � q � p p � k � � g � k � � i e � � d � f � g d p e � � �

� � k � � � � c d � � d � � � � � i � l d p � � � d � � � � � � d g k e � � k � � � � � � i � l d p � � � d � � � � � � � m � f � � d � � � � f g �� � � f e � j d g d p � � � f � c i e � � l f k � � d � � � e � � � � � � � e i � � c � k o � � � � p l f g � d � � i e � � d � f � g d p e � � � p � i m � g �p f � � e � � � � d i � j d p � � � f � c i � d � � g � � k � e � g k � g � � g d e d � � f � � p d � e � � � � d i i f g � p d g � � d � � j i d � c f g � j� � � � � i j � � � � i e f p � � m � c � � d � � i d � c � i d � f � d p � g � f � f � � j q c f � c � d g � � � c � � � c � � o d � � � g g f g �� g � � i � � � � � � o � � � � � � � � � � � � � � � � � q f � c d g f g � g f � � � d g k � i d � � � � � i � i � � � d g k d g � g p f i f � � kq � � k � f � � m � p � � � f � c i k � � f � g f � d p p d � � � � � c � i d � c � i d � f � d p � c � � � � � � c f g k � c � k � � f � g � o � � � ke � � � � d i � m

{ | � ~ } � ^ � ] � a � � _ } | � ^ \ ~ _ a ` � n � � � � d i i f g � f � d l � � � � � i e p � � � d � � m � c � � � d � � d g � i � � � � od � e � � � � � o e � � � � d i i f g � � c d � i d � � f � � � � � i e p � � m � c � � � � � f � � c d � i � � � e � � � � d i i f g � e � �   � � � �d � � l � � � p d � � � j � � ¡ � f � f g � � c � � � � � k f g d � � k � ¢ � � � � � o i d g � e � � e p � m � � c f � f � � c � � � e f � d � � � � � �p f � � � � z f g � � o � q d � � � g � f g � � � f g � m � � c � g � � � f � � c d � i d g � e � � � � d i i f g � e � �   � � � � f g l � p l �� � � � f g � d g k d � � � � � f g � p d � � � ¡ � d g � f � f � � � o k d � d � £ � f � g � p � m � � c f � f � � c � � � e f � � o � � � � � � � � g k d � d� � � � � � � � � � d g k k d � d � d � � � p f � � � � � � d g k � � � m � � c � p d � � f � � c d � i d g � e � � � � d i i f g � e � �   � � � �f g l � p l � � � p l f g � � � i e p � � � � i e � � d � f � g d p e � � � p � i � j o � � q c f � c � f i e p f � � f � � � g d f l � � � p � � f � g � i d �g � � � � � £ � f � g � � g � � � c m � c � � � i e p � � e � � � p � i � i d � f g l � p l � g � i � � f � d p k d � d � � c � � � �   � � � � o� � � � � � � � g g � i � � f � d p d g d p � � f � j p f � � � � ¤ ¤ � j � � � � o � � g � c � � f g l � p l � k f � � � � � � k d � d m � c f � f �q c � � � � c � � � e f � � o d p � � � f � c i k � � f � g d g k d g d p � � f � f � f i e � � � d g � m� p � c � � � c � c � d p � � � f � c i � k f � � � � � � k f g � c f � � � � � � � q f p p � o � � g � � e � � � � g � � g p � d � f g � o � d � � f � g � o � c �� � k � � c d � f � � � g � � d � � k f g d p d � � � � � o � q d � � � � � � � i j � c f � � i d p p o � d � � f � g i d � � � l � � � f i e � � � d g �o � � � c � � � � � � � � � o � c � � l � � d p p e � �   � � � m � g � g o � � � � g d � � p � � � i i � g d e e � � d � c � � � c f � e � � � p � i f � � �� � � � k � � f � g d g f g � £ � f � g � d p � � � f � c i d g k k d � d � � � � � � � � � � � � � p l � � c � e � � � p � i j d g k � c � g � d � � � c f �e � � � k � � f � g d g k d � � � i e � � � � g � � � � g � f � � e � � o � � i d g � � m � c � e � � � p � i f � � c d � f o � c � � g k � � p � f g �k � � f � g f � � d k j � c � g � o � � g g � d i � � g � � o � g � � � � g f g � f � � � f g � � � i d � � d � � � � � d g � f d p k f ¢ � � � g � � m� c � o � � � � � o � c f � � � � � � � f � � g c � q � � k � � f � g � � � k d p � � � f � c i � j d g k c � q � � d g d p � � � � c � f � � £ � f � g � � m¥ � q f p p � � � k � d g � i � � � � o k f ¢ � � � g � � � � c g f ¡ � � � o � � k � � f � g f g � d p � � � f � c i � � k f l f k � � d g k � � � g ¡ � � � jk � g d i f � e � � � � d i i f g � j k � e � c � � � � � � � d � � c � j d g k d e e p � � c � i � � d g � i � � � � o k f ¢ � � � g � e � � � p � i � m

¦ § ¨ © ª « ¬ ­ ® ¯ ° ± ² ³ ¬ ´ µ ¶ µ ¨ · ¸ ¯ ° ¹ º º º ° ± » © ¯ ¶ ¨ ¼ § ¨ ½ © · ¯ » « ¾ ¿ ¬ » ¸ ¿ » ° À ¸ ¬ ³ » « Á ¬ ¯ ª ¨ ¼ µ ² « ª  ² ¸ ´ ° § ¨   » ­ » à ² « Ä ° µ ± ° Å Æ Ç È Å ¶É ® » Á »  » ¿ ¯ · « » ¸ ¨ ¯ » Á Ê » « » © « » © ² « » ´ Ë ª ± ² ³ ¬ ´ µ ¨ · ¸ ¯ ¼ ¨ « ¯ ® » ¿ ¨ · « Á » § µ ¾ § È Ì ¹ Í Î É À Ì Å Æ Ï Ð ° ± » Á ¬ ­ ¸ ² ¸ ´ Ñ ¸ ²  ª Á ¬ Á ¨ ¼§ ¨ ½ © · ¯ » « Ñ Â ­ ¨ « ¬ ¯ ® ½ Á ° ² ¯ ¯ ® » À ¸ ¬ ³ » « Á ¬ ¯ ª ¨ ¼ µ ² « ª  ² ¸ ´ ° § ¨   » ­ » à ² « Ä ¶ à » « ½ ¬ Á Á ¬ ¨ ¸ ¯ ¨ · Á » ° ¿ ¨ © ª ° ½ ¨ ´ ¬ ¼ ª ° ² ¸ ´ ´ ¬ Á ¯ « ¬ Ë · ¯ »¯ ® » Á » ¸ ¨ ¯ » Á ¼ ¨ « » ´ · ¿ ² ¯ ¬ ¨ ¸ ²  © · « © ¨ Á » Á ² ¸ ´ Ê ¬ ¯ ® ¨ · ¯ ¼ » » ¬ Á ® » « » Ë ª ­ « ² ¸ ¯ » ´ ° © « ¨ ³ ¬ ´ » ´ ¯ ® ² ¯ ¯ ® ¬ Á ¿ ¨ © ª « ¬ ­ ® ¯ ¸ ¨ ¯ ¬ ¿ » ² © © » ² « ¬ ¸²   ¿ ¨ © ¬ » Á ¶

Page 2: Design and analysis of computer algorithms (dave mount, 1999)

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� g � g k � � � � d g k f g � � o � � � k k � � f � g � � � c g f ¡ � � � f � � � f � f � d p � � � � f g � d � p � � � � � � k e � � � � d i � m � gd k k f � f � g j f � f � f i e � � � d g � � � � � d � p � � � ¡ � f � � p � d g d p � � � � c � � � g g f g � � f i � � � o � c � � � k � � f � g �� q f � c � � � � � e � g � f l � e � � � � � � e f g � d g k � � � � f g � � m ¥ � q f p p � � � f g q f � c d � � l f � q � o � c � d g d p � � f �� � � c g f ¡ � � � j q c f � c q � � � � � l � � � k f g � c � e � � � � ¡ � f � f � � � � � � � � j � � � m � � � c d e � � � � � � z � o � h o � � i � � � f g o � � i d � f � g m

� ~ � � � } � } _ � ~ b � c � o � � i � p d � � c d � d � � k � � f l � k o � � � c � � � g g f g � � f i � � � o e � � � � d i i d � � o � � g � � ¡ � f � �� � i e p � � m ¥ c � g k � � f � g f g � d p � � � f � c i � j � c � i d f g e � � e � � � � o � c � d g d p � � f � f � � � � � � d � � g � �o � � � c � � � � g k f g � c � d p � � � f � c i � � � � g g f g � � f i � m � � g � � d � � d g d p � � f � f � e � � � d � p � � � � � k � g � � �f i e p � i � g � f g � � c � d p � � � f � c i d g k i � d � � � f g � n � � � � � g k � m � ¥ � q � � p k p f � � d � f i e p � q d � � o� � e � � � � g � f g � � � i e p � � o � g � � f � g � j q c f � c � d e � � � � � � c � � � � � g � f d p � � � q � c � d � � e � � e � � � f � � m � c f � f �� c � e � � e � � � � o � � � � � � � � � � � m� � � i e � � � f � d g d p � � f � f � � d � � k � g � q � � f i e p f o � f g � d � � � i e � f � g � j q c f � c c � p k f g i � � � � � � � g � � d p p �� d � � � m � � � f � f � f i e � � � d g � � � � g k � � � � d g k � c � � � d � � � i e � f � g � d g k � c � p f i f � d � f � g � � o d � � i e � � � f �d g d p � � f � m� ] � a \ _ ` � � } ~ _ � \ ~ b ¥ � d � � i � � � f g � � � � � � � k f g c � q � c � � � g g f g � � f i � � � � q � o � � p d � � � l d p � � �

� o m a ` � � \ � � ` ~ } ] ` } � ] � } � � ~ b � c � d � � � d p � � g g f g � � f i � � o � c � e � � � � d i k � e � g k � � g l d � f � � � � � g �

� � d g � o d � � � � � f g � c � f i e p � i � g � d � f � g � � � k f g � � � f � � � j � e � f i f � d � f � g � f g � � i e f p d � f � g j � e � � k � o� c � � g k � � p � f g � c d � k q d � � j � � � � m � c � � � o � � � j q � q f p p f � g � � � � � g � � d g � o d � � � � � m

� c �   � � � f � � d � f � g o � � � � g � f k � � f g � p d � � � f � � c d � f o f � � i d p p j � c � g d p i � � � d g � d p � � � f � c i f �o d � � � g � � � c m n � � e p � d � � i � � � � � g � � � g � k d � � � � � � g g f g � � f i � � o � � p d � � � f g e � � � m � � � � c � i � � �e d � � j � c � � � d � � � i e � f � g � d � � � � d � � g d � p � q c � g i d � f g � � � i e d � f � � g � � � � q � � g o � g � � f � g � � c d � c d l �� f � g f � � d g � p � k f ¢ � � � g � � � c d l f � � � m � � � � � d i e p � j � � e e � � � q � c d l � � q � e � � � � d i � j � g � q c � � �� � g g f g � � f i � f � � � � � � d g k d g � � c � � q c � � � � � g g f g � � f i � f � � � � � � � � m � � c � p d � � � �d p � � � f � c i i d � � � o d � � � � � � � d � � � f � � � � � d i � � � � � e c f � � f � d � � k d g k � � i e p � � d p � � � f � c i j d g k � c �d k k � k � � e c f � � f � d � f � g � � � � p � � f g d p d � � � � � � g � � d g � o d � � � � m � � � � � i d p p � � m � m j � � � � � c � � � � �d p � � � f � c i f � � c � o d � � � � � o � c � � q � m � � � d � � � � � i � � p d � � � � � c � � � p d � f l � k f ¢ � � � g � � � f g � � g g f g �� f i � � � � � i � i � � c � � � d � � � m � � � � i f g � � g � i f p p f � g � e � � d � f � g � e � � � � � � g k m

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Page 3: Design and analysis of computer algorithms (dave mount, 1999)

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Page 4: Design and analysis of computer algorithms (dave mount, 1999)

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Page 5: Design and analysis of computer algorithms (dave mount, 1999)

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Page 6: Design and analysis of computer algorithms (dave mount, 1999)

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Page 7: Design and analysis of computer algorithms (dave mount, 1999)

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Page 8: Design and analysis of computer algorithms (dave mount, 1999)

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L(0) = 1 L(1)=1 L(2)=2 L(3)=3 L(4)=5

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Page 9: Design and analysis of computer algorithms (dave mount, 1999)

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Page 10: Design and analysis of computer algorithms (dave mount, 1999)

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� c � g f � e d � � f � f � g � � c � d � � d � f g � � � p � i � g � � � c d � d � � p � � � � c d g d g k � � � d � � � � c d g � c � e f l � � m� c � g f � � � � � � � f l � p � � � � � � � d � c e d � � m� � f � � � � � � f � q f k � p � � � � d � k � k d � � c � � � � � � � � � o � c � o d � � � � � � f g � d p � � � f � c i � � � g i � k � � gi d � c f g � � � m � g � � � e p d g d � f � g f � � c d � f � � f g g � � p � � e � � i e d � � � � p � i � g � � d � d f g � � d � f g � p �e f l � � l d p � � j q c f � c � d g � � � � � � � k f g d � � � f � � � � o � � o d � � d � � � � � m � c � � � c � � d p � � � f � c i �� � i e d � � � q � � p � i � g � � f g � c � d � � d � m � c f � f � � � g � f k � � � k d g � � � � � � � � � � � f g � d p � � � f � c i j� f g � � f � � � � � g � � � c � � d � � d � � � � � d � � m � � � k � � � f i e p f � f � p � � � � � c � � � � � � i � � � � � � � � f � g � � d � � j� � � � c f � f � � � � d p p � g � � � � � g � � k m � � � f � � � � � � � � � � m� c f � d p � � � f � c i f � � � p � � � f g � c � � � � � � � � � � � � � j d g k � � � � f g � c � q � � � � � d � � m � c �e � � � d � f p f � � � c d � � c � d p � � � f � c i � d � � � d � � i e � � � f � d p p � p � g � � � � d � � � i f g � � c d � � c � e f l � � f �� c � � � g � d g k � i p � � f � � � � � � i � p � � i d p p o � � p d � � � m� \ � a \ ~ � � } b � � � � � � � � d p � � q � � � � � � � � � � f l � p � m � � f � d � p d � � f � d p k f l f k � � d g k � � � g ¡ � � � d p � � � f � c i m� c � d � � d � f � � e p f � f g � � � q � � � � d � � d � � � o � � � � c p � � ¡ � d p � f � � m � c � � d � � � � � � � k � � � � � � f l � p � m� c � g � c � � q � � � � � � k � � � d � � d � � d � � i � � � � k � � � � � c � � f g � � � � f i � m � � � � � � � � f � � c � � g p � � � � � � � � � � � f g � d p � � � f � c i � o � c � � � � c � � � m � c � k � q g � f k � f � � c � � � � � �� � � � f � � c � � g p � d p � � � f � c i � o � c � � c � � � � c d � � � ¡ � f � � � d k k f � f � g d p d � � d � � � � � d � � � f � g � � f g � � c �� � � � � � f � g � � d � � � j d g k � c � � f � f � � � � � � � � � � � m � c f � f � � � � d � � � � c � i � � � f g � e � � � � � � i � � � � �� c � � q � d � � d � � f g � � d � c f � k d � � d � m � p � c � � � c f � f � e � � � f � p � � � i � � � � d � � d � � f g � e p d � � j f �� d g g � � � � k � g � f g � � � � f i � m� \ ] � ~ � � } b � � d e � � � � f � � d � � k � g d g f � � k d � d � � � � � � � � � j � d p p � k d � � � � j q c f � c f � d g � £ � f � g � f i �e p � i � g � d � f � g � o d e � f � � f � � ¡ � � � � k d � d � � � � � � � � � m � e � f � � f � � ¡ � � � � � � e e � � � � � c � � e � � d � f � g �� o f g � � � � f g � d � � � j d g k k � p � � f g � � c � � p � i � g � q f � c � c � � i d p p � � � � � � l d p � � m � c � d e � d g � �� � f p � o � � � � � � f g � � � � f i � j d g k � c � i f g f i � i � � � � d g � � � � � � d � � � k f g � � p � � � � f i � m� � d e � � � � f � d g � � � � � � � � � � � f g � d p � � � f � c i j � � � f � f � � � � � � � � � � m

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Page 11: Design and analysis of computer algorithms (dave mount, 1999)

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� � d e � � � � q � � � � � � � � f p k f g � � c � c � d e � � � k � � � k f g � � l � � � � � � k � � � � � c d � � c � i d � f i � i� d g � � � � � � d � � � k � £ � f � g � p � � d g k � c � g � � e � d � � k p � � � � � d � � f g � � c � p d � � � � � � p � i � g � m � ¥ c � f �� � � � d � � � � c � i d � f i � i � d � c � � � c d g � c � i f g f i � i f � d g f i e p � i � g � d � f � g k � � d f p j � � � � c f � f �� c � � � � � � i d � f g � � c f � q � � � d � d g f g � e p d � � � � � � f g � d p � � � f � c i m �� o � � � � g p � q d g � � � � � � � d � � � c � � � i d p p � � � l d p � � � j d c � d e � d g d p p � q � � � � � k � � c f � f � � � � p � � � � f i � m � c � d e c d � � c � d k k f � f � g d p d k l d g � d � � � o � � f g � � � � k f g � � g � � � � � q c � � �� c � e � f � � f � � � o � p � i � g � � � c d g � � � m � d � c � c d g � � � o e � f � � f � � � � � � l d p � � � � d g � � e � � � � � � � k f g � p � � � � f i � m

� � � \ � � � � ` ^ ~ � � � � � � � ] � _ ~ � ` � � ] ~ \ ^ � � � } _ ` a b � c � o d � � � c d � � p � � � � � � � f g � d p � � � f � c i �d � � � c � o d � � � � � d � � � g k o � � i d g � � � d � � j � � � � � � � � � c d � � c f � i d � � � � c � � � � � � c d � q � � d g k � m d g q � � � � � f g � � p � � � � f i � � � � � d p p � c d � � c f � i � d g � d � � i e � � � f � � � � f � � p � o d � � � � p � � � f i � � �¥ � q f p p � f l � d g d � � � i � g � � c d � g � � � � � f g � d p � � � f � c i � d � � k � g � � i e d � f � � g � d g � � o d � � � � � c d g� c f � m� � � � ] � _ ~ � ` � � ] ~ \ ^ � � � } _ ` a � � a � � _ } | � b � c � i d g g � � f g q c f � c � c � d p � � � f � c i e � � i � � � � � c �

� p � i � g � � f � � d � � k � � p � p � � g � c � � � � � p � � � o � c � � � i e d � f � � g � � c d � � c � d p � � � f � c i i d � � �� � � q � � g � c � � p � i � g � � � � � � � � � � � k m

� � � � � g � � d p � e � � e � � � � � � � f g � d p � � � f � c i � d � � � � i e d � f � � g � � d � � k � d � d � � d p p � c � d p � � � f � c i � k f � �� � � � � k d � � l � � m ¥ � q f p p � � � � c d � � � � � e � f � g � � � f � � f g � e � � f d p � d � � � m � c f � k � � � g � � e � � � p � k � � c �e � � � f � f p f � � � o � � � � f g � d p � � � f � c i � q c � � � d � � f � g � d � � k � � � � i f g � k � � � � c � � � e � � d � f � g � m¥ � q f p p � c � q � c d � d g � � � i e d � f � � g � � d � � k � � � � f g � d p � � � f � c i o � � d � � ¡ � � g � � � � � � � � � � � � � � � � i � � �i d � � d � p � d � � � � p � � � � � i e d � f � � g � f g � c � q � � � � � � d � � m � c f � f � � � d p p � d k f £ � � p � � d � � f o � � � � c f g �d � � � � f � j � � � d � � � � � � e � � � o � d g g � � � d � � d k l d g � d � � � o d g � � g � q p � k � � d � � � � � c � e � � � � d i i f g �p d g � � d � � � � f g � � � � k j � � � c � i d � c f g � � � f g � � � � k j � � c � q � c � d p � � � f � c i � � � � d � � � � k � � f k f g �q c f � c � p � i � g � � � � � � i e d � � m � c � � g p � o d � � � c d � q � � d g � � e p � f � f � � c d � � c � d p � � � f � c i �� d � � f � g �d � � k � � � � i f g � k � � � c � � � � � p � � o f � � � � i e d � f � � g � m

� \ � _ ~ _ � ` � � \ \ � � a � � \ ` } b � g � � k � � � � e � � l � p � q � � � � � g k � j q � g � � k d g d � � � � d � � q d � � o i � k � p f g �� d g � e � � � f � p � � � � � � f g � d p � � � f � c i � � f g � � q � � d g g � � f i d � f g � q c d � � � � � � o � � � d � f l f � � d g d p � � � f � c ik � � f � g � � � o � c � o � � � � � i d � � i e p � � � m � g � � � i e d � f � � g � � d � � k � � � � f g � d p � � � f � c i d g k d g f g e � �� f � � j � d g � � l f � q � k d � � � � d � � p � � c � � � � c d � � � � � � � � � � d p p � k d � � � � � � � � � � � � m ¥ � � c f g � � o � c �

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Page 12: Design and analysis of computer algorithms (dave mount, 1999)

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d p � � � f � c i � � m � m � � � � � m � c � � q � � � d g � c � � � � e � � � � g � � � � � j � � � e � � � f l � p � m � p p f g e � �� � ¡ � � g � � � f g q c f � c � � � � � � � g � f g � � k � q g � c � p � o � � � d g � c d g k � c � � � q f � c � � � � �� � g � f g � � d p � g � � c � � f � c � � � d g � c m

� \ ] � ` � ^ \ b � d � c p � d o g � k � � � � � � � e � g k � � � d e � f g � f g � c � d p � � � f � c i q c � � � d p p � c � � � i e d � f � � g �c d l � � � � g i d k � m � � k � � g f � f � g � o d � � i e d � f � � g � � d � � k d p � � � f � c i j � c � d p � � � f � c i � � d � � f � g �� � c � e � � i � � d � f � g f � � � g � � d � � � � f � � � i e p � � � p � k � � � � i f g � k d � � c f � e � f g � m � c � k � e � c � o dp � d o g � k � � k f � � d g � � o � � i � c � � � � � � f � � c � g � i � � � � o k � � f � f � g � i d k � � � o d � m � � � d f g p � � c �� � g g f g � � f i � � o � c � d p � � � f � c i i � � � � � d � p � d � � � c f � p d � � � m

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� f � � � � � � � � f � f � g � � � � o � � � � � � f g � m� � � � c � d g d p � � f � j q � q f p p g � � k � � � � � � c � o � p p � q f g � � d � f � o d � � d � � � � � f g d � � � � � � � m � � � g � � c �� � � � � � � o d � f g d � � � � � � � � � � � c � p � g � � c � o � c � p � g � � � � e d � c � g � i � � � � o � k � � � � o � � i � c � � � � � � �d g � p � d o m� \ � � ] b � � f g d � � � � � � q f � c p � d l � � c d � c � f � c � d � p � d � � p � m � � � � b � � � f � � � c d � d � � i e p � � � � f g d � � � � � � � o c � f � c � � c d � � � p � d l � � j d g k � c f � f � � c � p d � � � � �

g � i � � � � o p � d l � � e � � � f � p � o � � d g � � f g d � � � � � � q f � c � c f � c � f � c � m � � o � p p � q � � c d � f o f � � c �g � i � � � � o p � d l � � � o � � i � � � � � � o c � f � c � � j � c � g � � � j f i e p � f g � � c d � � � p � m

� � q j c � � � f � � � � i d f g � � � � p � m� | \ � � \ � b � g � � � i e d � f � � g � � d � � k � � � � f g � d p � � � f � c i c d � q � � � � � � d � � � � g g f g � � f i � � � p � � � m � � � � b � g � f k � � d g � � � � � f g � d p � � � f � c i d g k f g � � � � � j d g k � � g � f k � � � c � � � � � p � f g � k � � f � f � g � � � � m

� � � � � k � g � � � � c � g � i � � � � o � � i e d � f � � g � � c f � d p � � � f � c i i d � � � f g � c � q � � � � � d � � j � c d �f � j � � f � � ¡ � d p � � � c � c � f � c � � o � c � k � � f � f � g � � � � m� � q i d g � p � d l � � i � � � � c � k � � f � f � g � � � � c d l � � � o � c � f g e � � � � g � f � � � � o k f � � f g � � g � i � � � � j� c � g � c � � � g � i � � � � � � � p k � � e � � � � g � � k f g d g � � o � k f ¢ � � � g � e � � i � � d � f � g � m � � � � d � ck f ¢ � � � g � e � � i � � d � f � g j � c � d p � � � f � c i i � � � e � � i � � � � c � g � i � � � � f g d g � � � � g � f d p p � k f ¢ � � � g �q d � m � c f � f i e p f � � � c d � � c � g � i � � � � o p � d l � � f g � c � k � � f � f � g � � � � f � d � p � d � � � j f i e p � f g �� � � � � p � i i d � c d � � c � c � f � c � � o � c � � � � � f � d � p � d � � p � � m¥ � � d g d e e p � � � � � � � � � �� � � � � � � � � � � � � � o � � � � � � � � h e d � � z � � f � p k f g �

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Page 13: Design and analysis of computer algorithms (dave mount, 1999)

� � � � � � � � � � � � � � � � � � � � � � �

� c f � � � i e p � � � � � c � e � � � o m� c f � � d g d p � � � � � � g � � d p f � � k � � � c � q � c d � � c � � � � � � � � � � � � f i � � � � � � � f � d p � � � � p � � � m

� _ ` \ ] � � _ � \ � � � } _ ` a b � c f � p � q � � � � � g k f i e p f � � � c d � f o q � c � e � � � � � � � g � i � � � � o d � � � � � c d g f g � p � � � � f i � j q � � d g g � � k � f � � � i d � f g � � � i e d � f � � g � d p � g � m � � � � q � � � g � f k � � � c � ¡ � � � � f � g� o q c � � c � � f � f � e � � � f � p � � � � � � � q f � c � � � � c � � � � � o � � i e d � f � � g � m � c � � d g � q � � f � � � � j � � � � g p �� g k � � l � � � � � � � � f � � f l � � f � � � i � � d g � � � m� � � ` } _ ` a � � � } b � � g � f g � � � � � d � � � i � � � c d � � d � c f g e � � f � d g f g � � � � � f g � c � � d g � � o � � i � � � � m � c �d p � � � f � c i � � � � � f g � � � � � � f i � m � o � f � � g � q g � � � � � � � j � c � g � c f � f i e p f � � � c d � � c � � � � � p � f g �� � � � f g � d p � � � f � c i f � � � � � f i � m� � � g � � c � � � � � � o d g � p � i � g � f g d g d � � d � � � � � � c � g � i � � � � o � p � i � g � � f g � c � d � � d � � c d �d � � p � � � � c d g � � � ¡ � d p � � f � m � c � � d � f � f k � d f � � � k � � � � i f g � � c � � d g � � o � l � � � � p � i � g � f g � c �d � � d � m � g � � � � � � g � q � c � � d g � � � o � c � � p � i � g � � j � � � � � � � � � � f i e p � � � e � f g � � d � c � p � i � g �� � � c � d e e � � e � f d � � p � � d � f � g � o � c � � g d p � � � � � k � � � e � � d � � d � � � � � � � i � � d � � f � g � � k � k f o � c � � �d � � k � e p f � d � � � � m � c � ¡ � � � � f � g f � c � q � � � g k � c � � d g � � o d g � p � i � g � q f � c � � � � � i e d � f g � f � � �� c � � � c � � � p � i � g � � � o � c � d � � d � � � � � d � � � � c � � p � i � g � � d � � � l � � � c � � d g � � � � � � � � � � � � � j q � � d gi d f g � d f g d g d � � d � � c d � � � � g � � � c � g � i � � � � o � � � � � � � g � � � � o � d � c � p � i � g � m � � g � f g � � � � � � � � �� c � o � p p � q f g � � c � � � d � � d � � m� � � � � � � � p k � � c � f g f � f d p f g e � � m � �� � f � � c � � g � f � � � � � � � k j d g k � �� � � � � � f � � c � f g � � � � � � � � l d p � �

� g q c f � c � � � � � � m� � � � � � � � � d � � o � � � � � k � q c f � c c � p k � � c � � � � � � k � � � e � � m� � � � � � � � g d � � d � � o f g � � � � � � m � � � � q f p p � � g � d f g � c � � d g � � o � f g � j q c � � � � � � � � � � � m� c � d p � � � f � c i f � � � i d � � d � p � � f i e p � j � � � k � � � e � f l � p � � p � l � � m � c � d p � � � f � c i � e � � d � � � � � � � � �� � g � � � � � � f g � � m ¥ � k � � c f � f g � q � � � � e � m � f � � � q � � � � � � � � � � � � � c � g � i � � � � o � p � i � g � � � o� �� � q c � � � � � � f � � ¡ � d p � � � m ¥ � � d g k � � c f � f g f � f d p f � f g � � � � � � � � j d g k � c � g o � � � d � c � j o � � i� � � j q � f g � � � i � g � � �� �� � � � � � � � � � m � c � � j f o � �� � � � � � � j � c � g � c � � c � p � i � g � � o � f �f g � � � i � g � � k j f g k f � d � f g � � c d � q � c d l � � � � g � g � i � � � m � � k � � � � i f g � � c � g � i � � � � o � p � i � g � �� c d � d � � p � � � � c d g � � � ¡ � d p � � � j q � � � e p d � � � � � � q f � c � c � � � i � o � p � i � g � � f g � c � � � � d � � d �� � � � � � � m � c f � f � k � g � � �   � � � � � � e f g � d � � g g f g � � � � d p � o � c � � p � i � g � � � o � m� � q � � � � g � q � � g � d f g � � c � � d g � � o � m � c f � i � d g � � c d � f o � � � �� � � � � � � c � g � c � � g d p e � � f � f � g� o � �� � � c � � p k � � d � e � � f � f � g � � � � f g � c � � g d p � � � � � k d � � d � m � c � � j q � � � � � �� � � � � � � �� � m � � � f � �� c d � � c f � � � e f � � � c � � g � f � � � � � � � k j g � �   � � � � c � � � � l d p � � m � c � � � f � d � � � � p � � � c � � � c � q � l � � m¥ � g � � k � � � � � d � � o � p f o � c � � � d � � k � e p f � d � � � j � f g � � q � k � g � � q d g � � c � i � � � l � � q � f � � � c � � d i �p � � d � f � g � o � m � � k � � c f � j q � k � � � � i � g � � � � � d o � � � � � e � f g � m � �

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Page 14: Design and analysis of computer algorithms (dave mount, 1999)

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Page 15: Design and analysis of computer algorithms (dave mount, 1999)

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Page 16: Design and analysis of computer algorithms (dave mount, 1999)

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Page 17: Design and analysis of computer algorithms (dave mount, 1999)

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Page 18: Design and analysis of computer algorithms (dave mount, 1999)

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o � � � c � � p � � d p e � � � p � i � � � � � � p l � k � e � f i d p p � j � d � c � � � e � � � p � i � c � � p k � � � � p l � k � e � f i d p p � m � � � ` � � _ ] � � � � ] ` � ~ � � � � � � � \ � ~ b � g f i e � � � d g � d � e � � � � � � c � � £ � f � g � � � o � n f � � c d � � c �� � � d p g � i � � � � o � � � e � � � p � i � � � � � � � p l � k � c � � p k � � d � i � � � d e � p � g � i f d p g � i � � � m

� } � _ ` a ~ b � g � f i e � � � d g � d � � d � o d p � � � f � c i k � � f � g f � � c � � � � k � � o d p � � � f � c i � o � � � c d � d � � � � � � � f g � � m� c � � � d � � d g � i � � � � o f i e � � � d g � e � � � p � i � c � � � m � i � g � � c � i � � � f i e � � � d g � c d � � � k � q f � c� £ � f � g � p � � � d � � c f g � o � � d � � � � � � f g � � � � � g � � d p p � d e d � � � � g f g p d � � � e f � � � � o � � � � m � � c f � f � q c d �� � � � � k f � � � � d g k o � g � � f � g � p f � � � � � � e � k � q c � g � � � e � � o � � i d � � d � � c m � � g i d g � f g � � d g � � �� � � k � g � � q d g � � � � g k d e f � � � � o � � � � � � d � � p � j � � � � d � c � � � � i � � c f g � � c d � f � � � f i f p d � � m � c f �d � f � � � o � � � � d i e p � f g � � g � � f � � � � � � d � � c m � � g � � f � � � k � � d � � � � � � � k d � p � g � � � � i � p � � � p � � m � c �� � � � � � d g k � � d g � � � � � � � g k � q g f g � � d p � g � � � ¡ � � g � � � � d � c � o q c f � c f � � g � � o o � � � � d � f �� � e � � j � j � j � m � � � � � d � � i d � � c � � � d � � p � � � � � � f g � f � p � � � � � � d � � � � o � i d p p � c d g � � � f g � � �� � e p f � d � f � g m � � � � c f � � � d � � g j f � f � � o f g � � � � � � � � � � i e � � � � f i f p d � f � f � � � � � q � � g � � � f g � � � c d � k �g � � i d � � c � � d � � p � m � g � � � i i � g i � � c � k � o i � d � � � f g � � c � k � � � � � � o � f i f p d � f � � � � � q � � g � q �� � � f g � � f � � � � � i e � � � � c � f � p � g � � � � � � i i � g � � � � � ¡ � � g � � m

� � ` a \ ~ } � � � � � ` � � � ~ \ � � \ ` � \ b � � � � � � c f g � � o � c d � d � � � � � � � f g � � d � � � ¡ � � g � � � � o � c d � d � � � � � m� f l � g � q � � � ¡ � � g � � � � � � � � � � � � � � � � � � � d g k � � � � � � � � � � � � � � � j q � � d � � c d � � f � d � � � � � � � � � � � � o � f o � c � � � f � d � � � f � � p � f g � � � d � f g � � � ¡ � � g � � � o � f g k f � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � c � c d � � � � � � � � � � � � � � � � � � � � m � � � � � d i e p � j p � � � � � � � � � � � � � � � �d g k p � � � � � � � � � � j � c � g � f � d � � � � � ¡ � � g � � � o � m� f l � g � q � � � � f g � � � d g k j � c � � � � � � � � � � � � � � � � � � � � � � � � � � o � d g k f � d p � g � � � � � � ¡ � � g � �� q c f � c f � � � � c d � � � � � ¡ � � g � � � o � d g k m � � � � � d i e p � j p � � � � � d � � � o � � � d g k p � � �� � � � � � � � � � � � � m � c � g � c � p � g � � � � � � i i � g � � � � � ¡ � � g � � f � � � � � � � � � � � m� c � � � g � � � � � i i � g � � � � � ¡ � � g � � n � � � p � i � � � � f � � c � o � p p � q f g � m � f l � g � q � � � ¡ � � g � � �� � � � � � � � � � � � � d g k � � � � � � � � � � k � � � � i f g � d p � g � � � � � � i i � g � � � � � ¡ � � g � � m � � � � � c d �f � f � g � � d p q d � � � g f ¡ � � m � � � � � d i e p � � c � � � � o � � � � � d g k � � � � � f � � f � c � � � � � � � � � � � � m

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Page 19: Design and analysis of computer algorithms (dave mount, 1999)

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Page 20: Design and analysis of computer algorithms (dave mount, 1999)

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with back pointers includedLCS Length Table

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LCS = BCB3221

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2211

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start here

X = BACDB

X: X:

Y: Y:

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Page 21: Design and analysis of computer algorithms (dave mount, 1999)

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Page 22: Design and analysis of computer algorithms (dave mount, 1999)

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Page 23: Design and analysis of computer algorithms (dave mount, 1999)

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Page 24: Design and analysis of computer algorithms (dave mount, 1999)

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Page 25: Design and analysis of computer algorithms (dave mount, 1999)

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Page 26: Design and analysis of computer algorithms (dave mount, 1999)

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Page 27: Design and analysis of computer algorithms (dave mount, 1999)

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Page 28: Design and analysis of computer algorithms (dave mount, 1999)

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� � � � � � � � � � � � � � m � � f � g � � c d � k � � e � � l � � � � f g k � � � f � g � � c d � � l � � � � � f d g � � p d � f � g � o d g � � f k � ke � p � � � g � � g � f � � � � o � z � c � � k � d g k � � � � f d g � p � � m� � f d g � � p d � f � g � d � � � o f g � � � � � � o � � d g � i � � � � o � � d � � g � m d g � � � � i � � � f � d p � � � f � c i � e � � d � � � �� � � � k � � � i e � � f g � d � � i e p � � e � p � � � g d p � c d e � f g � � � � f d g � p � � m � c � g d g d p � � � f � c i � d g � � d e e p f � k� � f d g � p � � � � � f d g � p � m � � � g � � c � � � � � � � � � � � o � c � � � f d g � � p d � f � g � � � � d � � d e c q c � � � l � � � f � � �d � � � c � � � f d g � p � � j d g k f g q c f � c � q � l � � � f � � � � c d � � d � f k � f o � c � � q � � � f d g � p � � � c d � � d � � i i � g� c � � k m � � � � � l � � c d � � c � k � d p � � d e c f � d o � � � � � � � m � � g � � d p � � � f � c i � o � � � � d l � � � f g � � � � � � � d g� � � � � k o � � � � d l � � � f g � � c � � � f d g � p � � � o d � � f d g � � p d � f � g m� g � � g � � d p j � f l � g d � � g l � � e � p � � � g j � c � � � d � � i d g � e � � � f � p � � � f d g � � p d � f � g � m � g o d � � j � c � g � i � � �f � � � e � g � g � f d p f g j � c � g � i � � � � o � f k � � m ¥ c f � c � � f d g � � p d � f � g f � � c � � � � � � � � � c � � � d � � i d g �� � f � � � f d � c d � d � � � � � k k � e � g k f g � � g � c � d e e p f � d � f � g m � g � � � f � � � f � g f � � � f i d � f g � � c d � � � � i � � �� e d � � o � � � c � f g � � � � � � � f g k � d q f g � � c � � � f d g � � p d � f � g j d g k � � � q d g � � � i f g f i f � � � c � d i � � g �� o f g � � � � � � � m � c f � � � � � � � � � � c � o � p p � q f g � f g � � � � � � f g � e � � � p � i � _ ` _ � � � � � \ _ a | } � � ` � \ � � � � � a � ` } � _ ] ` a � � ] } _ � ` b � f l � g d � � g l � � e � p � � � g k � � � � i f g � � c �

� � f d g � � p d � f � g � c d � i f g f i f � � � � c � � � i � o � c � e � � f i � � � � � � o f � � � � f d g � p � � m

A triangulation Another triangulation The dual tree

� f � � � � � z � � f d g � � p d � f � g � m� f l � g � c � � � k f � � f g � � l � � � f � � � � � j � / j � j q � k � � g � � c � � � � � � � � o � c � d � � � � f d � � k � � f d g � p � � � � c �q � f � c � o � g � � f � g

� � � � � � / � � � � �� � � / � � �� / � � � �� � � � �q c � � � �� � � / � k � g � � � � � c � p � g � � c � o � c � p f g � � � � i � g � � � � / m

� � � � \ ~ � � ` ^ \ ` � \ } � � _ ` ] � � � � \ \ ~ b � g � � � e p d g d � f � g � � c f g k � c � � f i f p d � f � � � o � � f d g � � p d � f � g � d g k� c � � c � � c d f g i d � � f � i � p � f e p f � d � f � g d p � � � f � c i f � � � � � � � � l � � c d � � � � c d � � o � g k d i � g � d p p � � � p d � � k� � � f g d � � � � � � � m � g � c � � d � � � o � c � � c d f g i d � � f � i � p � f e p f � d � f � g j � c � d � � � � f d � � k � f g d � � � � � � f � � c �� l d p � d � f � g � � � � o � � � c � i � p � f e p f � d � f � g j q c � � � � c � p � d l � � � o � c � � � � � � � � � � � e � g k � � � c � i d � � f � � � jd g k � d � c g � k � � o � c � � � � � f � d � � � � f d � � k q f � c d e � � k � � � � o d � � ¡ � � g � � � o � q � � � i � � � i d � � f � � � m� � � � � � c d � � c � � � f � d � f i f p d � � � � � � � e � g k � g � � c � � � j � � g � f k � � d g � � � � � � f k � k � � g l � � e � p � � � g� � � � � � � � � � � � � � � � j d g k � � � g � � f k � � o � c � e � p � � � g � � d � � � � � � m � � q � � g � f k � � d � � � � � k � f g d � �� � � � q c � � � � � � � g � k � f � � c � � � f d g � p � � � g � d f g f g � � f k � � � � � j q c � � � f g � � � g d p g � k � � d � � � c � g � k � �� o � c � k � d p � � � � j d g k q c � � � p � d l � � � � � � � � e � g k � � � c � � � i d f g f g � � f k � � � o � c � � � � � m � � � � � l � � c d �e d � � f � f � g f g � � c � e � p � � � g f g � � � � f d g � p � � f � � ¡ � f l d p � g � � � d � f g d � � � � � � q f � c p � d l � � j d g k l f � �l � � � d m � c f � f � f p p � � � � d � � k f g � c � � � � � � � � p � q m � � � � � c d � � l � � � � � f d g � p � f � d � � � � f d � � k q f � c d gf g � � � g d p g � k � � o � c � � � � � d g k � l � � � � k � � � o � c � � � f � f g d p e � p � � � g j � � � � e � o � � � c � k f � � f g � � f � c � k� � d � � f g � � f k � � � � � j f � d � � � � f d � � k q f � c d p � d o g � k � � o � c � � � � � m� g � � � � � � � � � c f � � � g g � � � f � g m � c � g � c � o � p p � q f g � � q � � � � � � l d � f � g � o � p p � q � d � f p � m � � � � � l � � c d �� c � d � � � � f d � � k � f g d � � � � � � c d � p � d l � � j d g k c � g � � � � � � � d g k d � k � � � � p � � � g � f g d � � � � � � � � � �f g � � � g d p g � k � � m � f g � � � d � c f g � � � g d p g � k � � � c � � � c d g � c � � � � � c d � � g � � k � � � g � � � f g � f � j � c � � �

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Page 31: Design and analysis of computer algorithms (dave mount, 1999)

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Page 32: Design and analysis of computer algorithms (dave mount, 1999)

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Page 33: Design and analysis of computer algorithms (dave mount, 1999)

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Page 34: Design and analysis of computer algorithms (dave mount, 1999)

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Page 36: Design and analysis of computer algorithms (dave mount, 1999)

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d � � f l f � � � m ¥ � � p d f i � c d � � c f � f � d o � d � f � p � � � c � k � p � � f m � m j f � c d � g � f g � � � o � � f g � d � � f l f � f � � � m� c f � f � � � � d � � � � � � � � � d g g � � c d l � d g � � � c � � d � � f l f � f � � � c d � � � d � � � � o � � � � � g f � c � � j � f g � �� � c � � q f � � � c � � � d � � f l f � f � � q � � p k f g � � � o � � � q f � c � m � f g � � � f � � � k � � g f � f � g � c � � � � � d � � f l f � �� � � g f � c j f � c d � d g � d � p f � � � g f � c � f i � � c d g � j d g k � � f � � d g g � � f g � � � o � � � q f � c d g � � o � c �d � � f l f � f � � f g � � � � � m � c � � j � � f � d o � d � f � p � � � c � k � p � m p � d � p � � d g k � � � � g � d f g � c � � d i �g � i � � � � o d � � f l f � f � � j f i e p � f g � � c d � � � f � d p � � � e � f i d p m� � ] _ � b � c � � � � � k � d p � � � f � c i � f l � � d g � e � f i d p � � p � � f � g � � � c � d � � f l f � � � � c � k � p f g � e � � � p � i m � � � � b � c � e � � � o f � � � f g k � � � f � g � g � c � g � i � � � � o d � � f l f � f � � m � � � � c � � d � f � � d � � j f o � c � � � d � �g � d � � f l f � f � � j � c � g � c � � � � � k � d p � � � f � c i f � � � f l f d p p � � e � f i d p m� � � � c � f g k � � � f � g � � � e j p � � � � d � � � i � � c d � � c � � � � � k � d p � � � f � c i f � � e � f i d p � g d g � � � � � od � � f l f � f � � � o � f � � � � � f � � p � � i d p p � � � c d g �� � j d g k q � e � � l � � c � � � � � p � o � � � m � � � � � � � � c � � � �� o d � � f l f � f � � � c d � k � g � � f g � � � o � � � q f � c d � � f l f � � � m � c d � f �

� � � � � � � � � � � � � � �� � � � � l � � c d � d g � � � p � � f � g o � � � � � d g � � i d k � f g � � d � � p � � f � g o � � � � � � f i e p � d k k f g �d � � f l f � � � j d g k l f � � l � � � d m � f g � � � f � f g � c � � e � f i d p � � c � k � p � � � � � c � d � � l � � p d f i � j f � o � p p � q �� c d � � � e � � k � � � d g � e � f i d p � � p � � f � g o � � � c � � l � � d p p e � � � p � i j q � � c � � p k � � � � � � c � k � p � �d g k � c � g d e e � g k � c � � e � f i d p � � c � k � p � o � � � � m � � � � � f g k � � � f � g � � f g � � �� � � �� � � j � c f � f �� � d � � p � q c d � � c � � � � � k � d p � � � f � c i k � � � m

� � ] � } _ � ` ] � � ` ] � ~ ] � � � � � � \ � b � c � � p d � � f � d p � � � � � � � � � � � � � � � � � � � f � d o d i � � � � e � f i f � d � f � ge � � � p � i m � � c f � o f � � � � � f g � d � � � � � j d g k � g k � f � � i � q c f � c � d g � � � d � � g m � c � � � c f � � i f �q � � � c � � k � p p d � � d g k q � f � c � � � e � � g k � j q c � � � � � d g k � � d � � f g � � � � � � m � � q d g � � � � � d � � d �l d p � d � p � d p � d k d � e � � � f � p � j � � � c d � d � g d e � d � � � c d � � d g � g p � � d � � � � � � � d p e � � g k � m ¥ c f � cf � � i � � c � � p k c � � d � � � � � c � � � d � � g � c d � � c f � f � � d p p � k � � � � g d e � d � � f � � c d � � d � c f � � i i � � � � �p � o � � � � � � � d � � g � � � f g f � � g � f � � � � m � � f � g � � e � � � f � p � � � � d � � e d � � � o d g f � � i � � i � p � f e p � � � e f � �� o d g f � � i m � � c f � � e � f i f � d � f � g e � � � p � i d � f � � � f g f g k � � � � f d p e d � � f g � d e e p f � d � f � g � m � � � � � d i e p � j� � � i d � q d g � � � � c f e � � i � � � � � � � � o f � � i � � g d � � � � � � o p f i f � � k � d e d � f � � m� g � � g � � d � � j f g � c � � � � � � � � � � � � � � � � � � � � � � � � � � � c � � � � � e f � � � d � � p � � c � � d i � j � � � � c � � c f � o f �d p p � q � k � � � d � � d g � � � � � � � � � � o d g f � � i o � � d o � d � � f � g � o � c � q � f � c � d g k d o � d � � f � g � o � c � l d p � � m� � j � � � i f � c � � c f g � � o � d � c � �   � � � d � � � f g � d � d � � � o � � p k j q c f � c � � � � d g e d � � f d p p � � i e � � � � �� � o � � � � d � f g � m� c � � � � � g d e � d � � e � � � p � i f � c d � k � � � � p l � j d g k f g o d � � f � f � d g � n � � � i e p � � � e � � � p � i � i � d g f g �� c d � � c � � � e � � � d � p � k � � � g � � � � f � � d g � £ � f � g � � � p � � f � g � m � � q � l � � j � c � � � f � d l � � � � f i e p � d g k� £ � f � g � � � � � k � d p � � � f � c i o � � � c � o � d � � f � g d p � g d e � d � � e � � � p � i m � � � � � � � � � � � k � g � � � � c � � � � � � � � � � � � � � � � � � m ¥ � � � � � � c � f � � i � f g k � � � � d � f g � � � k � � � o � � j d g k d k k � c � i f g � c f � � � k � � m� o � c � f � � i � � � j q � � d � � f � d p p m � � � � i � e � f g � � c � � � f � d g f � � i � c d � k � � � g � � � � f g � c � � � i d f g f g �� e d � � m ¥ � � d � � d � i � � c � o � c f � f � � i d � e � � � f � p � j � c � � � p p f g � � c � � g d e � d � � � g � f � � p � m � c f � f �f p p � � � � d � � k f g � c � � � � � � � � p � q m� � f � � d � � � � � � � � c d � � c � � � � � k � d p � � � f � c i f � � e � f i d p o � � � c � o � d � � f � g d p e � � � p � i m � f l � g d � � � iq f � c � d � � � � o � � p k j � f p l � � j d g k � � � g � � j � � � q � � p k � � l f � � � p � � d � � d � i � � c � � p k d � e � � � f � p � j� c � g � d � � d � i � � c � f p l � � d � e � � � f � p � j d g k � c � g d � i � � c � � � g � � d � e � � � f � p � m � � � f � q � � p k g � l � �� � g � � � � � � � � � d � � d p f � � p � p � � � � � p k � � � c d � � � � � � � p k � � e p d � � f � q f � c d g � ¡ � d p l � p � i � � o � � � g � � m� � � � d g d p � � � � � q c � � c � � � � � k � d p � � � f � c i i d � g � � � � � e � f i d p f g � c � � � � � d � � m � g � f k � � � c �� � d i e p � � c � q g f g � c � � � � � � m � o � � � q � � � � � � � � � � c � f � � i � � � � � j � c � g � � � q � � p k � � � � � d � �� c � f � � i � � o q � f � c � j � c � g � � j d g k � c � g � � f g � � � c � f � � i � o q � f � c � � � k � � � g � � � � � � � � q � � p k� � � � p � o � � � c � f � � i � o q � f � c � z � j o � � d � � � d p l d p � � � o � z � � � � � � � � � � � � � � � m� g � c � � � c � � c d g k j f o � � � c d k � � � g p � � � � � � � k � j d g k f � g � � � k � c � f � � i � o q � f � c � j � c � g � � �� � � p k � d � � � c � f � � i � � o q � f � c � � � � d g k � � o � � d � � � d p l d p � � � o � � � � � � � ¤ � � � � ¤ � m � c f � o � d � � � �

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� � � � f g � d p � � � f � c i � o � � � � � c k f � � � � � k d g k � g k f � � � � � k � � d e c � m � g � � f � f l � p � j d � � � � � f � d � � p p � � � f � g� o l � � � f � � � � � g � k � � j � � g g � � � � k � � d � � p p � � � f � g � o � k � � � m � � d e c � d � � � � � � � i � p � f i e � � � d g � � � �� d � � � � c � � d � � d l � � � � � � f � p � i d � c � i d � f � d p i � k � p o � � i d g � d e e p f � d � f � g e � � � p � i � m � d � f � d p p � jd g � � f i � � � � c d l � d � � � � o � �   � � � � j d g k � c � � � f � � � i � � � � g g � � � f � g � � � � � � p d � f � g � c f e � � � � f g � � � �d � � f � g � � � � q � � g e d f � � � o � �   � � � � j d � � d e c f � d � � � k q d � � � i � k � p � c f � m � � d i e p � � � o � � d e c � f gd e e p f � d � f � g f g � p � k � � � � � � � � � � � � � � d g k � � � � � � � � � � � � � � � � � � � � � � j � � � � d g k � � c � � � � � � � � o � � � � �� � � � � � � � j � � � � � � � � � � � � � � � � k o � � � c d e � k � � � � f e � f � g f g � � i e � � � � � d f k � k k � � f � g d g k � � � � � d e c f �f g o � � i d � f � g � � � � � i � j � � � � � � � � � � � � � � � � � � � � � f g � � c � k � p f g � � � � � � i � m � c � p f � � � o d e e p f � d � f � g f �d p i � � � � � � p � g � � � � l � g � � g � f k � � � g � i � � d � f g � f � m � � � � o � c � e � � � p � i � f g � � i e � � d � f � g d p � � d e c � c � � � � � c d � q � q f p p � � g � f k � � d � f � � � � � d � � � � c � �d � � � o f i e � � � d g � � � � � g � � � i � � � � o � c � � � d e e p f � d � f � g d � � d � m � � � � c � � i � � � j i d g � � o � c � � �e � � � p � i � o � � i � c � � d � f � � � f p k f g � � p � � � � o � � i q c f � c i � � � � � i e p � � d p � � � f � c i � d � � � c � g � � f p � m

� � ] � | ~ ] ` ^ � _ a � ] � | ~ b � � � � o � � � c d l � � g � � � g � � � � k � c � g � � f � g � � o k f � � � � � k d g k � g k f � � � � � k � � d e c � f g � � c � � � � � � � � � j� � q � q f p p � f l � d ¡ � f � � � l � � l f � q c � � � m n p � d � � � � d k � � � � f � g � m � j m j d g k � z m � f g � h o � � o � � � c � �f g o � � i d � f � g m� \ � ` _ } _ � ` b � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � g � f � � � � o d � g f � � � � � � j � d p p � k � c �

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Page 40: Design and analysis of computer algorithms (dave mount, 1999)

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Page 41: Design and analysis of computer algorithms (dave mount, 1999)

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1

32

4

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Adjacency matrix Adjacency list (with crosslinks)

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� �

Page 42: Design and analysis of computer algorithms (dave mount, 1999)

� � � � � � � � � � � � � � � � � � � � � � �

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s

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33

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21

1

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Shortest path tree(one of many possible)

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Page 43: Design and analysis of computer algorithms (dave mount, 1999)

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Q: e, b

Q: d, e, b

Q: (empty) Q: b, f, g

Q: c, d, eQ: a, c, d

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f

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1 1 1

1 11111

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deb, f, g

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Page 44: Design and analysis of computer algorithms (dave mount, 1999)

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Page 45: Design and analysis of computer algorithms (dave mount, 1999)

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Page 46: Design and analysis of computer algorithms (dave mount, 1999)

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Page 47: Design and analysis of computer algorithms (dave mount, 1999)

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� � � � � � � m � o � c f � � k � � f � d � � � � j o � � q d � k j � � � � � � � � k � � j � c � g � � � � � � � � � m � o � c � � k � � f � d� d � � � k � � � c � g � � � � � � � � � m � � � � b � � � � � � � j o � � q d � k j d g k � d � � � k � � � j � c � e � � � o o � p p � q � k f � � � � p � o � � i � c � e d � � g � c � � f �p � i i d m � � m � m o � � d o � � q d � k � k � � � � � � � j � f � d k � � � � g k � g � � o � j d g k � � � � � � � d � � � � g f � cf g � � � l d p f � � � g � d f g � k q f � c f g � � � j f i e p � f g � � c d � � c d � d g � d � p f � � � g f � c � f i � m � � � � d � � � � �� k � � � � � � � q � � g � q � c d � � c � � q � � f i � f g � � � l d p � d � � k f �   � f g � m ¥ c � g q � q � � � e � � � � � � f g �� j � q d � g � � q c f � � � � � c � � q f � � � � � � � q � � p k � � d � � � � � k � � � j f i e p � f g � � c d � � q d � � � d � � � k� � o � � � � m � � � d � � � � c � f g � � � l d p � d � � k f �   � f g � j � i � � � c d l � d p � � � g f � c � k � � o � � � � m

� \ � � ] b � g � f k � � d k f � � d e c � � � � � � � d g k d g � � � � o � � � � � o � � � m � c d � d � � � p � f o d g k � g p �� c � � � � o � � � � � c d � d � d � � � k � � m � � � � b � � � � o � c � � � f � d � d � � � k � � � � � � � j � c � g � f � d g d g � � � � � � � o � j d g k � � o � p p � q f g � � � � �� k � � � o � � i � � � � q � � � � d � � � p � m� � � ¥ � � c � q � c � � � g � � d e � � f � f l � m � � e e � � � � c � � � d � � g � � d � � � k � � � m � � � c � p � i i d d � � l � j� d � c � o � c � � � i d f g f g � � � e � � � o � k � � � j � � � � j o � � q d � k j d g k � � � � � d p p c d l � � c � e � � e � � � � � c d �� c � � � � o � � i l � � � f � � � q f � c c f � c � � � g f � c f g � � f i � � � l � � � f � � � q f � c p � q � � � g f � c f g � � f i � m� c � � d p � g � d g � e d � c j � g f � c � f i � � k � � � � d � � i � g � � � g f � d p p � j f i e p � f g � � c � � � � d g � � g � � � � p � m

� \ � ] � \ b � � � d � � � k � � � i � d g � g � � � � p � � m � � � � � � � c � � p k g � � f g o � � � c d � � c � � � f � � � i � � f i e p �� � p d � f � g � c f e � � � q � � g � c � � � � � � � � o � d � � � k � � � d g k � c � � � � � � � � o � � � p � � m � � � � � d i e p � j d � � �� � � � i d � � g p � c d l � d � f g � p � � d � � � k � � j d g k � c � � � i d � d g � q c � � � o � � i � g � � e � � d g � � e � g � g � f d pg � i � � � � o � f i e p � � � � p � � f g � c � � � d e c m� � f i f p d � � c � � � � i d e e p f � � � � � g k f � � � � � k � � d e c � j d g k f � g � � c d � k � � e � � l � m

� _ � \ � } \ ^ � � � � � _ � � � ] � | b � � � � � � � � � � � � � � � � � � � � � f � � o � � g � d p p � k d � � � o � � � c � � � � � � � � d � f � � f gi d g � d e e p f � d � f � g � q c � � � � c � � � d � � e � � � � k � g � � � � � � k � � f g � � � g � � � d f g � � m � � � � � d i e p � j f o � c � � �d � � d � � � f � � � o � d � � � � � � � e � � o � � i � k j d g k � � � � d f g � d � � � i � � � e � � � � k � � � c � � � d � � � � � m � m f g� � g � � � � � � f � g � � � c d l � � � � � f p k � c � � � � � � � � � � � o � � � � � � � � f p k � c � � � � � g k � � � � j � � � � � � � d g k �� c � � p � � � � f � d p q f � f g � q c f p � � � � f g � � d p p � c � q f g k � q � � m � g � � g � � d p d � � � � � � � � � � � � � � � � � � � � � � � � �f � d � � � f g q c f � c l � � � f � � � d � � � d � � � d g k � c � � k � � � � � � � i � d g � � c d � � d � � � i � � � � � � � i e p � � � k� � o � � � � d � � � � � � f g � m� � � � � � � � � � � � � � � � � o d � � � f � d p f g � d � � � k � � f g � � o � c � l � � � f � � � � o � c � � � � � � � c � c d � o � � � d � c� k � � � � � � � j � d e e � d � � � � o � � � � f g � c � � � k � � f g � m � � � � � c d � f g � � g � � d p j � c � � � i d � � � i d g � p � � d p� � e � p � � f � d p � � k � � � o � � d � f l � g � � � m� � � � i e � � � d � � e � p � � f � d p � � k � � f g � f � d � � � d p p � l � � � � d � � j � f l � g � � � m � � � c � e � � l f � � � p � i i d jo � � � l � � � � k � � � � � � � f g d � � � j � c � � g f � c � f i � � o � f � � � � d � � � � c d g � c � � g f � c � f i � � o � m � c � � j f �� � £ � � � � � � � � e � � � c � l � � � f � � � f g � � l � � � � � � k � � � o � g f � c f g � � f i � m � � k � � c f � q � � � g d � � � � f e e � kk � q g � � � � j d g k q c � g � d � c l � � � � � f � � g f � c � k q � d k k f � � � � c � o � � g � � o d p f g � � k p f � � m � c � � g d pp f g � � k p f � � � � k � � q f p p � � � c � � g d p � � e � p � � f � d p � � k � � m � c f � f � � f l � g � � p � q m � � � � � � � �

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Page 48: Design and analysis of computer algorithms (dave mount, 1999)

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Page 51: Design and analysis of computer algorithms (dave mount, 1999)

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i d � f i � i � g f � c � f i � � o d p p � c � l � � � f � � � f g � c � � � i e � g � g � j d g k � � � � � c � � � f g k � � � � d � f g �� � k � � m � c � g � c f � � � k � � f � d � � e � p � � f � d p � � k � � o � � � c � � � i e � g � g � k f � � d e c m

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Page 52: Design and analysis of computer algorithms (dave mount, 1999)

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2

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� c � � � d � � g f � � c d � � � � � q d � d � � l � � g i � g � � � e e � � � � k i � g � e � p � d � � g � � f i � j d g k q d � � � �� e � g � f � p � o � � c d g k p f g � d p p � � p � e c � g � � � g g � � � f � g � m � o d � � i e d g � q d g � � k � � � � g g � � � d � � p p � � � f � g� o f g � � d p p d � f � g � � � d g e � f l d � � f g � � � g d p e c � g � � � � � � i j � � � � q d � � � ¡ � f � � k � � � p d q � � � � � g g � � �� c � i f g � c � i f g f i � i � � � � i d g g � � j q c f � c f � � p � d � p � d � e d g g f g � � � � � m m m � � f � f � �� � i � � � i e d g f � � k f � � � l � � � k � c d � � c � � � � � p k d � � � d p p � � � k � � � � c � f � � � g g � � � f � g � � � � � � � � e � g f g �d g � q � � � � � f g � � d p p d � f � g m � � � c d g f g � � d p p d � f � g � � � l � k g � e � � e � � � � � c � � � c d g � � d � � d � d gf g � � � i � k f d � � e � f g � o � � � � g g � � � f � g � m � g � � d i e p � f � � c � q g f g � c � � � � � � � � p � q m � g � c � p � o � j� � g � f k � � o � � � f g � � d p p d � f � g � � c d � � c d � p f � d � � c � � � � g � � � � o d � � � ¡ � d � � m � � � � i � � c d � d p p� k � � p � g � � c � d � �   � � � � � � p f k � d g k f � � d g � � � m � � f � � d � � � � � � � � c d � � c � � � � � � o d g � � � o � � � c f �� � g � � � � d � f � g f � z � d � � c � q g � g � c � p � o � � m � � q � l � � j f o � � � f g � � � k � � � d g � q f g � � d p p d � f � g d � � c �� � g � � � j q c � � � k f � � d g � � � � � d � c � o � c � � � c � � o � � � e � f g � � f � � � � � m � � f � g � q e � � � f � p � � � � � g g � � �� c � � � � l � e � f g � � q f � c d � � � d p � � � � � o � o � � � � � � � � � � � � z m � c f � f � � � � � � � � c d g � c � � � m� g � � g � � d p j � c � e � � � p � i � o k � � � � i f g f g � � c � p � q � � � � � � � f g � � � � � g g � � � f � g � � � � � � � q � � g d � f l � g� � � � o g � k � � j d � � � i f g � � c d � � � � d � � d p p � q � k d k k f � f � g d p g � k � � � � d p p � k � � � � � � � � � � � � � � f � � d p p � k

Page 53: Design and analysis of computer algorithms (dave mount, 1999)

� � � � � � � � � � � � � � � � � � � � � � �

Cost = 3

1

Cost = 2 sqrt(2) = 2.83

MST SMT

Steiner point

� f � � � � z z � � � f g � � f g f i � i � � � � m

� c � � � � � � � � � � � � � � � � � � � � � � � � o � � � c � � � � m � g f g � � � � � � f g � o d � � f � � c d � d p � c � � � c � c � � � f � d� f i e p � � � � � k � d p � � � f � c i o � � � � � � � d � q � q f p p � � � � � p � q � j � c � � � e � � � p � i f � i � � c c d � k � � jd g k f g o d � � f � � n � c d � k m � � � � c � q d � j � c � � � � � � d p � � k � f � � d � c � � d i � f � � � � � � g � c � e � f g � d �� � q c � � c � � � c � e c � g � � � i e d g � q d � � � ¡ � f � � k � � � � � � � � � � � � � � � f g i d � f g � � � g g � � � f � g � m �

� \ ` \ � _ � ] � � � � ] � | b ¥ � q f p p e � � � � g � � q � � � � � � � d p � � � f � c i � � � � � � � d p � � d g k n � f i � � d p � � � f � c i � � o � �� � i e � � f g � d i f g f i � i � e d g g f g � � � � � m h � � d p p � c d � d � � � � � � � � � � � � � � � f � � g � � c d � � � f p k � d� � p � � f � g � � � � e � d � � k � � p � � � f g � � c � � c � d e � � � � � � � � g � � d p p � p � � d p p � � e � f i d p � c � f � � � d i � g � d p p� e � f � g � d � � d � c � � d � � m � g f i e � � � d g � � c d � d � � � � f � � f � � o � � � � k � d p � � � f � c i � f � � c d � � g � � � c � �i d � � d � c � f � � j � c � � g � l � � � � g i d � � � � c f � � c � f � � m � � o � � � e � � � � g � f g � � c � � � d p � � � f � c i � j p � � � �� � l f � q � � i � � d � f � o d � � � d � � � � o � � � � � � � � m � c � � d � � d p p ¡ � f � � � d � � � � e � � l � m� \ � � ] b

� � o � � � � � � � q f � c l � � � f � � � c d � � � d � � p � � � � k � � � m� � c � � � � � f � � � d � g f ¡ � � e d � c � � � q � � g d g � � q � l � � � f � � � � o d o � � � � � � � m� � k k f g � d g � � k � � � � d o � � � � � � � � � � d � � � d � g f ¡ � � � � � p � m � � � d � f g � � � � � k � � � g � c f �

� � � p � � � � � � � � � d o � � � � � � � m� � � � � � � � � � � � d g � g k f � � � � � k j � � g g � � � � k � � d e c q c � � � � k � � � c d l � g � i � � f � � k � � q � f � c � �� q c f � c i d � � � e � � f � f l � j g � � d � f l � � � � � � � � m � c � f g � � f � f � g � � c f g k � c � � � � � k � � � d p � � � f � c i � f �� f i e p � j q � i d f g � d f g d � � � � � � � o � k � � � � j q c f � c q f p p f g f � f d p p � � � � i e � � j d g k q � q f p p d k k � k � � �� g � d � d � f i � j � g � f p � � ¡ � d p � � c � � � m ¥ � � d � � c d � d � � � � � � � � � f � � � � � � f o � f � d � � � � � �� o � k � � � f g � � � � � � m ¥ � � d � � c d � d g � k � � � � � � � � � � � f � � � � � f o � � � � � � � � � f � l f d � p � m � g� � c � � q � � k � j � c � � c � f � � � � � � � f � d � d o � � c � f � � � � d k k � � � c d � � � d g � � f p p � � � � � � g k � k � � o � � i d g � � m � � � � � c d � f o � f � l f d � p � f � � d g g � � � � g � d f g d � � � p � m � � � g � � f � � � � � k � d p � � � f � c i � e � � d � � �� � � � e � d � � k p � d k k f g � d g � � � � � � k � � � � � c � � � � � � g � � e d g g f g � � � � � m

{ | \ ` _ ~ ] ` \ ^ a \ ~ ] � \ � ¥ � � � g � f k � � � c � � c � � � � � f � d p f � � � � � � � c f g k k � � � � i f g f g � q c � � c � � d g � k � �f � � d o � � � g � � m � � � � � � d � � � � � � � o � c � l � � � f � � � � � � m � � � � � � � � � � � f �   � � � d e d � � f � f � g� o � c � l � � � f � � � f g � � � q � k f �   � f g � � � � � � � � m � g � k � � � � � � � � � � � � � � � c � � � � f o � g � � g k e � f g � f � f g �d g k � c � � � c � � f � f g � � � m � f l � g d � � � � � � � o � k � � � � j q � � d � � c d � d � � � � � � � � � � � � f o g � � k � �f g � � � � � � � � � c � � � � m � � f � g � � c d � k � � � � � q c � � � � e � � � f g � � � � � d � � f i e � � � d g � � � � c f � e � � � p � i m� o q � c d l � � � i e � � � k d e d � � f d p � � j d g k q � q f � c � � � g � q q c f � c � k � � � � d g � � d k k � k � c d � k �� � � f g k � � � d � � � p � f g � c � � � � � � g � � � j d g � � k � � � c d � � � � � � � � d � � � e � � � f g � � � � f � d e � � � f � p �� d g k f k d � � m� g � k � � � o � f � d � � � � � � � � � � � � � � f g � d � � � j f o d i � g � d p p � k � � � � � � � � f g � � c � � � � j f � c d � � c �i f g f i � i q � f � c � � � c � p f � c � � k � � i d � g � � � � � g f ¡ � � f o � c � � � d � � k � e p f � d � � � k � � q � f � c � � � m� g � � f � f � g � d � � � c d � � f g � � d p p � c � � k � � � � c d � � � � � � d � � � e � � � f g � � � � k � g � � f g k � � � d � � � p � j � c � g� c � p f � c � � � � � k � � � � � � � f g � d � � � f � d g d � � � d p � c � f � � m � c � i d f g � c � � � � i q c f � c k � f l � � � � � cd p � � � f � c i � f � � c � o � p p � q f g � m � � � � � � g � f d p p � � d � � � c d � q � � d g d p q d � � d � � i � g � � � � d k k f g � � c �

z

Page 54: Design and analysis of computer algorithms (dave mount, 1999)

� � � � � � � � � � � � � � � � � � � � � � �

i f g f i � i q � f � c � � k � � � c d � � � � � � � � d � � � q c f � c � � � e � � � � � m � � � f � � � d � � k f g � � i e p � � � � � g � � d p f � � j� � � c d � f � � d g � � d e e p f � k � � � � � c d p � � � f � c i � m �� � � � \ � � ] b � � � � � � � � � � � � d � � g g � � � � k j � g k f � � � � � k � � d e c q f � c � � d p � l d p � � k q � f � c � � � g

� c � � k � � � m � � � � � � d l f d � p � � � � � � � � o � � f m � m d � � � � � � � o � � i � � � � j p � � � � � � � � � � �d g � � � � � c d � � � � e � � � � � j d g k p � � � � � � � � � d p f � c � � k � � � � � � � f g � � c f � � � � m � c � g � c � � k � �� � � � � f � � � � � o � � � m � � � � b � � q f p p � f i e p f o � � c � e � � � o � � d � � � i � � c d � d p p � c � � k � � q � f � c � � d � � k f � � f g � � m � � � � �d g � � � o � � � m � o � � g � d f g � � � � � � � c � g q � d � � k � g � m � � e e � � � � c d � g � � � � � g � d f g �� � � � � m ¥ � q f p p k � � f l � d � � g � � d k f � � f � g m

A

y

u v u v

y

u v

y6

8

7

4

9

4 4

x8

x x

T’ = T - (x,y) + (u,v)T + (u,v)

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Page 55: Design and analysis of computer algorithms (dave mount, 1999)

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k � f g � d � � � j � d � c f � � � d � f � g � � o � c � � � � � � k � � q c f p � p � � e � � d g � � e � � o � � i � k f g � � � � � �� f i � m � c � ¡ � � � � f � g f � c � q i d g � f � � � d � f � g � d � � � � ¡ � f � � k f g � � g � � d p � ¥ � � p d f i � c d � � c � � �d � � g � l � � i � � � � c d g � p � � � f � � � d � f � g � g � � k � k m � � � � � q c � j p � � � k � g � � � � c � g � i � � � � o

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� � i e � g � g � � d � � � i � � � d � � m � d � c � o � c � � � � i e � g � g � � j q f p p i � � � � q f � c d � p � d � � � g � � � c � �� � i e � g � g � m � o � � � q d � k � � c � g � i � � � � o � � i d f g f g � � � i e � g � g � � � � � p k � � d p � q d � � � f o � c � � d p pi � � � � � � � � � c � � � j � � � g � l � � c f � c � � � c d g � � � � f o � c � � i � � � � f g e d f � � � m � c � � j � c � g � i � � � � o� � i e � g � g � � k � � � � d � � � � � d � p � d � � c d p o q f � c � d � c f � � � d � f � g m � f g � � q � � � d � � q f � c � � � i e � g � g � � j� c f � � d g c d e e � g d � i � � � p � � � f i � j � g � f p � g p � � g � � � i e � g � g � � � i d f g � m � c � � j � c � � � � d p � � g g f g �� f i � f � � � � � � � � p � � � � � f i � m � � d f g j � f g � � � f � � � g g � � � � k j � f � d � � i e � � � f � d p p � g � p d � � � �� c d g � j � � q � � d g q � f � � � c f � i � � � � � � � f g � � p � d � � � � p � � � � m � c � � d p p � c � � � d p � � � f � c i � c d l �� c � � d i � d � � i e � � � f � � � g g f g � � f i � m

; < = > ? @ < A � B � < � I < � D @ > � < � I H > < @ uJ K L M N O P Q R S � � � W � S X Y Y Y Z� \ ` \ � ] � [ \ � ] � � ~ b h � � d p p � c d � � � � � l � � d p p � � d p f g � c f � � � � � � � f � � � e � � l f k � d � g k � � � � d g k f g � � o

� c � � � � p � � � � k f g k � � f � g f g � d g k d g d p � � f g � � £ � f � g � d p � � � f � c i � m � � o d � � c � i d f g � � e f � � c d l �f g � p � k � k f � � � � � � � p d � � k � � � � � � f g � j k � g d i f � e � � � � d i i f g � j � � � � k � d p � � � f � c i � j d g k d � f � � g� � d e c d p � � � f � c i � m

� ] } \ � _ ] � � � � � � \ � } b � � � d � � � g p � � � � e � g � f � p � o � � i d � � � f d p � c d � c d � � � � g � � l � � � k f g � p d � � � � � g� p d � � d � � f � g i � g � � m � � q � l � � f � f � d p q d � � d � � � k f k � d � � � � � � c � � � � � � � � � � d � � � � � � f g � f � c � f g � �� � i � � o � c � � � e f � � q � c d l � � � l � � � k m � � � � c � � p d � � q � � e d � � � f g � c � p � � � � � � � � � � � � f � g � o � � � c �� � p � l d g � � c d e � � � � � � � � d k m � c � � g p � i d � � � f d p � c d � q d � g � � � � l � � � k f g � h q d � � c � i d � � � f d p� g � � g � � � � � � f l � f g k � � � f � g d g k � d � � l � d � � d p � � � f � c i m

� | \ ] } � | \ \ } b h � � d p p � c d � � c � � � d i f � � p � � � k � � � � � j � p � � � k � g � � � � j � � � � � � d � � d p p � q � k d � c � � � � og � � � � � o � � g � d g k � d � � � m � � � � c f � � c � � � � � q � f � � k � q g � c f g � � p f � � � � � � } ] ` } � � � � � � ] ~ b � m � m j � � i i d � f � g � � � � c d � � � j � � � j � c � d � � � � � c � � � � i o � � � � � � � �

� � g � � � j d g k i d � � � d o � q � o � c � i d   � � � � � � � � � g � � � p f � � � � � � � � � � � � � � p � � � m� \ � ` _ } _ � ` ~ b � � � � � d i e p � j q c d � f � d � � � � g � � � i e � g � g � � � d i f g f i � i � e d g g f g � � � � � m � � � � � �

� � g � � � q c � � c � � � c � � � g � � e � f � k � � g � k o � � � � d e c � � � k f � � d e c � j q � f � c � � k � � � g q � f � c � � k m� ] ~ _ � � \ � � ] ~ b � � � c d � j d � � � � q f � c l � � � f � � � c d � � � � k � � � j d g k � c � � � p � i i d m� | � � } ^ \ ~ � � _ � } _ � ` ~ � � ] � a � � _ } | � ~ b � � � � d � c d p � � � f � c i � e � � f o � � c � f g e � � d g k d g � d � � � i e �

� f � g � � g f g e � � � � � � � � � � � j � c � � � � e � � d g k d � � i e � � � f � � � g g f g � � f i � m � p � � q � f � � k � q g d� � f � o k � � � � f e � f � g � o � c � d p � � � f � c i � o � p p e � � � k � � � � k � f � p � � � f i e � � � d g � � c d g � � i � � c f g � � c d �i d � � � � � g � � � � � � � � m � � � k � g � � � � p � � g � c f � m � � f � f i e � � � d g � � c d � � � � c d l � d g f g � f i d � �q � � � f g � � g � q p � k � � � o � c � d p � � � f � c i m

� o � � � � � � k � � c f � j � c � g � � � � � � l � � � � c f g � � g � � � � � c � d � � c � � � � � q � p p � c d � � � � k � g � � g � � k � �p � � � d � f � k � � f g � � c � � � d i m � p � � j o � � � d � c d p � � � f � c i j � � � � � � � c d � � � � � d g q � � � f � � � � � gd g � � d i e p � q f � c � � � � � g � � p � f g � � � � � � c � d � � c � � � m � i � � f � d � f � g � o � c � � � � � � o � c � d p � � � f � c if � g � � � � d p p � f i e � � � d g � m ¥ c d � f � f i e � � � d g � f � � c d � � � � � g k � � � � d g k c � q � c � d p � � � f � c i q � � � �f g � � � g d p p � j � f g � � � c f � f � q c d � � � � q f p p g � � k � � d g � q � � ¡ � � � � f � g � d � � � � � c � d p � � � f � c i m

� � � _ � ~ b � ` ] � � ~ _ ~ b � � i i d � f � g � j d � � i e � � � f � � j � � � � � � � g � � � m � � e � � � d � p � d � � � g � ¡ � � � � f � g � c d � � � �¡ � f � � � � � p l f g � d � � i i d � f � g � � � � � � � � � g � � j � � d g d p � � f g � d g d p � � � f � c i � � � � g g f g � � f i � m� � � } _ ` a b� ] ~ _ � � � � } _ ` a b h � l f � q � c � � d � f � � � � � f g � d p � � � f � c i � o � � i � � � � � � � � � � � j � � d e � � � � j

� � f � � � � � � � m � p � � � � � d p p � c d � � f l � g d p f � � � o g � i � � � � j � c � � � � c � i d p p � � � � d g � � o � � g kf g � � � � f i � m

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� f g � � � � � � � � � f i � m ¥ � � c � q � k � c d � � c f � � d g � � � � � k � � � � � � f g � � � � � � f g � c �� d g � � � � � � f g � � � � � f i � m

� � � � \ } � � � } b � � � � d g � � � � g � i � � � � � g f o � � i p � k f � � � f � � � � k � l � � � � i � � d g � � f g � � �� � e � � � � k � f i � m

� � ` ] � _ � � � a � ] � � _ ` a b ¥ � � � l � � � k � n d p � � � f � c i � o � � � � � � � ¡ � � g � � � � � � � � � � � k � j � c d f gi d � � f � � i � p � f e p f � d � f � g d g k i f g f i � i q � f � c � � � f d g � � p d � f � g � � � � c � � � � � � � � � � � � � k � m ¥ � d p � �e � � � � g � � k � g d e � d � � � g d c � i � q � � � m � c � � � d � � � � i i � g � n � � � � � � � � � � m h � � d p p � c d � � nf g l � p l � � � � g � � � � � � f g � d � d � p � f g d � � � � � i � � e i d g g � � j � � � i � i � f � d � f � g � d g � � � � � k � �� � f p k f g � � c f g � � f g d � � e � k � q g i d g g � � m� � \ \ ^ � � � a � � _ } | � ~ b ¥ � � � l � � � k � � ¢ i d g � � k f g � j d � � f l f � � � � c � k � p f g � j d g k o � d � � f � g d p � g d e �� d � � m � p � � j d p p � c � � � d p � � � f � c i � q � � d q d � � d p � � � � d i e p � � � o � � � � k � d p � � � f � c i � m � � � � k �d p � � � f � c i � d � � � � g � � d p p � l � � � � f i e p � �  � � � i d � � � c � � � � � � c � f � � d � � d � c � � d � � � m � c � i d f gf � � � � f � e � � l f g � � c � f � � � � � � � � g � � � m� � ] � | ] � a � � _ } | � ~ b ¥ � � � � d g � � k f � � � � � f g � � � d e c d g k k f � � d e c � � e � � � � g � d � f � g � j d k   d � � g � �p f � � � d g k d k   d � � g � � i d � � f � � � m ¥ � c d l � d p � � e � � � � g � � k d g � i � � � � o d p � � � f � c i � m� � � b � � d l � � � � d � � d e c f g � � � d k � c � � � � � � � k � � m h � � d p p � c d � � c f � f � � f i f p d � � � � � � j � � �

f g � � � d k � o � � � � f g � � g e � � � � � � � k l � � � f � � � f g d � � d � � j q � � � � d ¡ � � � � m � � � � d g � �d e e p f � k � � � i e � � f g � � c � � c � � � � � � e d � c � o � � i d � f g � p � � � � � � � f g d k f � � d e c q f � c � � �� k � � q � f � c � � f g � � � � � � � f i � m

� � � b � � d l � � � � � d � � d e c f g k � e � c � � � � � � � k � � m � � � o � p o � � i d g � e � � � p � i � f g � � � � � �� f i � m� � � � \ � \ } \ � } _ � ` b � � � � � i f g � q c � � c � � d � � d e c � � k f � � d e c c d � d � � � p � m � � � � � o � �

d g � � d � � � k � � m �� � � � � � a _ � ] � � � � } b � � � � � c � l � � � f � � � � o d � � � f g � � e � p � � f � d p � � k � � m � � � � � f � � � � g f � cf g � � l � � � � � � e � p � � f � d p � � k � � m �� � ` ` \ � } \ ^ � � � � � ` \ ` } ~ b � � � d � d � � d e c f g � � f � � � � g g � � � � k � � i e � g � g � � m � � d � c� � � � � � � f � d � � g g � � � � k � � i e � g � g � m �� } � � ` a � � � � � ` \ ` } ~ b � � � d � d k f � � d e c f g � � f � � � � � � g � p � � � g g � � � � k � � i e � g � g � � m� � c f � q d � � c � q � f � k d p � � � f � c i � c d � � � l � � � � k � c � k f � � d e c � k � � � m �� � � b � i e � � � � c � i f g f i � i � e d g g f g � � � � � � o d q � f � c � � k � � d e c m ¥ � e � � � � g � � k � c � � �

d p � � � f � c i � � c d � � � g f g � � � � � � � p � � � � � f i � � � � � � d p � � j n � f i � � j d g k � d � � l � d � �d p � � � f � c i � m

� ~ ~ _ � � \ � � \ ~ } _ � ` ~ b � � � � c � � p k � � e � � � d ¡ � � � � f � g � o � c � o � � i q � � � � c � � � � c � � i � d p � � � f � c i � gd � f l � g f g e � � j � c � q f g � f g � � � i � k f d � � � � � � p � � m � � p p e f � � � c � d p � � � f � c i d � � d g k � i j � � � d � � � � q d g �� � � � � p � d � g � c � i d p p m � � � � � i � i � � � � � i e p � � d p � � � f � c i � � � m � m � c d f g i d � � f � i � p � f e p f � d � f � g � �i d � � g p � d � � � � � � � k � � g � f � � � d � f � g m � � � � c � � p k d p � � � � e � � � d � � � f � � � o � c � � � � d g � q � � e � � � p � i �� d � d e e � d � f g � c � e � d � � f � � e � � � p � i � � m � p � c � � � c � c � d g � q � � � d � � � c � � � j � � i � � o � c � e � � � p � i �i d � � � ¡ � f � � d p � � � o � c � � � c � m � � g � � q � � � � d � � � � � � f e e f g � d e � � � p � i � � � � � � � � f g � d g d g � q � � �f o � � � k � g � � � � � � c � d g � q � � � f � c � d q d � m� c � � � c � � e � � � p � i � q f p p f g l � p l � � f l f g � e � � � o � j � � p l f g � � � i i d � f � g � � � � � � � � � � g � � � j k � � f � g f g �d g k d g d p � � f g � d p � � � f � c i � m � c � � � d � � i � � c i � � � � g e � � k f � � d � p � m � � � � � � d � � d � p � d � � d � � � e p �� o ¡ � � � � f � g � � c d � f g l � p l � d l � � � � f i e p � i � k f � � d � f � g � � d c � i � q � � � e � � � p � i j � � � � � � � � � � �� g k � � � � d g k � c � � � p � � f � g � � � d p p � c � c � i � q � � � e � � � p � i � � f g � p � k f g � � c � e � d � � f � � e � � � p � i � � m� � c � � ¡ � � � � f � g � � � � d p p � f g l � p l � d � p f � c � i � k f � � d � f � g � � d e � � � p � i � c d � q � c d l � q � � � � k � g m

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d p � � d k � � � � g � c d � � � � d k � c � � � � � � � d � � c f � d g � � � � � d p � � � f � c i o � � � g k f g � � c � � � � � � e d � c �o � � i d � f g � p � � � � � � � l � � � � � � � d p p � � c � � l � � � f � � � j d � � � i f g � � c d � � c � � � d e c c d � g � � k � � q � f � c � � m� � e e � � � � c d � � c � � � d e c c d � � k � � q � f � c � � j d g k q � q f � c � � � � i e � � � � c � � c � � � � � � e d � c � o � � i d� f g � p � � � � � � � l � � � � � � � d p p � � c � � l � � � f � � � f g � c � � � d e c m� � � c � q d � j � c � � � d � � � � c � � o � � i � p d � f � g � � o � c � � c � � � � � � e d � c e � � � p � i m � g � i d � q d g �   � � � � c �� c � � � � � � e d � c � � � q � � g d � f g � p � e d f � � o l � � � f � � � m � � � d p � � � f � c i � o � � � c f � e � � � p � i d � � l d � f d g � �� o � c � � f g � p � � � � � � � � d p � � � f � c i � c d � q � q f p p e � � � � g � m � c � � � f � d p � � d � f g � p � � f g � e � � � p � i j q c f � c� d g � � � � p l � k f g � c � � � d g � e � � � k f � � d e c m � i e � � f g � d p p � e d f � � � c � � � � � � e d � c � � d g � � � � p l � k � �f � � � d � f g � d � f g � p � � � � � � � � d p � � � f � c i � l � � d p p l � � � f � � � j � � � � c � � � d � � � � c � � � p � � d p i � � c � k � � c d �d � � o d � � � � m� c f g � � o � c � l � � � f � � � d � � f � f � � j d g k � c � q � f � c � � � � e � � � � g � � c � � � � � � o � � d l � p f g � o � � i � g � � f � � � �d g � � c � � � g � g � � f � � � g � � k � � � � d g � � � c � � � c � � o d c d l f g � f g � g f � � � � � � � m ¥ c � g � k � � q � f � c � � d � �e � � � � g � j q � k � � g � � c � � � � � � � � o d e d � c � � � � � c � � � i � o � k � � q � f � c � � d p � g � � c � e d � c m � � � g �� c � � � � � � � � � � � � q � � g � q � l � � � f � � � j � d g k � j � � � � � � � � � � � c � p � g � � c � o � c � i f g f i � i p � g � � ce d � c o � � i � � � � m � � � � � � � � � � � � � g � f k � � f g � e d � c � o � � k � � � o � � i � � � f � � � p o m �� c � � � � � � � � � � � � � � � � � � � � � � � � � e � � � p � i f � d � o � p p � q � m ¥ � d � � � f l � g d k f � � � � � k � � d e c q f � c� � � � � � � � � � � k � � q � f � c � � � � � � � � � d g k d k f � � f g � � f � c � k � � � � � � � � � � � j � � � m � c � e � � � p � if � � � k � � � � i f g � � c � k f � � d g � � o � � i � c � � � � � � � l � � � � � � � � l � � � l � � � � � f g � c � � � d e c m� � f � e � � � f � p � � � c d l � � � d e c � q f � c g � � d � f l � � k � � � j � � � f g � � k � � o � � � c � � c � � � � � � e d � c � � � � q � p pk � � g � k j q � g � � k � � d k k � c � � � ¡ � f � � i � g � � c d � � c � � � � � g � � � � p � � q c � � � � � � d p � � � � f � g � � d � f l �

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Page 66: Design and analysis of computer algorithms (dave mount, 1999)

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Page 67: Design and analysis of computer algorithms (dave mount, 1999)

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¥ � d � � � � � � c d � d o � � � � c � � � c e d � � � o � c � � o � � � � � p � � e � c d � � � � � � � � � � � � � � m � c � e � � � o f � � �f g k � � � f � g � g � m � � � � � l � � c d � d o � � � � c � f g f � f d p f � d � f � g � e d � � � � q � c d l � � � � � � � � � � � � � m � g� � g � � d p j e � f � � � � � c � � � c e d � � � c � � � � c � c � p � � e j � c � f g k � � � f � g c � e � � c � � f � � � p p � � � � c d � � � � � � � � �� � � � � � � � � m � o � � � � c � � � c e d � � � c � � � � c � c � p � � e j q � c d l � k � g � d � � p d � d � f � g � g � c � � k � � � � � � � � � � �� � f g � � q � k � � � p d � d � f � g � d p � g � d p p � c � � k � � � � m � c � � d o � � � � c � � � c e d � � q � c d l �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �h � � d p p o � � i � f   � � � � d � � d p � � � f � c i � c d � � � � � � f � g � l � � p � � � � c d g � � � � � � � � � f g � � � d � c � f i � q � k � d� � p d � d � f � g � c � � � � � f � � � d e d � c � c d � q f � g � � � � � f � � l d p � � � m � c � � j � � � � � f � f g o d � � � ¡ � d p � � � � � � � � � j� � i e p � � f g � � c � f g k � � � f � g e � � � o m� g � � i i d � � j d o � � � � e d � � � � � c � � � � c � c � o � � p � � e j d p p l � � � f � � � � c d � d � � � � k � � � d q d � � d p � g � � c �� c � � � � � � e d � c � � � � � o � � i � c � � � � � � � c d l � � c � � � � � � � � k f � � d g � � l d p � � � � � � � � k f g � � � � m � c � � j d o � � �� c � � � � � � � � f � � � d � f � g � o � c � o � � p � � e j d p p l � � � f � � � � c d l � � c � � � � � � � � k f � � d g � � l d p � � � � � � � � k f g� � � � m

� � � � ] _ � ~ � | � � } \ ~ } ] } | ~ b ¥ � � � g � f k � � � c � � � g � � d p f � d � f � g � o � c � � c � � � � � � e d � c e � � � p � i j � � � � i �e � � f g � � c � � � � � � e d � c � � � � q � � g d p p e d f � � � o l � � � f � � � m � � � � � � � � � � � � d k f � � � � � k � � d e c q f � c� k � � q � f � c � � m � o � � � � � � j f � d g � k � � � o � j � c � g � c � q � f � c � � o � c f � � k � � f � k � g � � � k � � � � � � mh � � d p p � c d � � c � � � � � � o d e d � c f � � c � � � i � o � k � � q � f � c � � d p � g � � c � e d � c m � c � � � � � � � � � � � � q � � g� q � l � � � f � � � � � � � � � f � � c � � � � � � o � c � i f g f i � i � � � � e d � c � � � q � � g � c � i m ¥ � q f p p d p p � q � � �c d l � g � � d � f l � � � � � � k � � � j � � � q � q f p p g � � d p p � q � � � c d l � d g � g � � d � f l � � � � � � � � p � � m¥ � � � g � f k � � � c � e � � � p � i � o k � � � � i f g f g � � c � � � � � � o � c � � c � � � � � � e d � c � � � q � � g d p p e d f � � � ol � � � f � � � f g d q � f � c � � k k f � � � � � k � � d e c m ¥ � q f p p e � � � � g � d � � � � d p � � � f � c i j � d p p � k � c � � � � � � � � � � � � � � � � � � � � � � � m � c f � d p � � � f � c i f � � d � � k � g � � � � � � � � � � � � � � � � � � m� � � � c f � d p � � � f � c i j q � q f p p d � � � i � � c d � � c � k f � � d e c f � � � e � � � � g � � k d � d g d k   d � � g � � i d � � f � j� d � c � � � c d g � c � i � � � � � i i � g d k   d � � g � � p f � � m � p � c � � � c d k   d � � g � � p f � � � d � � � � g � � d p p � i � � �� £ � f � g � o � � � e d � � � � � d e c � j � � � � f g � d p p � c � f g � � � � l � � � � � k f � � d g � � � q f p p � � ¡ � f � � � � � � � � � � d � � j� � � c � � d l f g � � f � g � �   � � � f � � k c � � � m � � � d � � � � c � d p � � � f � c i f � i d � � f � � � d � � k j q � q f p p � i e p � �� � i i � g i d � � f � g � � d � f � g j � � f g � � j � d g k � � � k � g � � � l � � � f � � � � d � c � � � c d g � j � j d g k � d � q �� � � d p p � k � m

` � � } � � � � ] } b � c � f g e � � f � d g i d � � f � � � o � k � � q � f � c � � j q c f � c d � � � d � � k � g � c � � k � �q � f � c � � f g � c � k f � � d e c m ¥ � p � � � � / k � g � � � � c � � g � � � f g � � q � d g k � � p � i g � � o � m

� � / ����

� f o � � � j� � � � � � f o � �� � d g k � � � � � � � j� � f o � �� � d g k � � � � � �� � m

� � � � f g � � � / � � f o � c � � � f � g � � k � � j f g � � f � f l � p � i � d g � � c d � � c � � � f � g � k f � � � � p f g � � � � q � � g� c � � � � q � g � k � � j d g k c � g � � � c � k f � � � � � � � � f � f g � g f � � m � c � � � d � � g o � � � � � � f g � � � � � � f � � c d �� c � � � � � � o � � � � f g � o � � i d g � l � � � � � � � � c � � p k � � � j � f g � � q � c d l � g � k f � � d g � � � � � � d l � p m � � � �� c d � f g k f � � d e c � f � f � e � � � f � p � � � c d l � � � p o � p � � e � k � � � j d g k � � � � � � � � i d � � � g � � d p p � � � g � g � � � � m� � � d g g � � � � g � � d � f l � � � f g � � � c � � � d � � g � g � � d � f l � � � � � � � � p � � � j d g k f o f � f � e � � f � f l � j � c � g � c � � �f � g � e � f g � f g � � f g � f � d � e d � � � o d g � � c � � � � � � e d � c m� c � � � � e � � q f p p � � d g k f � � d g � � i d � � f � � � � � / q c � � � � � / � � � � � � � j � c � � c � � � � � � e d � c� � � � o � � i l � � � � � � � � � m h � � � l � � f g � � c � � c � � � � � � e d � c � q f p p d p � � � � d g f � � � � m � � c � p e � � k � � c f � j

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Page 68: Design and analysis of computer algorithms (dave mount, 1999)

� � � � � � � � � � � � � � � � � � � � � � �

q � q f p p d p � � � � i e � � � d g d � � f p f d � � i d � � f � � � � � � � � � m � c � l d p � � � o � � � � � � � � q f p p � � d l � � � � � � c d �f � � � i � q c � � � d p � g � � c � � c � � � � � � e d � c o � � i � � � � m � o � c � � c � � � � � � e d � c � � d l � p � k f � � � � p � o � � i �� � � q f � c � � � e d � � f g � � c � � � � c d g � � � c � � l � � � f � � � j � c � g � � � � � � � � q f p p � � � � � � � � � � � m ¥ � q f p p � � �p d � � � � c d g � � f g � � c � � � l d p � � � f � q f p p � � e � � � f � p � � � � � � � g � � � � � � d g � � c � � � � � � e d � c f g � � � � f i � m

� � � � ^ � { ] � ~ | ] � � � � a � � _ } | � b � c � � p � � k � ¥ d � � c d p p d p � � � f � c i k d � � � � d � � � � � c � � d � p � ¤ � � � m ¥ d � � c d p pq d � f g � � � � � � � k f g � c � q � d � � � ¡ � � � � f � g � o � � d � c d � f p f � � k � � � � i f g � o � � � d � c e d f � � o l � � � f � � � � d g k� j q c � � c � � � � d g � � d � c � m � p � � k � � d p f � � k � c d � � c � � d i � � � � c g f ¡ � � � � � p k � � � � � k � � � � i e � � �� c � � � � � � e d � c � q f � c � g p � i f g � � l d � f d � f � g � m � c � � p � � k � ¥ d � � c d p p d p � � � f � c i � � g � f g � � � � � f i � m� c � � � g f � � � o � c � d p � � � f � c i f � f g � g k f g � d � p � l � � o � � i � p d � f � g o � � � c � � c � � � � � � e d � c � � � e � � � p � i m� � � d e d � c � � � � � � � � � � � � � � � � q � � d � � c d � � c � l � � � f � � � � � � � � � � � � � � � � � d � � � c � � � � � � � � � � � � � � � � � � � � � o � c f � e d � c m � � � � � c d � d e d � c � � g � f � � f g � � o d � f g � p � � k � � c d � g � f g � � � i � k f d � � l � � � f � � � m� � � � � � ] } _ � ` b � � � g � � � �� / � � � � � c � � c � � � � � � e d � c o � � i � � � � � � � c � c d � d g � f g � � � i � k f d � �

l � � � f � � � � g � c � e d � c d � � � c � � � g o � � i � c � � � � � � � � � � � � � � � m� g � � c � � q � � k � j q � � � g � f k � � d e d � c o � � i � � � � q c f � c � f � c � � � � g � f � � � � o � c � � f g � p � � k � � � � � � � j � �f � l f � f � � � � i � f g � � � i � k f d � � l � � � f � � � d p � g � � c � q d � j � � � � c � � � f g � � � i � k f d � � � d g � g p � � � � c � � � go � � i d i � g � � � � � � � � � � � � m � c � e d � c f � o � � � � � l f � f � d g � � � � � � � � o � c � � � l � � � f � � � j d g k � � k � � �f g d g � � � k � � m � � � � � d i e p � j f g � c � k f � � d e c � c � q g f g � c � o � p p � q f g � � � � � � j g � � f � � c � q � c � l d p � �� o � � �

� � � c d g � � � d � � l d � f � � m

i j(k-1)dij

1

3

4

1 4

1

2

2

28

(4)3,2

(3)d3,2

(2)d3,2

(1)d3,2

(0)d3,2 (no path)

k

(k-1)dik(k-1)dkj

Vertices 1 through k-1

d

= INF

= 12

= 12

= 12

= 7

(3,1,2)

(3,1,2)

(3,1,2)

(3,1,4,2)

� f � � � � � � � p � � k � ¥ d � � c d p p � � � i � p d � f � g m

� � � � ^ � { ] � ~ | ] � � � � ^ ] } \ [ � � \ b � � q k � q � � � i e � � � � � �� / d � � � i f g � � c d � q � c d l � d p � � d k � � � i �e � � � k � c � e � � l f � � � i d � � f � � � � � � � � c � � � d � � � q � � d � f � � d � � � j k � e � g k f g � � g � c � q d � � � c d � q �i f � c � � � � o � � i l � � � � � � � � l � � � � � � j d � � � i f g � � c d � � c � f g � � � i � k f d � � l � � � f � � � d � � � c � � � g o � � i� � � � � � � � � � � � � ` � } a � } | � � � a | � ] } ] � � b � c � g � c � � c � � � � � � e d � c o � � i � � � � � � � � � g p � f g � � � i � k f d � � l � � �

� f � � � � � � � � � � � � � � d g k c � g � � � c � p � g � � c � o � c � � c � � � � � � e d � c f � � � � � �� / m� � a � } | � � � a | � b � f � � � � � � � � l � � c d � d � c � � � � � � e d � c k � � � g � � e d � � � c � � � � c � c � � d i � l � � � � �

� q f � � j � � q � � d g d � � � i � � c d � q � e d � � � c � � � � c � � � d � � p � � g � � m � � c � d � � � i e � f � g � c d �� c � � � d � � g � g � � d � f l � � � � � � � � p � � f � � � f g � � � � k c � � � m � � c d � f � j q � � � o � � i � � � � j d g k � c � go � � i � � � � m � g � � k � � o � � � c � � l � � d p p e d � c � � � � d � � c � � � d � e � � � f � p � q � � c � � p k � d � � � c �� c � � � � � � e d � c o � � i � � � � j d g k � c � � c � � � � � � e d � c o � � i � � � � m � f g � � � o � c � � � e d � c � � � � �f g � � � i � k f d � � l � � � f � � � � g p � f g � � � � � � � � � � � � � j � c � p � g � � c � o � c � e d � c f � � � � � �� � � � � � �

/ m

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Page 69: Design and analysis of computer algorithms (dave mount, 1999)

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Page 71: Design and analysis of computer algorithms (dave mount, 1999)

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� \ � _ ~ _ � ` � � � � \ � ~ b d g � � o � c � e � � � p � i � � c d � q � c d l � k f � � � � � � k f g l � p l � � � � � � � � � � � � � � o � g � o � � i� � d g � � c � � � g k � c � � c � � � � � � e d � c j � g k � c � i f g f i � i � � � � � e d g g f g � � � � � j � g k � c � i f g f i � iq � f � c � � � f d g � � p d � f � g m � � � � d � c � � � � � c g f � d p � � d � � g � j i � � � � n � � � i e p � � � e � � � p � i � � c d � q � q f p pk f � � � � � q f p p � � e c � d � � k d � k � � f � f � g e � � � p � i � m � e � � � p � i f � � d p p � k d � � � � � � � � � � � � � � � f o f � �� � � e � � f � d � f i e p � � � � � � � � � g � � � � � � � � i d � � c f g � � o � c f � d � � � � � � � d p � � j � � � j d � � � e � � � �   � � � � m¥ � q f p p e c � d � � i d g � � e � f i f � d � f � g e � � � p � i � f g � � � i � � o k � � f � f � g e � � � p � i � m � � � � � d i e p � j � c �i f g f i � i � e d g g f g � � � � � k � � f � f � g e � � � p � i i f � c � � � � f l � g d q � f � c � � k � � d e c � d g k d g f g � � � � �� j k � � � � c d l � d � e d g g f g � � � � � q c � � � q � f � c � f � d � i � � � � �� c f � i d � � � � i p f � � d p � � � f g � � � � � � f g � o � � i � p d � f � g � o � c � e � � � p � i m � � k � � � g � � d � � o � � � c � q � f � c �� o � c � i f g f i � i � e d g g f g � � � � � j d g k f � k � � � g � � � l � g d � � o � � � c � � k � � � � o � c � � e d g g f g � � � � � � c d �d � c f � l � � � c f � q � f � c � m � � q � l � � j � � �   � � q f p p � � � � � c � q � c d � � � � � d f g e � � � p � i � � � � � � � � � � � p l � k� £ � f � g � p � m � o q � � c � q � c d � � c � � f i e p � k � � f � f � g e � � � p � i � d g g � � � � � � p l � k � £ � f � g � p � j � c � g � c �i � � � � � g � � d p � e � f i f � d � f � g e � � � p � i � � � � d f g p � � d g g � � � � � � p l � k � £ � f � g � p � � f � c � � m

� ] ` a � ] a \ [ \ � � a ` _ } _ � ` � � � � \ � ~ b � � � � � l � � c d � d k � � f � f � g e � � � p � i � d g d p � � � � � c � � � c � � o d � dp d g � � d � � � � � � � g f � f � g e � � � p � i m ¥ � � � � p k k � � g � d p d g � � d � � �

� � � � � � � � � � c d � d � � � o q � f � c � d � i � � � � � �� c f � � � � � � g � f � � � � o e d f � � j � c � � � � � � p � i � g � f � d � � d e c � � m � m � c � d k   d � � g � � i d � � f � � g � � k � k d �d � � � f g � � o � p p � q � k � � d g f g � � � � � � � g � � k � k d � d � f g d � � g � i � � � m � � � � � � f � i d � � � � i � � � d g � �

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� � � � � � � � � � c d � d � f i e p � e d � c � o p � g � � c d � p � d � � � � �� f l � g d � � d e c � d g k f g � � � � � � c � q q � � p k � � � � � � � � � g f � � � q c � � c � � f � f � f g � c � p d g � � d � � � �� � � i f � c � � � � � � d � � c f g � � c � � � d e c o � � d � f i e p � e d � c � j � g � f p � g k f g � � g � � o p � g � � c d � p � d � � � m � o� � � � g k � g � � c � g � � � � d g d � � � e � d g k � � � i f g d � � m � � q � l � � j f o g � � � c � g � � � i d � � e � g k d p � � � o� f i � � � d � � c f g � � � � e � � f d p p � f o � f � p d � � � j p f � � � � j d g k g � � � � c e d � c � � f � � � � m � � f � � � n � � �� g � � g � q � � c � d g � q � � m � g o d � � j q � q f p p � c � q � c d � � f � � n � � � i e p � � � m� g q c d � o � p p � q � j q � q f p p � � f g � � � k � � f g � d g � i � � � � o � p d � � � � m ¥ � q f p p   � i e � d � � d g k o � � � c� � � q � � g � c � � � � i � � p d g � � d � � � d g k � k � � f � f � g e � � � p � i � � j � � � o � � � � � e � � e � � � � � c � � i � d g � c �� d i � � c f g � � m � � o � � � � f l f g � d p p � c � � � � c g f � d p k � � g f � f � g � j p � � � � � d � d � f � d � � � � q c d � � c � � � g � � d p� p d � � � � p � � � p f � � d � d g f g � � f � f l � p � l � p m b � c f � f � � c � � � � � o d p p k � � f � f � g e � � � p � i � � c d � � d g � � � � � � � f g e � p � g � i f d p � f i � m ¥ � q f p p

� � g � � d p p � � � o � � � � � c � � � e � � � p � i � d � � � f g � � � d � � � � � � � £ � f � g � p � � � p l d � p � � m � � p � c � � � c � c f �i d � � � d g � � d � � � � d � f � g f g i d g � � d � � � m �

� b � c f � f � � c � � � � � o d p p k � � f � f � g e � � � p � i � � c d � � d g � � � � �� � � f g e � p � g � i f d p � f i � m � ¥ � q f p p� f l � d k � � g f � f � g � o � c f � � � p � q m � � c f � � p d � � � � g � d f g � n d � d � � � � � � m � c � � j f � � � g � d f g � dg � i � � � � o � d � � e � � � p � i � j � � � f � d p � � � � g � d f g � d g � i � � � � o e � � � p � i � � c d � d � � � � p f � l � k � �� � l � � � c d � k � � � � p l � m � c � � � � i � n k � � � � � � i � d g � g � � e � p � g � i f d p � m � � f � f g d p p � � c �� � � i i � d g � � g � g k � � � � i f g f � � f � e � p � g � i f d p � f i � � m � � � f � f � � f � i � � � f g � � f � f l � � � � � e p d f g� c � � � g � � e � o � � i � c � e � � � e � � � f l � � o l � � f � � d � f � g m

� � | ] � ^ b � g � e f � � � o f � � g d i � j � � � d � � c d � e � � � p � i f � � n � c d � k k � � � � � � i � d g � c d � f � f � c d � k� � � � p l � m h d � c � � f � i � d g � � c d � f o q � � � � p k � � p l � � c f � e � � � p � i f g e � p � g � i f d p � f i � j � c � g q �� � � p k � � p l � � � � � n e � � � p � i � f g e � p � g � i f d p � f i � m � � � � � c d � o � � d e � � � p � i � � � � � n c d � k jf � k � � � g � � c d l � � � � � f g � c � � p d � � � n m � f g � � f � f � q f k � p � � � p f � l � k � c d � d p p � n e � � � p � i � d � �g � � � � p l d � p � f g e � p � g � i f d p � f i � j f � f � q f k � p � � � p f � l � k � c d � g � � n � c d � k e � � � p � i f � � � p l d � p �f g e � p � g � i f d p � f i � m

� � � � � � � \ } \ b � e � � � p � i f � � n � � � i e p � � � f o � � � f � f � f g � n j d g k � � � f � f � � n � c d � k m � c d � f � j� n � � n � n � c d � k m

� z

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� � � � � � � � � � � � � � � � � � � � � � �

� c � � � � � � � � p � q f p p � � � � d � � � � g � q d � � c d � � c � � � � � n j � n j � n � c d � k j d g k � n � � � i e p � � � � � n �� � � � � p � � � m ¥ � � d � � � � � � � � � d � � � q � k � g � � � g � q q c � � c � � d p p � o � c � � � � � i e p � � f � � � p d � � � � d � �k f � � f g � � � � q c � � c � � � c � � d � � d p p � � p l d � p � f g e � p � g � i f d p � f i � m � c � � � d � � � � i � e � � � p � i � f g � c �� � � � � � c d � q � q f p p g � � k f � � � � � m � g � f � � � � � � � � � � � � � � � � � j q c f � c d � � � q c � � c � � � q � � � d e c �d � � f k � g � f � d p � e � � d � � g d i f g � � o � c � f � l � � � f � � � m � � f � � g � q g � c d � � c f � e � � � p � i f � f g � n j � � �f � f � g � � � g � q g � � � � f g n m � c � � � c � � f � � � � j q c f � c � � d g k � o � � � � � � � �� � � � � � � � � � � � � � � � � � m� g � c f � e � � � p � i � � � d � � � f l � g d � � � p � d g o � � i � p d q f � c ¡ � d g � f � � � � � � d g k � � d g k � � � q d g � � �� g � q q c � � c � � � c � o � � i � p d f � � � � � � � o d p � � m � c f � e � � � p � i f � � � � � g k � c � � � � e � � o � c f � � � � � � � j� � � i d � � � k f � � � � � � k f g d g d k l d g � � k � � � � � � � g � � i e p � � f � � � c � � � � m

Easy

Harder

NP

P

NP-Hard

One way that things ‘might’ be.

Hamiltonian Cycle

Graph Isomorphism?

MSTStrong connectivity

Satisfiability

Knapsack

QBF

NPC

No Ham. Cycle

� f � � � � � � � � � � � � � � � � o n j � n j d g k � � p d � � k � � i e p � � f � � � p d � � � � m � � � ` � � _ ] � � _ � \ � \ � _ � � ] } _ � ` ] ` ^ � \ � } _ � � ] } \ ~ b � � o � � � � d p � f g � d � � � � � c � � p d � � � o � n � � � i e p � � �

e � � � p � i � j f � f � f i e � � � d g � � � f g � � � k � � � � c � g � � f � g � o d l � � f � � d � f � g d p � � � f � c i m d g � p d g � � d � �� � � � � g f � f � g e � � � p � i � � c d � i d � � � l � � � c d � k � � � � p l � j � � � � c � � c d l � � c � e � � e � � � � � c d � f � f �� d � � � � � � �� � q c � � c � � d � � � f g � f � f g � c � p d g � � d � � m � g � f k � � � c � o � p p � q f g � e � � � p � i j � d p p � k � c � � � � � � � � � � � � � � � � � � � � � � � � m � f l � g d g � g k f � � � � � k� � d e c � j k � � � � c d l � d � � � p � � c d � l f � f � � � l � � � l � � � � � � � d � � p � � g � � m � � c � � � f � d � f i f p d � e � � � p � i� g k f � � � � � k � � d e c � j d g k � c � � � f � d p � � d l � � � f � g q c f � c d � � � q c � � c � � � c � � � f � d e d � c � c d � l f � f � � d p pl � � � f � � � m � ¥ � � d g k � � � � f � � � c f � e � � � p � i d � d p d g � � d � � � � � � � g f � f � g e � � � p � i j q c � � � � c � p d g � � d � �f �

� � � � � � � � � c d � d � d i f p � � g f d g � � � p � � �q c � � � � � � k � g � � � � d g � g � � k f g � � o d � � d e c � d � d � � � f g � m � c � � d i f p � � g f d g � � � p � e � � � p � i � � � i �� � � � i � � c c d � k � � j d g k � c � � � f � g � � g � q g e � p � g � i f d p � f i � d p � � � f � c i o � � � c f � e � � � p � i m � � �� � d i e p � j � c � � � � � � � � p � q � c � q � � q � � � d e c � j � g � q c f � c f � � d i f p � � g f d g d g k � g � q c f � c f � g � � m� � q � l � � j � � e e � � � � c d � d � � d e c k f k c d l � d � d i f p � � g f d g � � � p � m � c � g f � q � � p k � � d l � � � � d � � i d � �� � � o � � � � i � � g � � � � � g l f g � � � � � o � c f � m � c � � q � � p k � f i e p � � d � � � c � � � � p � f � � � � � � � � � � � � � � � � � � � � m¥ � � � � p k � c � g f g � e � � � � c � � � d e c j d g k � c � � � � c d � � c f � f � f g k � � k d p � � d p � � � p � d g k � c d � f � l f � f � �d p p � c � l � � � f � � � � o � c � � � d e c � � d � � p � � g � � m � c � � j � l � g � c � � � c q � � g � q � o g � � £ � f � g � q d � � �� � � � � c � � d i f p � � g f d g � � � p � e � � � p � i j � c � � � f � d l � � � � £ � f � g � q d � � � � � �� � � c d � d � f l � g � � d e cf � f g � m � c � � f l � g � � � p � f � � d p p � k d � � � � �� � � � � m � c f � f � � � i � e f � � � � o f g o � � i d � f � g q c f � c d p p � q �� � � � l � � f o � � c d � d � f l � g � � � f g � f � f g d p d g � � d � � m � � � o � � i d p p � j � f l � g d p d g � � d � � � j d g k � f l � g � � � j d � � �� � � � � � � � � � � � � � � � f � d g d p � � � f � c iq c f � c � f l � g � d g k d � � � f g � � d p p � k � c � � � � � �� � � � � j � d g l � � f o � � c d � � f � f g � c � p d g � � d � � � � � f g �� c f � � � � � f � � d � � d � c � p e m � o � f � g � � f g � � c � g � c � � � f � g � � c f g � � � l � � f o � m

� �

Page 75: Design and analysis of computer algorithms (dave mount, 1999)

� � � � � � � � � � � � � � � � � � � � � � �

HamiltonianNonhamiltonian

� f � � � � � � d i f p � � g f d g � � � p � m

� � � � � c d � g � � d p p p d g � � d � � � c d l � � c � e � � e � � � � � c d � � c � � d � � � d � � � � l � � f o � m � � � � � d i e p � j� � g � f k � � � c � o � p p � q f g � p d g � � d � � �

� � � � � � � � � � c d � d � g f ¡ � � � d i f p � � g f d g � � � p � �� � � � � � � � � c d � g � � d i f p � � g f d g � � � p � � �

� � e e � � � � c d � d � � d e c � f � f g � c � p d g � � d � � � � � m ¥ c d � f g o � � i d � f � g q � � p k � � i � � g � � f l � � �� c d � q � � p k d p p � q � � � � l � � f o � � c d � � f � f g k � � k f g � c � p d g � � d � � � � c � � � � � p k � f l � � � d g � � d i e p �� o � c � � g f ¡ � � � d i f p � � g f d g � � � p � j d g k q � � � � p k l � � f o � � c d � f � f � d � d i f p � � g f d g � � � p � j � � � q c d �� � � � � o � � � � f � � d � � � � � p k � c � � � f l � � � � � � � g l f g � � � � � c d � � c f � f � � c � � � � � � g � � � c � � � � � p k� f l � d g � � c � � � � � p � � c d � f � � � � � d i f p � � g f d g j � � � � c f � k � � � g � � i � d g � c d � � c � � � f � g � � d g � � c � �� � � p � � � i � q c � � � � c d � f � � d i f p � � g f d g m � c � � � � � p k � � � � � p f � � � l � � � � � c � � � � � p � � o p � g � � c j� � � � c f � q � � p k g � � � � d � d p p � £ � f � g � j � f g � � � c � � � d � � � e � � � f � p � � � � p � � f g � � g � � d p m � c � � j f � f �c d � k � � f i d � f g � � c d � � � i � � g � � � � p k � f l � � � � � i � f g o � � i d � f � g � c d � q � � p k d p p � q � � � � � £ � f � g � p �� � g l f g � � � � � � � p l � � � c d � d � f l � g � � d e c f � f g � c � p d g � � d � � m

� | \ � � ] ~ ~ � b� \ � ` _ } _ � ` b � � � g � � n � � � � � c � � � � � o d p p p d g � � d � � � � c d � � d g � � l � � f � � k � � d e � p � g � i f d p

� f i � d p � � � f � c i m¥ c � f � � c � � � � � d p p � k � � n � � d � c � � � c d g � � n � � � c � � � f � f g d p � � � i � n � � � � k o � � � g � g k � � � � i f g �f � � f � e � p � g � i f d p � f i � � m � c f � � � o � � � � k � � d e � � � � d i � � g g f g � � g d � � � � � � � � � � � � � � � � � � � � � � � �� c d � � d g i d � � � � � � � � � m � d � f � d p p � j � � � c d � � i e � � � � � � � p k g � g k � � � � i f g f � � f � d p p � � � � � � � c � l d p � �� o � � � � f � � d � � j d g k � c � g l � � f o � � c d � � c � � � � f g � f � f g � c � p d g � � d � � f g e � p � g � i f d p � f i � m ¥ � c d l �d l � f k � k f g � � � k � � f g � g � g k � � � � i f g f � i c � � � m � � q � � p k � � � � l � � � k f g d � � � � � � � g � � i e p � � f � �� c � � � � � � o � � i d p p d g � � d � � � c � � � � m� f � � n j � n f � d � � � � o p d g � � d � � � � d � � k � g � � i � � � i e p � � f � � i � d � � � � � � c � � � i e p � � f � � � o l � � f � �� d � f � g � m � � � � � l � � c d � n � � n m � g � � c � � q � � k � j f o q � � d g � � p l � d e � � � p � i f g e � p � g � i f d p � f i � j� c � g q � � d g � � � � d f g p � l � � f o � i � i � � � � c f e f g e � p � g � i f d p � f i � m � � � � o � � i d p p � j q � k � g � � � l � gg � � k � � � � � d � � � � f � � d � � � � � � p l � � c � e � � � p � i j q � � d g � � p l � f � f g e � p � g � i f d p � f i � d g � q d � � m� � q � l � � f � f � g � � � g � q g q c � � c � � n � � n m � � � � � i � � g � � d � � g d � p � � � � c f g � � c d � � c f � � c � � p k � �� � m � g � � c � � q � � k � j   � � � � � f g � d � p � � � l � � f o � � c d � � � � c d l � d � � � � � � � � � p � � f � g k � � � g � � c � p e � � �f g � g k f g � � c � d � � � d p � � p � � f � g l � � � i � � c m � � � � � e � � � � � � p f � l � � c d � n �� � n j � � � g � � g � c d �d e � � � o � o � c f � m � � � � � f i � q � q f p p k � � g � � c � g � � f � g � � o � n � c d � k d g k � n � � � i e p � � � m

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Page 76: Design and analysis of computer algorithms (dave mount, 1999)

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[ \ ] ^ b c d e � z ¤ j � c � � � � c � � � � f � g z ¤ m � m� � � � ] � � b � d � � � f i � q � f g � � � k � � � k d g � i � � � � o � � g � � e � � j � g � c � q d � � � k � � g f g � � n � � � i e p � � � g � � � m

� g e d � � f � � p d � j � c � o � p p � q f g � � � g � � e � � d � � f i e � � � d g � m� \ � _ ~ _ � ` � � � � \ � ~ b d � � e � � � p � i � o � � q c f � c � c � d g � q � � f � � f � c � � � � � � � g � m � n � � � i e p � � �

e � � � p � i � d � � � � e � � � � � k d � k � � f � f � g e � � � p � i � j d g k c � g � � � d g � � � c � � � c � � o d � p d g � � d � �� � � � � g f � f � g e � � � p � i � j d � � � i f g � � c d � � c � f g e � � c d � � � � g � g � � k � k d � d � � � f g � m ¥ � � � � � � �f g e � � � d � � � � f g � � m � � � � � d i e p �

� � � � � � � c d � d � d i f p � � g f d g � � � p � �� � � � � � � � � � � � c d � d � � � o � � � � d � i � � � � � � b f � � c � � p d � � � o d p p k � � f � f � g e � � � p � i � q c f � c � d g � � � � p l � k f g e � p � g � i f d p � f i � j � � o � �� � i � � � g � � d g � � m � � � � � d i e p � � � � � n � � � � � f � g � � � g � q g � d g k � � � e � � � � k g � � � � � � �f g n m� \ � } _ � � ] } \ b f � d e f � � � � o � l f k � g � � � c d � d p p � q � � � � � � � �� � f g e � p � g � i f d p � f i � � c d � d � � � f g �f � f g d � f l � g p d g � � d � � m � � � � � d i e p � j � � e e � � � � c d � � c � p d g � � d � � f � � c � � � � � o � d i f p � � g f d g� � d e c � m � � � � g l f g � � � � i � � g � � c d � d � � d e c f � f g � c f � p d g � � d � � j q � � � � p k � � e e p � � c �� � � � f � � d � � � � g � f � � f g � � o d � � ¡ � � g � � � o l � � � f � � � d p � g � � c � � � � p � m � � f � � d � � � � d � � � � � � c �d k   d � � g � � i d � � f � � � k � � � � i f g � � c d � � c f � f � d p � � f � f i d � � � � � p � f g � m � c � � � o � � � � � � � n m

� b f � k � � g � k � � � � � c � � p d � � � o d p p p d g � � d � � � � c d � � d g � � � � �� � � f g e � p � g � i f d p � f i � m � � � �� c d � � f g � � d p p p d g � � d � � � f g n � d g � � � � p l � k f g e � p � g � i f d p � f i � j � c � � � d g � � � � d f g p � � �l � � f � � k f g e � p � g � i f d p � f i � j � � q � c d l � n � � n m � � q � l � � j � n d p � � � � � i � � � c d l � � � i �e � � � � � c d � k e � � � p � i � � � � � p l � j � � � c d � � � m

[ \ ^ � � } _ � ` ~ b � c � � p d � � � o � n � � � i e p � � � e � � � p � i � � � g � f � � � � o d � � � � o k � � f � f � g e � � � p � i � � p d g � � d � � � �� d � � � � � � � o � c � � p d � � � n � � c d � g � � g � � g � q � c � q � � � � p l � � £ � f � g � p � j � � � f o � c � � � q � � � de � p � g � i f d p � f i � � � p � � f � g o � � � l � g d � f g � p � � n � � � i e p � � � e � � � p � i j � c � g � l � � � e � � � p � i f g � nq � � p k � � � � p l d � p � f g e � p � g � i f d p � f i � m � � � � � d � p f � c � c f � j q � g � � k � � f g � � � k � � � � c � � � g � � e � � od � � k � � � f � g m� � o � � � k f � � � � � f g � � � k � � � f � g � j p � � � �   � � � � � g � f k � � � c � o � p p � q f g � ¡ � � � � f � g m � � e e � � � � c d � � c � � �d � � � q � e � � � p � i � j � d g k � m � � � � g � q � � � � � � � � � � g � p � � � p f � l � d � p � d � � � � c d � f � f � f i e � � � f � p � � �� � p l � e � � � p � i � f g e � p � g � i f d p � f i � m � � � q d g � � � e � � l � � c d � � � d g g � � � � � � p l � k f g e � p � g � i f d p� f i � m � � q q � � p k � � � k � � c f � � ¥ � q d g � � � � c � q � c d �

� � �� n � � � � �� n � �� � k � � c f � j q � � � � p k e � � l � � c � � � g � � d e � � f � f l � j

� � � n � � � � � n � �� g � � c � � q � � k � j � � � c � q � c d � � f � g � � � � p l d � p � f g e � p � g � i f d p � f i � j q � q f p p � � e e � � � � c d � � c � � �f � d g d p � � � f � c i � c d � � � p l � � � f g e � p � g � i f d p � f i � j d g k � c � g k � � f l � d � � g � � d k f � � f � g � � � c � q f g �� c d � � � d g � � � � p l � k f g e � p � g � i f d p � f i � m� � q k � q � k � � c f � � � � e e � � � � c d � q � c d l � d � � � � � � � f g � � c d � � d g � � p l � d g � f g � � d g � � � o e � � � p � i� f g e � p � g � i f d p � f i � m � c � g d p p q � g � � k � � k � f � � � � c � q � c d � q � � d g � � � � c f � � � � � � � � f g � � �� � p l � e � � � p � i � f g e � p � g � i f d p � f i � m � c � � q � c d l � � � � k � � � k � e � � � p � i � � � e � � � p � i � m � � f �f i e � � � d g � � � g � � � c � � � � c d � � c f � � � e e � � � k � � � � � � � f g � f � � � d p p � d � � � � � � � m ¥ � � g � q � � � � � � � g � p �� � p f � l � � � c d � � � d g g � � � � � � p l � k f g e � p � g � i f d p � f i � j � c � � q � d � � � � � � g � f d p p � e � � l f g � � c d � � c �� � � � � � � f g � � d g g � � � � f � � j f i e p � f g � � c d � � � d g g � � � � � � p l � k f g e � p � g � i f d p � f i � m � � � � � � � � c d �� � � � g k � � � � d g k � c f � j � c f � � c � � d � f � � � c f g k d p p � � k � � � f � g � m �

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� � ] � � � \ b � � � � � � ] � _ � _ } � ] ` ^ � � _ � � \ � � � \ � b � � � � � � � g � f k � � d g � � d i e p � � � i d � � � c f � � p � d � � � m� c � o � p p � q f g � e � � � p � i f � q � p p � � g � q g � � � � � n � � � i e p � � � j d g k c � g � � f � f � � � � � g � p � � � p f � l � k � c d �� c � e � � � p � i � d g g � � � � � � p l � k f g e � p � g � i f d p � f i � m � � � � � � _ ` a � � � � � b � f l � g d � � d e c � j � d g � d � c � o f � � l � � � f � � � � � p d � � p � k q f � c � g � � o z k f ¢ � � � g �

� � � p � � � � j � � � c � c d � g � � q � d k   d � � g � l � � � f � � � c d l � � c � � d i � p d � � p m � p � � f g � d � f � � � f g l d � f � � � e d � � f � f � g f g � e � � � p � i � j q c � � � � c � � � f � d � � g � � � d f g � � c d � � q � � �   � � � �� d g g � � � � d � � f � g � k � � � c � � d i � � � � � o � c � e d � � f � f � g m � c � � � � i � � � p � � f g � � � � i � � o � � i � c �� � f � f g d p d e e p f � d � f � g q c f � c q d � f g i d e k � d q f g � m � q � � � � g � � f � � � c d � � c d � � d � � i i � g � � � k � �� c � � p k � � � � p � � � k q f � c k f ¢ � � � g � � � p � � � m � � f � q � p p � g � q g � c d � e p d g d � � � d e c � � d g � � � � p � � � k q f � c� � � p � � � j d g k � c � � � � � f � � � d e � p � g � i f d p � f i � d p � � � f � c i o � � � c f � m � � � k � � � � i f g f g � q c � � c � � z� � p � � � d � � e � � � f � p � � � l � g o � � e p d g d � � � d e c � � � � � i � � � � � c d � k d g k � c � � � f � g � � g � q g e � p � g � i f d p� f i � d p � � � f � c i m � g � c � � � � � � � � p � q q � � f l � � q � � � d e c � m � g � q c f � c � d g � � � � p � � � k q f � c z� � p � � � j d g k � g � � c d � � d g g � � m

3-colorable Not 3-colorable Clique cover (size = 3)

� f � � � � � ¤ z � � � p � � f g � d g k p f ¡ � � � l � � m� c � z � p e � � � p � i q f p p e p d � � c � � � p � � o e � � � p � i � j q c f � c q � � � � � g � p � � � � e � � � � � g � � � �� � p l d � p � f g e � p � g � i f d p � f i � m � � � � � � e � � � p � i � j � � g � f k � � � c � o � p p � q f g � e � � � p � i m � f l � g d� � d e c � � � � � � � j q � � d � � c d � d � � � � � � � o l � � � f � � � � � � � o � � i � d � � � � � � f o o � � � l � � � e d f � � ol � � � f � � � � � � � � � � � � � � � � m � c d � f � j � c � � � � � � d e c f g k � � � k � � � � f � d � � i e p � � � � � d e c m� � _ � � \ � � � \ � � � � � � � b � f l � g d � � d e c � d g k d g f g � � � � � � j � d g q � � g k � � � � � � � � � o l � � � f � � �

� � � � � � � � � � � j � � � c � c d � � � � � � � j d g k � c d � � d � c � � f � d � p f ¡ � � � o � m� c � � p f ¡ � � � � l � � e � � � p � i d � f � � � f g d e e p f � d � f � g � � o � p � � � � � f g � m ¥ � e � � d g � k � � � � � q � � g � q �g � k � � f o � c � � d � � � f i f p d � � g � � � c � � � � � p � � � � � � k f g � c � � d i � � � � � e m ¥ � q d g � � � � g � q q c � � c � �f � f � e � � � f � p � � � � p � � � � � d p p � c � l � � � f � � � f g � � � � � � � e � m� � e e � � � � c d � � � � q d g � � � � � p l � � c � � l e � � � p � i j � � � d o � � � d q c f p � � o o � � f � p � � � � ¢ � � � j � � �� � f p p � d g g � � � g k d e � p � g � i f d p � f i � d p � � � f � c i o � � � c � � l e � � � p � i m � � q � d g � � � e � � l � � c d � � l f � p f � � p � � � g � � c d l � d e � p � g � i f d p � f i � � � p � � f � g � � � � � g � q � c d � z � p f � � n � � � i e p � � � jd g k c � g � � � � e � � � � � � p f � l � � c d � z � p �� n m � � � o � � p � c d � � c � � � f � � � i � � � g g � � � f � g � � � q � � g � c � � l e � � � p � i d g k � c � z � p e � � � p � i m � c � � j � � � q d g � � � � c � q � c d �

� z � p �� n � � � � l �� n � �q c f � c � � � q f p p � c � q � � e � � l f g � � c � � � g � � d e � � f � f l �

� � l � n � � � z � p � n � �� � k � � c f � j � � � d � � � i � � c d � � � � c d l � d � � � � � � � d � � � � � � � f g � � � � � � � � � m � f l � g d � � d e c � d g kd g f g � � � � � � j � c f � � � � � � � � f g � � � � � � g � � � � � f o � c d � d � p f ¡ � � � � l � � � o � f � � � d g k o d p � � � � c � � q f � � j

� �

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� � � � � � � � � � � � � � � � � � � � � � �

d g k o � � � c � � i � � � j � c f � � � � � � � � f g � � � g � f g e � p � g � i f d p � f i � m � � q � d g q � � � � � c f � � d p p � � � k �� � � � � � � f g � � � � � p l � � c � q � p p � � g � q g c d � k z � p e � � � p � i � ¥ � q d g � � � q � f � � d e � p � g � i f d p � f i �� � � � � � � f g � o � � z � p j d g k � c f � � � � � � � � f g � f � d p p � q � k � � � d p p � c � � � � � � � � f g � � � � � � � � � o � � d g �� � d e c � d g k d g � f g � � � � � � m� � � c e � � � p � i � f g l � p l � e d � � f � f � g f g � � c � l � � � f � � � � e f g � � � � � � e � m � c � � g p � k f ¢ � � � g � � c � � � f �� c d � f g � g � e � � � p � i � c � g � i � � � � o � p f ¡ � � � f � � e � � f � � k d � e d � � � o � c � f g e � � d g k f g � c � � � c � � � c �g � i � � � � o � � p � � � p d � � � � f � � � � k d � z m � g � c � � p f ¡ � � � � l � � e � � � p � i j o � � � q � l � � � f � � � � � � � f g � c �� d i � � � � � e � c � � i � � � � � d k   d � � g � � � � d � c � � c � � m � g � c � z � � � p � � f g � e � � � p � i j o � � � q � l � � � f � � �� � � � f g � c � � d i � � � p � � � � � � e j � c � � i � � � g � � � � d k   d � � g � m � g � � i � � � g � � j � c � e � � � p � i � d � �d p i � � � � c � � d i � j � � � � c � � � ¡ � f � � i � g � d k   d � � g � � g � g � d k   d � � g � f � � � d � � p � � � l � � � � k m¥ � � p d f i � c d � q � � d g � � � � � � � c � z � � � p � � f g � e � � � p � i � � � c � � p f ¡ � � � � l � � e � � � p � i d � o � p p � q � m� f l � g d � � d e c � o � � q c f � c q � q d g � � � k � � � � i f g � f � � z � � � p � � d � f p f � � j � � � e � � � c � e d f � � � � z � q c � � �� k � g � � � � � c � � � i e p � i � g � � o � m ¥ � � d g � c � g o � � k � c � e d f � � � � z � f g � � d � � � � � � � f g � o � � � p f ¡ � �� � l � � m � c f � f � f p p � � � � d � � k f g � c � � � � � � � � p � q m

_G H

_

Not 3-colorable

H

Not coverable

G

Coverable by 3 cliques3-colorable

� f � � � � � � p f ¡ � � � � l � � � f g � c � � � i e p � i � g � m� � ] _ � b � � � d e c � f � z � � � p � � d � p � f o d g k � g p � f o f � � � � i e p � i � g � � c d � d � p f ¡ � � � � � l � � � o � f � � z m

� g � � c � � q � � k � j� � z � p f ¢ � � � z � � � l � � � � � b � � � � o � z � � � p � � d � p � j � c � g p � � � � � � � � � � � � � c � � c � � � � � p � � � p d � � � � m ¥ � � p d f i � c d �

� c f � f � d � p f ¡ � � � � l � � � o � f � � z o � � � j � f g � � f o � d g k � d � � k f � � f g � � l � � � f � � � f g � � j � c � g� � � � � �� � � � � � � f g � � d k   d � � g � l � � � f � � � � d g g � � c d l � � c � � d i � � � p � � � q c f � c f i e p f � � � c d �� � � � � � � � � � m � c � � � l � � � e d f � � o k f � � f g � � l � � � f � � � f g � � d � � d k   d � � g � f g � m� � � � � e e � � � � c d � d � p f ¡ � � � � l � � � o � f � � z j k � g � � � k � � � � � � � � m � � � � � � � � � � z � � f l � � c �l � � � f � � � � o � � � � p � � � m ¥ � d � � � � � � c d � � c f � f � d p � � d p � � p � � f g � o � � � j � f g � � f o k f � � f g � � l � � � f � � �� d g k � d � � � � � c f g � � j � c � g � � � � � � � � � � � � f g � � � c � � d � � f g d � � i i � g � p f ¡ � � � j f i e p � f g �� c d � � � � � � �� � � � � � m � � g � � j � q � l � � � f � � � q f � c � c � � d i � � � p � � d � � g � � d k   d � � g � m

� � � ` � � _ ] � � } _ � \ � \ ^ � � } _ � ` b ¥ � g � q � d � � � c f � f g � � f � f � g � o � � k � � f g � � g � e � � � p � i � � d g � � c � �� c � � � � c � c � � � � � o d � � � � � � � f g � � d p p j d g k e p d � � f � � g i � � � o � � i d p o � � � f g � m � � � f � � � c d � f g � c �� � d i e p � d � � l � j q � � � g l � � � � k d g f g � � d g � � � o � c � z � � � p � � f g � e � � � p � i � � � f g � � d g � ¡ � f l d p � g �f g � � d g � � � o � c � p f ¡ � � � l � � e � � � p � i � � � z � m� \ � ` _ } _ � ` b ¥ � � d � � c d � d p d g � � d � � � f m � m k � � f � f � g e � � � p � i � � � f � � � � � � � � � � � � � � � � � � � � � � � �

� � p d g � � d � � � � � q � f � � � g � � � � � � � f o � c � � � f � d e � p � g � i f d p � f i � � � i e � � d � p � o � g � � f � g � j� � � c � c d � o � � d p p � j � � � � f o d g k � g p � f o � � � � � � � m

� g � c � e � � l f � � � � � d i e p � q � � c � q � k � c d �z � p � � � l �

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� � � � � � � � � � � � � � � � � � � � � � �

� g e d � � f � � p d � q � c d l � � � � � � � � � z � m � � � � � c d � f � f � � d � � � � � � i e p � i � g � d � � d e c f g � � �� f m � m e � p � g � i f d p � � f i � � � m � m � f e � � � d g k � � � f g � c � d k   d � � g � � i d � � f � � m � c � � � f � � � i e � � d � p � f ge � p � g � i f d p � f i � m� g � � f � f l � p � j � d � f g � � c d � � � � � � � i � d g � � c d � � f o � � f � � � p l d � p � f g e � p � g � i f d p � f i � j � c � g � �f � � � m� � c f � f � � � � d � � � d e � p � g � i f d p � f i � � � � � � � � f g � o � � � � � � � p k � � d e e p f � k � � � � � � � �k � � � � i f g � q c � � c � � � � � � � � � j � � � ¡ � f l d p � g � p � q c � � c � � � � � � m � c � � j f g � � g � � � o e � p � g � i f d p� f i � � � i e � � d � f p f � � j � � f � � g � c d � k � � � � c d g � � m� c � q d � f g q c f � c � c f � f � � � � k f g � n � � � i e p � � � g � � � f � � � d � � p � � c � � � g l � � � � m ¥ � � � � d p p � c d l �� � � � g � � l f k � g � � � c d � � � f � g � � � � p l d � p � f g e � p � g � i f d p � f i � j d g k c � g � � � c � � � k � � � f � g f � � ¢ � � � f l � p �� ¡ � f l d p � g � � � � d � f g � � � f g � � � � f � g � � p f � � p � � � � � � � p l d � p � f g e � p � g � i f d p � f i � j � c � g � � f � d p � �g � � p f � � p � � � � � � � p l d � p � f g e � p � g � i f d p � f i � m� � c � � j � c f � f � c � q e � p � g � i f d p � f i � � � k � � � f � g �� d g � � � � � k � � � c � q � c d � e � � � p � i � d � � d � c d � k � � � � p l � d � � g � q g k f £ � � p � e � � � p � i � m� \ � � ] b � o � � � � � � d g k � � � � � c � g � � � � m� \ � � ] b � o � � � � � � d g k � � �� � � c � g � � �� � m� g � f i e � � � d g � o d � � d � � � � � � k � � f � f p f � � f � � c d � f � f � � � d g � f � f l � m � g � � c � � q � � k �� \ � � ] b � o � � � � � � d g k � � � � � � � c � g � � � � � � m� c � � � d � � g f � � c d � f o � q � o � g � � f � g � � � � � d g k � � � � d � � � � i e � � d � p � f g e � p � g � i f d p � f i � j � c � g� c � f � � � i e � � f � f � g � � � � � � � f � � � i e � � d � p � f g e � p � g � i f d p � f i � d � q � p p m � � � c � � p k � � g � � � k � c d �� � � � � � � � � � � � c � � � � i � � � k � � � f � g � q c � � � i � � � � � c � � � � � � � � � � � c � � � � i � � � d g � o � � i d � f � g � m� c � k f � � f g � � f � g f � � � � � p � j � � � e � � e p � � d � f g � � � c � � � � � � � � � f g � � i e p � � f � � � c � � � � � c � � p k � � d q d � �� o � c f � m

� � � � � � � \ } \ ` \ ~ ~ b � c � � � � � o � n � � � i e p � � � e � � � p � i � d � � d p p e � � � p � i � f g � c � � � i e p � � f � � � p d � �� n j o � � q c f � c f � f � � g � q g � c d � f o d g � � g � f � � � p l d � p � f g e � p � g � i f d p � f i � j � c � g � c � � d p p d � � jd g k � � g l � � � � p � j f o d g � � g � f � g � � � � p l d � p � f g e � p � g � i f d p � f i � j � c � g g � g � d � � m � c f � f � i d k �i d � c � i d � f � d p p � o � � i d p � � f g � � c � g � � f � g � o e � p � g � i f d p � f i � � � k � � � f � g � m� \ � ` _ } _ � ` b � p d g � � d � � � f � � � � � � � f o

� � � � � o � � d p p � � � � n �� \ � ` _ } _ � ` b � p d g � � d � � � f � � � � � � � � � � � f o � � � n j d g k � f � � n � c d � k m� g d p � � � g d � f l � � d g k � � � d p p � � d � f � � q d � � � � � c � q � c d � d e � � � p � i f � � n � � � i e p � � � f � � � � � �� � d g � f � f l f � � m� \ � � ] b � f � � n � � � i e p � � � f o

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� c � � � d � � g f � � c d � d p p � � � � � � d � � � � k � � f � p � � � � � � � f g � � � � f � � n � � � i e p � � � d g k c � g � �� n � c d � k � d g k c � g � � � � � � d g � f � f l f � � � � � f � � � k � � f � p � � � � j f i e p � f g � � c d � � f � � n � c d � k m� c f � � f l � � � � d q d � � � e � � l � � c d � e � � � p � i � d � � � n � � � i e p � � � j � g � � q � � g � q � c d � � g � e � � � p � if � � n � � � i e p � � � m � g o � � � � g d � � p � j f � d e e � d � � � � � � d p i � � � f i e � � � f � p � � � e � � l � � c d � � g � e � � � p � if � � n � � � i e p � � � j � � � d � � � � c � k � � g f � f � g � d � � � c d � q � c d l � � � � � d � p � � � � � k � � � � � � � e � � � p � i f g� n � � � c f � e � � � p � i m � c � � � d � � f g � g f � � p � i d g � � � � c e � � � p � i � j � � c � q � d g q � � l � � c � e � � � k �� c f � � ¥ � q f p p � d p � d � � � � � c f � g � � � � f i � q f � c � � � � � � c � � � � i m � � � � c � q � k � c d � � c � � � f � � g �e � � � p � i � d p p � k � � � � � c � � � o � � � � � p � d g � d � f � � d � f p f � � � � c d � f � � n � � � i e p � � � m � � e � � l � d � � � � g k

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e � � � p � i f � � n � � � i e p � � � j d p p q � g � � k � � k � f � � � � c � q � c d � � � � e � � � p � i f � f g � n � d g k c � g � �f � f � � � k � � f � p � � � � � � � j d g k � c � g � � � c � q � c d � q � � d g � � k � � � � � � � � � � � g � � d p p � � � i � � g � q g� n e � � � p � i � � � � � � e � � � p � i m � � o � p p � q � � c d � � � � e � � � p � i f � � ¡ � f l d p � g � � � � � � � q f � c � � � e � � �� � � � p l d � f p f � � f g e � p � g � i f d p � f i � � m � c f � f � f p p � � � � d � � k f g � c � � � � � � � � p � q m

Proving a problem is NP-hard

Known NPCYour reductionYour problem

Proving a problem is in NP

NP

Resulting structure

NPC NPCSAT SAT

P P

NP

NPC

NP

P

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; < = > ? @ < r � B C D D � � E � � < D @ < u w G H � > � < @ � < H ? = > I D G EJ K L M N O P Q R S � � � X � S X Y Y Y Z[ \ ] ^ b � � � � f � g z ¤ m m � c � � � k � � � f � g � f l � g c � � � f � � f i f p d � j � � � g � � � c � � d i � d � � c � � � k � � � f � g � f l � gf g � c � � � � � m[ \ � ] � b � � o d � q � f g � � � k � � � k � c � k � � g f � f � g � � o � n � � � i e p � � � g � � � m h � � d p p � c d � q � i � g � f � g � k � c �

o � p p � q f g � � � e f � � b f � � c � � � � � o k � � f � f � g e � � � p � i � � � � p d g � � d � � � � � c d � d � � � � p l d � p � f g e � p � g � i f d p � f i � m� b f � � c � � � � � o k � � f � f � g e � � � p � i � � � � p d g � � d � � � � � c d � � d g � � l � � f � � k f g e � p � g � i f d p � f i � j � � � ` � � _ ] � � \ ^ � � } _ � ` b � � � � � � i � d g � � c d � � c � � � f � d e � p � g � i f d p � f i � � � i e � � d � p � o � g � �

� f � g � � � � c � c d � � � � � f o d g k � g p � f o � � � � � � � m � i � � � f g � � f � f l � � � � c f g � d � � � � � c f � j f �� c d � f o q � c d k d � � � � � � � f g � � � � � p l � � � f g e � p � g � i f d p � f i � j � c � g q � � � � p k � � � f � � � � � p l �� � f g e � p � g � i f d p � f i � mn � p � g � i f d p � � k � � � f � g � d � � � � d g � f � f l � j � c d � f � j f o � � � � � � d g k � � � � � � j � c � g � � � � � � m

� � � ] � ^ b � f � � n � c d � k f o o � � d p p � � � � n j � � � � � m � c � � j f o q � � � � p k � � p l � � f g e � p � g � i f d p� f i � j q � � � � p k � � p l � d p p � n e � � � p � i � f g e � p � g � i f d p � f i � m

� � � � � � � \ } \ b � f � � n � � � i e p � � � f o � � � � � � n d g k � � � � f � � n � c d � k m� c � f i e � � � d g � � � o � n � � � i e p � � � e � � � p � i � � c � � p k g � q � � � p � d � m � o d g � � n � � � i e p � � � e � � � p � i �� d g k � � g � � d p p � d g � � n � c d � k e � � � p � i � f � � � p l d � p � f g e � p � g � i f d p � f i � j � c � g � l � � � � n � � � i e p � � �e � � � p � i � d g k f g o d � � � l � � � e � � � p � i f g � n � f � d p � � � � p l d � p � f g e � p � g � i f d p � f i � m � g l � � � � p � jf o q � � d g e � � l � � c d � d g � � n � � � i e p � � � e � � � p � i � d g k � � g � � d p p � d g � e � � � p � i f g � n � � d g g � �� � � � p l � k f g e � p � g � i f d p � f i � j � c � g � l � � � � n � � � i e p � � � e � � � p � i � d g k � � g � � d p p � � l � � � � n � c d � ke � � � p � i � � d g g � � � � � � p l � k f g e � p � g � i f d p � f i � m � c � � d p p � n � � � i e p � � � e � � � p � i � d � � � ¡ � f l d p � g �� � � g � d g � � c � � � f g � c d � � c � � d � � � f � c � � d p p � � p l d � p � f g e � p � g � i f d p � f i � j � � g � g � d � � � m� g d p � � � g d � f l � q d � � � � c � q � c d � d e � � � p � i f � � n � � � i e p � � � f � � � � � � � � d g � f � f l f � � � o � � m� \ � � ] b � f � � n � � � i e p � � � f o

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f g � � � � � � j k � � � � c d l � d � � � � � � � � � o � l � � � f � � � � � � c � c d � o � � � d � c k f � � f g � � � � � � � � j� � � � � � � m � g � � c � � q � � k � j k � � � � c d l � d � l � � � � � � � � � � � q c � � � f g k � � � k � � � � � d e c f �� � i e p � � � m

� \ � } \ � � � � \ � � � � � b � � � � � � � � � � f g d g � g k f � � � � � k � � d e c � � � � � � � f � d � � � � � � � o l � � � f � � �� � � � � � � c � c d � � l � � � � k � � f g � c d � d � p � d � � � g � � g k e � f g � f g � � m � c � � � � � � � � � �� � � � � � � � � � f � � f l � g d g � g k f � � � � � k � � d e c � d g k d g f g � � � � � � j k � � � � c d l � d l � � � � �� � l � � � o � f � � � �

� � � _ ` ] } _ ` a � \ } � � � � b � � � � � � � � � � � � � � f g d � � d e c � � � � � � � f � d � � � � � � � o l � � � f � � � � �� � � c � c d � � l � � � l � � � � � f g � c � � � d e c f � � f � c � � f g � � � � f � d k   d � � g � � � � � i � l � � � � � f g � � m� c � � � � � � � � � � � � � � � � � � � � � � � � � f � � f l � g d � � d e c � � � � � � � d g k d g f g � � � � � � j k � � � �c d l � d k � i f g d � f g � � � � � o � f � � � �

� � g � � � � g o � � � � c � � p f ¡ � � � � � � � � � e � � � p � i q f � c � c � � p f ¡ � � � � � l � � � � e � � � p � i � c d � q �k f � � � � � � k f g d g � d � p f � � p � � � � � � m � c � � p f ¡ � � e � � � p � i � � � � � � � � g k d � f g � p � � p f ¡ � � � o � f � � � j d g k� c � � p f ¡ � � � � � l � � e � � � p � i � � � � � � � e d � � f � f � g � c � l � � � f � � � f g � � � � � � � e � j � d � c � o q c f � c f � d � p f ¡ � � m¥ � c d l � k f � � � � � � k � c � o d � � � � c d � � p f ¡ � � � d � � � o f g � � � � � � f g d e e p f � d � f � g � k � d p f g � q f � c � p � � � � � f g � m� c � l � � � � � � � l � � e � � � p � i d � f � � � f g l d � f � � � � � � l f � f g � d e e p f � d � f � g � m � � � � � d i e p � j � � � c d l � d� � i e � � � g � � q � � � d g k d e � � � � d i � c d � � c � � � � � c � f g � � � � f � � � o � c � � � i i � g f � d � f � g p f g � � m � � � d l �� c � � e d � � � o f g � � d p p f g � � c � e � � � � d i � g � l � � � � � i e � � � � f g � c � g � � q � � � j f � � � £ � � � � � f g � � d p p f �� g d p p � c � � � i e � � � � � o � � i f g � d l � � � � � � � l � � m � � � i � c � � � g � k � � d p p � c � p f g � � � d g � � � � � � � k m� � i f g d � f g � � � � f � � � � o � p f g o d � f p f � � p � � d � f � g e � � � p � i � m � � � � � d i e p � j � � e e � � � q � q d g � � � � � p � � �q c � � � � � e p d � � d � � � � o � � � � � d � f � g � � � � c � c d � � l � � � c � � � � f g � c � � f � � f � q f � c f g � i f g � � � � � o � c �g � d � � � � � � � � � d � f � g m ¥ � � � � d � � d � � d e c f g q c f � c � q � p � � d � f � g � d � � d k   d � � g � f o � c � � d � � q f � c f g� i f g � � � � � o � d � c � � c � � m � i f g f i � i � f � � k k � i f g d � f g � � � � q f p p � � d i f g f i � i � � � � o p � � d � f � g �� � � c � c d � � l � � � � � c � � p � � d � f � g f � � � d � c d � p � q f � c f g � i f g � � � � o � � i � g � � o � c � � � � f � � � m� c � � � � � � e � � � p � i f � � � l f � � � p � � p � � � p � � � p d � � k � � � c � f g k � e � g k � g � � � � e � � � p � i � � � � � f l � gd � � d e c � k � � � f � c d l � d � l � � � � � � � � � � � � c d � f � � � i e p � � � p � � � � � � � � � � � � � m � � f � g � � ¡ � f � � d � � p � d �� c d � � c � l � � � � � � � l � � e � � � p � i f � � � p d � � k m � � q � l � � j � c � o � p p � q f g � p � i i d i d � � � � c f � � � g g � � � f � g� p � d � d � q � p p m� \ � � ] b � f l � g d g � g k f � � � � � k � � d e c � � � � � � � q f � c l � � � f � � � d g k d � � � � � � � � � � � o � f � �

� m � c � o � p p � q f g � d � � � ¡ � f l d p � g � � f � � � f � d � p f ¡ � � � o � f � � � o � � � c � � � i e p � i � g � j � m� f f � � � f � d g f g k � e � g k � g � � � � � o � f � � � o � � � m� f f f � � � � � f � d l � � � � � � � l � � � o � f � � � � o � � � m � � � � b� _ � � � _ _ � b � o � � f � d � p f ¡ � � o � � � j � c � g o � � � d � c � � � � � � j � � � � � f � d g � k � � � o � f i e p � f g �

� c d � � � � � � f � g � � d g � k � � � o � j f i e p � f g � � c d � � � f � d g f g k � e � g k � g � � � � o � � � m

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� � � � � � � � � � � � � � � � � � � � � � �

GG G

in the complement of GV’ is an IS of size k V-V’ is a VC of size n-k

in Gin GV’ is CLIQUE of size k iff iff

� f � � � � � p f ¡ � � j � g k � e � g k � g � � � � j d g k � � � � � � � l � � m

� _ _ � � � _ _ _ � b � o � � f � d g f g k � e � g k � g � � � � o � � � j � c � g o � � � d � c � � � � � � j � � � � � f � g � � d g� k � � � o � j f i e p � f g � � c d � � l � � � � k � � f g � f � f g � f k � g � � � d l � � � � � f g � � � � j f i e p � f g �� c d � � � � � f � d � o � � � m� _ _ _ � � � _ � b � o � � � � f � d l � � � � � � � l � � o � � � j � c � g o � � d g � � � � � � � � c � � � f � g � � k � �� � � � � f g � j f i e p � f g � � c d � � c � � � f � d g � k � � � � � � � f g � j f i e p � f g � � c d � � � f � d � p f ¡ � �f g � m � � f � d g f g k � e � g k � g � � � � o � � � m

� c � � j f o q � c d k d g d p � � � f � c i o � � � � p l f g � d g � � g � � o � c � � � e � � � p � i � j q � � � � p k � d � f p � � � d g � p d � �f � f g � � d g d p � � � f � c i o � � � c � � � c � � � m � g e d � � f � � p d � j q � c d l � � c � o � p p � q f g � m� | \ � � \ � b � � � � � f � � n � � � i e p � � � m� � � � � � � b � c � � � � � f � � d � � � � g � f � � � � o � c � � l � � � f � � � f g � c � � p f ¡ � � m � f l � g � � � c d � � � �

� f � � d � � q � � d g � d � f p � l � � f o � f g e � p � g � i f d p � f i � � c d � d p p e d f � � � o l � � � f � � � f g � c � � � � d � �d k   d � � g � m

� � � � � � � � b ¥ � q d g � � � � c � q � c d � � f l � g d g f g � � d g � � � o � c � � � e � � � p � i � � � � � j q �� d g e � � k � � � d g � ¡ � f l d p � g � f g � � d g � � � o � c � � � � � � e � � � p � i � � � � � � � f g e � p � g � i f d p � f i � m� � i e � � � d g � ¥ � k � g � � � g � q q c � � c � � � c d � d g f g k � e � g k � g � � � � j d g k q � k � g � � c d l � � f i �� � � � i e � � � f � m � � f l � g � d g k � j � � � � � � � d g k � � � � j d g k � � � e � � � c � e d f � � � � � � � � m � �� c � d � � l � p � i i d j � c f � f g � � d g � � f � � ¡ � f l d p � g � m

� | \ � � \ � b � f � � n � � � i e p � � � m� � � � b � c � � � � � f � � d � � � � g � f � � � � o � c � � l � � � f � � � f g � c � l � � � � � � � l � � m � f l � g � � � c d � � � � f � �

� d � � q � � d g � d � f p � l � � f o � f g e � p � g � i f d p � f i � � c d � � l � � � � k � � f � f g � f k � g � � � � g � � o � c � � �l � � � f � � � m

� � � � � b ¥ � q d g � � � � c � q � c d � � f l � g d g f g � � d g � � � o � c � � � e � � � p � i � � � � � j q � � d g e � � k � � �d g � ¡ � f l d p � g � f g � � d g � � � o � c � � e � � � p � i � � � � � � � f g e � p � g � i f d p � f i � m ¥ � � � � � � � �d g k � � � � � m � � � c � p � i i d d � � l � j � c � � � f g � � d g � � � d � � � ¡ � f l d p � g � m

� � � _ ` ] } _ ` a � \ } b � � q f � c l � � � � � � � l � � j k � i f g d � f g � � � � f � d g � � d i e p � � o d � � d e c � � l � � f g � e � � � p � i m� � � � � c � � � g k f � f � g f � d p f � � p � k f ¢ � � � g � j � d � c l � � � � � f � d k   d � � g � � � � c � i � i � � � � � o � c � k � i f g d � f g �� � � j d � � e e � � � k � � � d � c � k � � � � f g � f g � f k � g � � � � d � c i � i � � � � o � c � k � i f g d � f g � � � � m � � l f � � � p � jf o � f � � � g g � � � � k d g k c d � d l � � � � � � � l � � � o � f � � � j � c � g f � c d � d k � i f g d � f g � � � � � o � f � � � � � c �� d i � � � � � o l � � � f � � � � j � � � � c � � � g l � � � � f � g � � g � � � � � d � f p � � � � � m � � q � l � � � c � � f i f p d � f � � � � � � � � � �� c d � f o � f g � n � � � i e p � � � j � c � g � � f � p f � � p � � � � � � n � � � i e p � � � d � q � p p m � c � i d f g � � � � p � � o� c f � � � � � f � g f �   � � � � c f � m

� ¤

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� | \ � � \ � b � � f � � n � � � i e p � � � m� � � � � d p � c � e � � � o c d � � q � e d � � � m � f � � � q � � c � q � c d � � � � � n m � c � � � � � f � � d � �   � � � � � g � f � � �� o � c � � � � � � � � � f g � c � k � i f g d � f g � � � � m � g e � p � g � i f d p � f i � q � � d g k � � � � i f g � q c � � c � � � l � � �l � � � � � f � f g � � � � f � d k   d � � g � � � d l � � � � � f g � � m� � � � q � � c � q � c d � d g � � f � � f g � � n � � � i e p � � � e � � � p � i f � � � k � � f � p � � � k � i f g d � f g � � � � m ¥ � � c � � � �l � � � � � � � l � � d g k � c � q � c d � � � � � � m ¥ � q d g � d e � p � g � i f d p � f i � o � g � � f � g j q c f � c � f l � g d gf g � � d g � � � o � c � l � � � � � � � l � � e � � � p � i � � � � � j e � � k � � � � d g f g � � d g � � � � � � � � � � o � c � k � i f g d � f g �� � � e � � � p � i j � � � c � c d � � c d � d l � � � � � � � l � � � o � f � � � f o d g k � g p � f o � � c d � d k � i f g d � f g � � � � � o� f � � � � m� � q � � q � � � d g � p d � � � � � q � � g � c � � � e � � � p � i � � � c � � � � k f ¢ � � � g � � f � � c � � � g k f � f � g m � g � � � l � � � � k � � f � f g � f k � g � � � d l � � � � � f g � � � m � g � � � � l � � � l � � � � � f � � f � c � � f g � � � � f � d k   d � � g �� � d l � � � � � f g � � � m � c � � � c � � � d g � p d � f � g i � � � � � i � c � q i d e � c � g � � f � g � o � f g � f k � g � � � �� d k   d � � g � � m � � � d � � � f g � f k � g � � f � d e � � e � � � � � o � k � � � j d g k d k   d � � g � � f � d e � � e � � � � � o l � � � f � � � j� c f � � � � � � � � � � c d � � c � � � k � � � f � g o � g � � f � g i � � � � � i � c � q � � � g � c � � k � � � � o � c � � � d e c f g � �l � � � f � � � � � � c � c d � d g f g � f k � g � � k � � f � i d e e � k � � d g d k   d � � g � l � � � � � m� c f � � � � � � � � � � c � o � p p � q f g � f k � d � q c f � c k � � � g � � ¡ � f � � q � � � � m ¥ � q f p p f g � � � � d l � � � � � f g � � � c �i f k k p � � o � d � c � k � � � o � c � � � d e c m � g � � c � � q � � k � j o � � � d � c � k � � � � � � � j q � q f p p � � � d � � d g � q� � � � � � � � � � � � j � d p p � k � �

� j d g k � � e p d � � � c � � k � � � � � � � q f � c � c � � c � � q � � k � � � � � � � �� � d g k

� � � � �� � m � c � o d � � � c d � � q d � f g � f k � g � � � � k � � � � � � � c d � g � q � � � g � � e p d � � k q f � c � c � o d � � � c d �

� f � d k   d � � g � � � � c � � � � � � � e � g k f g � l � � � � � � �� m ¥ � � � f p p g � � k � � k � i f g d � � � c � g � f � c � � � � m � �

k � � c f � j q � q f p p p � d l � � c � � k � � � � � � � f g � c � � � d e c d � q � p p m � � � � � � � � c � � � � � p � f g � � � d e c m� c f � f � � � f p p g � � ¡ � f � � � � � � � � � � c � � � c m � � � g � d g � � � � � � � � � � � � � � � � � � g � � c d � f � f g � f k � g � � � g �� k � � � m � o � f � f � � p d � � k f � � d g � g p � � � k � i f g d � � k f o f � f � f g � p � k � k f g � c � k � i f g d � f g � � � � m � f g � �f � f � g � � f g � f k � g � � � d g � � k � � � j f � k � � � g � � g � � k � � � � f g � c � l � � � � � � � l � � m � � � � � k � g � � � � c �f � � p d � � k l � � � f � � � f g � j d g k p � � � k � g � � � � c � g � i � � � � o f � � p d � � k l � � � f � � � m � c � g � i � � � � o l � � � f � � �� � � � ¡ � � � � o � � � c � k � i f g d � f g � � � � q f p p � � � � � � � � m� � q q � � d g � f l � � c � � � i e p � � � � � k � � � f � g m � f l � g � c � e d f � � � � � � o � � � c � � e � � � p � i j q � � � � d � �d � � d e c � � d � o � p p � q � m � g f � f d p p � � � � � m � � � � d � c � k � � � � � � � f g � q � � � � d � � d g � q l � � � � � � �

�f g � � d g k d k k � k � � � � � � � �

� � d g k � � � � �� � f g � � m � � � � k � g � � � � c � g � i � � � � o f � � p d � � k l � � � f � � �

d g k � � � � � � � � � m � � � e � � � � � � � � � m � c f � � � k � � � f � g f p p � � � � d � � k f g � c � o � p p � q f g � � � � � � m � � � �� c d � � l � � � � � � e � d g � � e � � o � � i � k f g e � p � g � i f d p � f i � m

f

Gk=3

G’k’=3+1=4

� f � � � � � � � i f g d � f g � � � � � � k � � � f � g m� � � � � d � p f � c � c � � � � � � � � g � � � � o � c � � � k � � � f � g j q � g � � k � � � c � q � c d � � c d � d l � � � � � � � l � � � o� f � � � f o d g k � g p � f o � � c d � d k � i f g d � f g � � � � � o � f � � � � m � f � � � q � d � � � � � c d � f o � � f � d l � � � � �

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� � l � � o � � � j � c � g � � � � � � � � � f � d k � i f g d � f g � � � � o � � � � m � � � � � l � � c d �� � � � � � � � � � � � � � � � � � � � �

� � o � c f � ¡ � d g � f � � f � p � � � � c d g � � j � c � g q � � d g d k k d g � l � � � f � � � q � p f � � � � i d � � � c � � f � � � ¡ � d p � �� � m �� � � � � � c d � � � � f � d k � i f g d � f g � � � � j � � � � � � � � � l � � c d � d p p � c � f � � p d � � k l � � � f � � � d � � f g � � � d g k� � � c � � d � � k � i f g d � � k m � � � � g k j � d � c � o � c � � e � � f d p l � � � f � � � � �

� f g � � � � � � � � e � g k � � � d g � k � �� � � � � f g � f i e p � f g � � c d � � f � c � � � � � � f � f g � c � l � � � � � � � l � � � � m � c � � � �

� f � k � i f g d � � k � �� c � � d i � l � � � � � f g � � � � f g d p p � j � d � c � o � c � g � g f � � p d � � k � � f � f g d p l � � � f � � � � f � f g � f k � g � � � d � p � d � �� g � � k � � f g � j d g k c � g � � � f � c � � f � f � f g � � � � � p � � d p p � o f � � g � f � c � � � � d � � f g � � m � g � f � c � � � d � � j� f � � f � c � � f g � � � � � d k   d � � g � � � d l � � � � � f g � � � m � c f � f � � c � q g f g � c � � � e e d � � � o � c � o � p p � q f g �� � � � � m

vertex cover for G dominating set for G’

vertex cover for Gdominating set for G’ using original vertices

� f � � � � z � � � � � � g � � � � o � c � � � � � � � � k � � � f � g m � g l � � � � p � j q � � p d f i � c d � f o � � c d � d k � i f g d � f g � � � � � � � � o � f � � � � � � � � � � c � g � c d � d l � � � � �� � l � � � � � o � f � � � m � � � � � c d � d p p � � f � � p d � � k l � � � f � � � � o � � i � � � � � f g � c � k � i f g d � f g � � � � m � f � � � jp � � � � � � � � � � � � � � � � c � � � i d f g f g � � l � � � f � � � m ¥ � i f � c � � � � � � � p d f i � � i � � c f g � p f � � � � � � f �d l � � � � � � � l � � o � � � m � � � � c f � q f p p g � � g � � � � � d � f p � q � � � j � � � d � � � � � � � i d � c d l � l � � � f � � � � c d �d � � g � � e d � � � o � c � � � f � f g d p � � d e c � m� � q � l � � j q � � p d f i � c d � q � g � l � � g � � k � � � � � d g � � o � c � g � q p � � � � d � � k � e � � f d p l � � � f � � � f g � � � � m� g e d � � f � � p d � j f o � � i � l � � � � � � �

� � � � � � j � c � g i � k f o � � � � � � � � � e p d � f g � � �� q f � c � m � ¥ � � � � p k

c d l �   � � � d � � d � f p � � � e p d � � k f � q f � c � m � � � � � � l � � c d � � c � l � � � � � � �� f � d k   d � � g � � � � g p � � d g k

� j � � f � k � i f g d � � � f � � � p o d g k � c � � � � � c � � � q � l � � � f � � � m � � � � f g � � f g � � � d k j q � � � f p p k � i f g d � �� j � j d g k � �

� � � � � d � � � � c d � � k � � � � � f g � � � � d g k � �� � m � c � � � � � � e p d � f g � � � �

� q f � c � q �k � i f g d � � � c � � d i � l � � � f � � � � d g k e � � � g � f d p p � i � � � � m � � � � � k � g � � � � c � � � � � p � f g � � � � d o � � � � c f �i � k f � � d � f � g m � � c f � f � � c � q g f g � c � p � q � � i f k k p � e d � � � o � c � � � � � � m �¥ � � p d f i � c d � � � f � d l � � � � � � � l � � o � � � m � o j � � � c � � � g � � d � � � c � � � q � � � d g � k � � � � � � � � o �� c d � q d � g � � � � l � � � k � g � f � c � � � g � � � q d � f g � � � � c � g � c � � e � � f d p l � � � � � � �

� q � � p k g � � � �d k   d � � g � � � d g � l � � � � � � o � � � f g � � j � � g � � d k f � � f g � � c � c � e � � c � � f � � c d � � � � q d � d k � i f g d � f g �� � � o � � � � m

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� � � � d g � � � � � � � � o f g e � � � k � � � j � c � � � � � f � � � d � � � � o e d � c � q c f � c   � f g � d � c f g e � � � k � �� � j � � j � � � � � � f � � � � � e � � � f l � � � � e � � � k � � � � j � � j � � � � � � � c � c d � � � � � � c � � � c � � � e d � c �l f � f � � l � � � l � � � � � f g � c � � d k � � � � � d � � p � � g � � m

� � g � � � � � � � � o e d � c � � c d � � � d � � � g � c � f g e � � � k � � � d g k � g k � g � c � � � � e � � � k � � � j d g kl f � f � d p p � c � l � � � f � � � � o � c � � d k � � � � � d � � p � � g � � j i � � �   � f g � � � � � � e � g k f g � f g e � � � � �� � � � � � e � g k f g � � � � e � � � m � � g � � c � � q � � k � j d e d � c � c d � � � d � � � � g f g e � � � � i � � � � � f � � g� � � e � � � � m �

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� | \ [ \ ^ � � } _ � ` b � � � � � � f l � d i � � � o � � i d p k � � � � f e � f � g � o � c � � � k � � � f � g m h � � d p p � c d � q � d � � � f l � g d� � � p � d g o � � i � p d � f g z � � � m ¥ � � � � d � � d k f � � d e c � d � o � p p � q � m � � � � d � c l d � f d � p � � � d e e � d � f g �f g � j q � � � � d � � d � � � � � � � � � � � � j g d i � k � � m ¥ � d p � � � � � d � � d l � � � � � g d i � k � � � � c � � g k f g �l � � � � � � m � � � � d � c � p d � � � � j q � � � � d � � d � � n � � d k � � � q c � � � f g e � � � d g k � � � e � � � d � � p d � � p � k q f � c� c � � c � � � p f � � � d p � � o � m � � c � � � k � � f � � g f i e � � � d g � j d � p � g � d � � d � c f g e � � d g k f � � � � � � � � e � g k f g �� � � e � � d � � p d � � p � k � c � � d i � m �¥ �   � f g � c � � � l � � � f � � � q f � c � c � � d k � � � � d � o � p p � q � m � � � � d � c l d � f d � p � � � j � � g � f k � � d p p � c � � p d � � � �� � � � � � � � � � � f g q c f � c � � d e e � d � � d � d p f � � � d p � � g � � i e p � i � g � � k � m � � f g � � � � d g � k � � � � � c � f g e � �p d � � p � k q f � c � � f g � c � � d k � � � o � � � � j d g k f g � � g � � d p   � f g � c � � c � � � � e � � � o � d k � � � � / p d � � p � k � �q f � c � c � f g e � � � o � d k � � � � / � � q f � c � c f � � d i � p d � � p m � f g d p p � j   � f g � c � � � � e � � � o � c � p d � � � d k � � �� � � � c � g � � � l � � � � � l d � f d � p � � � � � m � � o � c f � f � � c � p d � � l d � f d � p � j � c � g   � f g f � � � � � f g � � � d k m � � c �� � � � p � f g � � c d f g � o � k � � � f � � d p p � k � c � � � � � � � � � o � � l d � f d � p � � � m � � � i d � � � � g k � c d f g f g � � d � � p �� c � � d i � q d � j � � � � c f � � f i �   � f g f g � � c � � d k � � � � o � � � c � � p d � � � � f g q c f � c � � d e e � d � � m � c f � f �� d p p � k � c � � � � � � � � � � o � � � � m � c � � � � � p � f g � k f � � d e c f � � c � � � � e � � � o � c � � � k � � � f � g m � � � � � l �� c d � � c � � g � f � � � � g � � � � � � f � g � d g � � e � � o � � i � k f g e � p � g � i f d p � f i � j � � � f i e p � f g � e � � � f g � � c �o � � i � p d j � � � d � f g � � c � d e e � � e � f d � � l � � � f � � � j d g k d k k f g � � c � d e e � � e � f d � � � k � � � � � � c � k f � � d e c m� c � o � p p � q f g � p � i i d � � � d � p f � c � � � c � � � � � � � � g � � � � o � c f � � � k � � � f � g m� \ � � ] b � c � � � � p � d g o � � i � p d � f � � d � f � � d � p � f o d g k � g p � f o � c � k f � � d e c � e � � k � � � k � � � c �

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� c f � e d � c q f p p � � d � � d � � c � l d � f d � p � l � � � � � � � j � c � g q f p p � � d l � p d p � g � � f � c � � � c � � � � � e d � c� � o d p � � e d � c o � � � � j k � e � g k f g � � g q c � � c � � f � f � � � � � j � � � e � � � f l � p � j f g � c � d � � f � g i � g � jd g k � c � g f � q f p p � � g � f g � � q f � c � � j � c � g � � j d g k � � � g j � g � f p � � d � c f g � � � m � � � c d e d � c q f p pl f � f � � d � c l d � f d � p � l � � � � � � � d � � p � � g � � m� � � d � � � � c f � f � d � d � f � o � f g � d � � f � g i � g � j q � � g � q � c d � o � � � d � c � p d � � � j � f � c � � � j � j � � z� o f � � p f � � � d p � q f p p � � � � � � m � c f � i � d g � � c d � o � � � d � c � p d � � � j � f � c � � � j � j � � z j e d � c � q f p pd � � � i e � � � � � d l � p � c � � � � c � c � � � � � � � e � g k f g � � d k � � � m � � q � l � � j q � c d l � d � � � � k f g � c �d � � l � � p d f i � c d � f g � c f � � d � � f � f � e � � � f � p � � � l f � f � � l � � � l � � � � � f g � c � � d k � � � � � d � � p � � g � � m� c � � � l � � � l � � � � � f g � c � � � d e c f � l f � f � � k � � d � � p � � g � � j f i e p � f g � � c d � � c d � d � d i f p � � g f d ge d � c m

� � � e e � � � � c d � � c d � d � d i f p � � g f d g e d � c m ¥ � d � � � � � � c d � � c � o � � i � o � c � e d � c i � � � � �� � � � g � f d p p � � c � � d i � d � � c � � g � k � � � � f � � k f g � c � e � � l f � � � e d � � � o � c f � e � � � o m � g e d � � f � � p d � j� c � e d � c i � � � l f � f � � c � l d � f d � p � l � � � f � � � f g f g � � � d � f g � � � k � � o � � i � � � g � f p � � j � � � d � � � � o� c � q d � f g q c f � c � c � � � l � � � f � � � d � �   � f g � k � � � � � c � � m� p � � � � � � � l � � c d � o � � � d � c l d � f d � p � l � � � � � j � c � e d � c q f p p e � � � � � k d p � g � � f � c � � � c � � � � �e d � c � � � c � o d p � � e d � c m � o f � e � � � � � k � d p � g � � c � � � � � e d � c j � � � � c � � � � � � � e � g k f g � l d � f d � p �

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; < = > ? @ < r � B s v v @ D � I u w > I D G s � � D @ I > � u E B � C w G H � x �J K L M N O P Q R S � U � W S X Y Y Y Z[ \ ] ^ b c d e � z � � � e � c � � � � c z � m � � f g � h m� � � _ ` a � _ } | � � � � � � � \ } \ ` \ ~ ~ b ¥ f � c � n � � � i e p � � � g � � � q � c d l � � � � g � c d � � c � � � d � � i d g � f i �

e � � � d g � � e � f i f � d � f � g e � � � p � i � � c d � d � � p f � � p � � � � � ¡ � f � � c d � k � � � � p l � � � d � � p � m � f g � � � c � � � d � �f i e � � � d g � e � � � p � i � j q � � d g g � � � f i e p � � f l � � e d � � c f � e � f g � j � f g � � e � � e p � k � g � � k � � p � � f � g � � �� c � � � e � � � p � i � m � � q k � q � � � e � q f � c � n � � � i e p � � � g � � � � ~ \ � � � } \ � � � � � \ ~ \ ] � � | b � l � g � g � c � o d � � � � � e d � d p p � p � � i e � � � � � � c f � d e e � � d � c f � l f d � p � � g p �

o � � � c � � i d p p � � � f g � � d g � � � � o � c � � � e � � � p � i � m� \ � � _ ~ } _ � ~ b � � � � � � � � � � f � d � � � d � � � � o � � e � � k � � f g � d l d p f k � � p � � f � g j � � � � c � � � d � � g � � � d � d g � � � �c � q � p � � � f � f � � � � e � f i d p m � c f � f � q � � � c q c f p � f o d p p � p � � o d f p � j � � f o p d � � � o � e � f i d p f � � f �

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\ � � � � � ] ` � \ � � � ` ^ ~ b � � � � n � � � i e p � � � e � � � p � i � c d l � � � � g � � d � � k d � k � � f � f � g e � � � p � i � o � �� c � � � � � f � d p � � d � � g � m � � q � l � � � g k � � p � f g � i � � � � o � c � � � e � � � p � i � f � d g d � � � d p � e � f i f � d � f � ge � � � p � i m � � � � � d i e p � j � c � � � n � e � f i f � d � f � g e � � � p � i f � � � � g k � c � � f i e p � � � � p � � o i f g f i � i� � � � f g d k f � � d e c j � c � � � e � f i f � d � f � g e � � � p � i f � � � � g k � c � l � � � � � � � l � � � o i f g f i � i � f � � j� c � � p f ¡ � � � e � f i f � d � f � g e � � � p � i f � � � � g k � c � � p f ¡ � � � o i d � f i � i � f � � m � � � � � c d � � � i � � f i � �q � d � � i f g f i f � f g � d g k � � i � � f i � � q � d � � i d � f i f � f g � m � g d e e � � � f i d � f � g d p � � � f � c i f � � g � � c d �� � � � � g � d p � � f � f i d � � d g � q � � j � � � g � � g � � � � � d � f p � � g � � o � c � � i d p p � � � � f � � m� � q k � q � i � d � � � � c � q � � � k d g d e e � � � f i d � f � g d p � � � f � c i f � � ¥ � k � � g � � c � � � � � � � � � � � � od g d e e � � � f i d � f � g d p � � � f � c i d � o � p p � q � m � f l � g d g f g � � d g � � � � o � � � e � � � p � i j p � � � � � � � � � c �� � � � � o � c � � � p � � f � g e � � k � � � k � � � � � d e e � � � f i d � f � g d p � � � f � c i j d g k p � � � � � � � � � � c � � e � f i d p� � p � � f � g m ¥ � q f p p d � � � i � � c d � � � � � � d � � � � � f � � p � e � � f � f l � l d p � � � m � � � d i f g f i f � d � f � g e � � � p � iq � q d g � � � � � � � � � � � � � � � � i d p p j d g k o � � d i d � f i f � d � f � g e � � � p � i q � q d g � � � � � � � � � � � � � � �� i d p p m � � � d g � f g e � � � f � � j q � � d � � c d � � c � d e e � � � f i d � f � g d p � � � f � c i d � c f � l � � � � � � � � � � � �� � � j f o o � � d p p � j � � � � q � c d l �

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� d g � � n � � � i e p � � � � d g � � d e e � � � f i d � � k j � � � � c � � d � f � � � � g k f � d � � p � q � � � q f g � � o � g � � f � g� o m � � � � � d i e p � j � c � � � � � � l � � e � � � p � i � d � � g � � d p f � d � f � g � o � c � l � � � � � � � l � � e � � � p � i � j� d g � � d e e � � � f i d � � k � � q f � c f g d o d � � � � � o p g m ¥ � q f p p g � � k f � � � � � � c f � d p � � � f � c i j � � � f �f � � � l � � � k f g � h m

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Page 95: Design and analysis of computer algorithms (dave mount, 1999)

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� g o d � � j i � � c p f � � � n � � � i e p � � � e � � � p � i � j � c � � � d � � � � p p � � � f � g � � o e � � � p � i � q c f � c d � � � � � p f � l � k �� � � � c d � k � � d e e � � � f i d � � d g k d � � � ¡ � f l d p � g � f g � c � � � g � � � c d � f o d g � � g � � d g � � d e e � � � f i d � � kf g e � p � g � i f d p � f i � � c � g � c � � d p p � d g � � m � c f � � p d � � f � � d p p � k � � � � � � � � � � � � � � m ¥ � q f p p g � �k f � � � � � � c f � o � � � c � � m � � £ � � f � � � � d � � c d � � c � � � e f � � o d e e � � � f i d � f � g d p � � � f � c i � q � � p k � p pd g � � c � � � � � � � � m

� \ � } \ � � � � \ � b ¥ � � � � f g � � � c � q f g � � c d � � c � � � f � d g d e e � � � f i d � f � g d p � � � f � c i o � � l � � � � � � � l � �q f � c d � d � f � � � � g k � o � j � c d � f � j � c f � d p � � � f � c i q f p p � � � � d � d g � � � k � � � g k d l � � � � � � � l � � q c � � �� f � � f � d � i � � � � q f � � � c d � � o � c � � e � f i � i m h � � d p p � c d � � c � l � � � � � � � l � � e � � � p � i � � � � � � � � g kd � � � � o l � � � f � � � � � � c � c d � � l � � � � k � � f g � c � � � d e c � � � � c � � � g � � o � c � � � l � � � f � � � m� � q k � � � � g � � � d � � � � � g k f g � d g d e e � � � f i d � f � g d p � � � f � c i m � c � � � � � d e e � � d � c f � � � � � �� � i � � c f g � � c d � � � � i � p f � � d � � � d � � g d � p � � � � � k � � � d � � � � j d � � � � � � � � � m � � � � � g � � � � � c d � i d g �� f i e p � c � � � f � � f � � j q c � g g � � � e � f i d p j � d g � o � � g � � e � � l � k � � � � � p � � � � � � e � f i d p m� � � � f � d g l � � � � f i e p � d p � � � f � c i j � c d � � � d � d g � � � � d g d e e � � � f i d � f � g q f � c f g d o d � � � � � o � o � �� c � l � � � � � � � l � � e � � � p � i m � � f � � d � � k � g � c � o � p p � q f g � � � � � � l d � f � g m � g � f k � � d g d � � f � � d � � � k � �� � � � � f g � c � � � d e c m � g � � o f � � � q � l � � � f � � � � � � � � � f g � c � � � l � � j � � � q � k � g � � � g � q q c f � c� g � m � c � f k � d � o � c f � c � � � f � � f � f � � � � f i e p � e � � � � � � l � � � f � � � f g � � � c � l � � � � � � � l � � m � � � � � d g g � �� � � i � � c � � � e f k � � � c d g � c f � � � � c � g q � � � i � l � d p p � k � � � � c d � d � � f g � f k � g � � � � d g k � � � f g � �� c � � d � � g � q d p p � � l � � � k � j d g k � � � � � � � � g � c � � � i d f g f g � � k � � � m � � � � l � � � � g � l � � � � � � c d � i � � �� � f g � c � � � l � � j q � e � � � f g � � � � � � � l � � j � � f � f � � d � � � � � � � � c d � � c � � � l � � q � � � g � � d � � f � d �i � � � � q f � � � c � � f � � � o � c � � e � f i � i � � l � � m

G and opt VC The 2-for-1 Heuristic

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Page 96: Design and analysis of computer algorithms (dave mount, 1999)

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Page 99: Design and analysis of computer algorithms (dave mount, 1999)

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Optimum: {S5, S6}

Greedy: {S1, S2, S3, S4, S5, S6}S6

S5

S4 S3 S2 S1

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Page 100: Design and analysis of computer algorithms (dave mount, 1999)

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� � � � � � _ � ] } _ � ` � � a � � _ } | � b � � � d e e � � � f i d � f � g d p � � � f � c i f � � d � � k � g d � f i e p � � � � � k � d p � � � f � c i� c d � e � � k � � � � d � � � � p � g � � � k f � � d g � � � � � � � c d � f � g � � i � � � � c d g � q f � � � c � � e � f i � i � � � � p � g � � �k f � � d g � � m ¥ � � � � f g � � p � � � f g � � c � � c � � � � � � � g � � � � � � � � � � l � � � � � f g � c � � � d e c � � c � p � q � � p � o �l � � � � � j � d � j f g � c � � � � � � � � p � q � m � i e � � � � c � k f � � d g � � � � � � q � � g � c f � l � � � � � d g k d p p � c � � � c � �l � � � f � � � f g � c � � � d e c � � � � � � � � m � g � f k � � � c � l � � � � � � c d � f � o � � � c � � � o � � i � c f � � � g � � � � � c � � e e � �� f � c � l � � � � � d � k f � � d g � � � z f g � c � � � � � � � m � c f � � c � � � � � p � g � � � l � � � � � o � � � � � � m ¥ � q � � p k p f � �� � � � p � � � � c � g � � � � � g � � � � � d � � � � � k � � � � c f � k f � � d g � � m � � p � � � �   � � � i d � � f � � c � g � � � � � g � � � j� d p p � k � � m � c � g q � � � i e � � � � c � k f � � d g � � � o � � i � d � c l � � � � � f g � c � � � d e c � � � c � � � � � � � � o � �d g k � � m � � � � � � � q c � � � k d � c � k p f g � � f g k f � d � � q c f � c l � � � f � � � d � � � p � � � � � � q c f � c � � g � � � � m � � d f g� d � � � c � � � � � p � g � � � l � � � � � � � � � � c � g � � � � � g � � � � � c � � � g � � � l � � � � � f g � k � � j d g k d � d f g � � i e � � �� c � k f � � d g � � � o � � i � d � c l � � � � � � � f � � g � d � � � � � � g � � � m h � e � d � � c f � � g � f p d p p � � � g � � � � c d l � � � � g� � p � � � � k m � g � c � � � d i e p � j � c � z � � g � � � � d � � � c d k � k j d g k � c � � g d p � � � � p � g � � � k f � � d g � � f � � � m

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Page 103: Design and analysis of computer algorithms (dave mount, 1999)

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5

9

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7

6

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5

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8 95

14 19

12

6 14

19

23

5

9

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6

11

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Greedy Cost = 11 Optimumum Cost = 7

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Page 104: Design and analysis of computer algorithms (dave mount, 1999)

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� d g � � � � � � � � e f g � � p l f g � d g � e � � � p � i f � � � e � � k � � � d � f i e p � d g k � p � d g i d � c � i d � f � d po � � i � p d � f � g m � � � � � d i e p � j � c f � i f � c � f g l � p l � k � � � � f � f g � � c � e � � � p � i d � d g � e � f i f � d � f � ge � � � p � i � g � � d e c � j � � � � j � � � � � f g � � m� � \ ] } \ a � � ^ � � � a | ^ \ ~ _ a ` ~ b � � o � � �   � i e f g � f g d g k � � d � � f g � � � k f g � j f � f � f i e � � � d g � � � � � � f gq f � c d � � � k � � � � c k � � f � g m � o � � � � � � � � c k � � f � g f � � d � � k � g d � d k e d � d k f � i � � m � m � � c d � � �� f l � � g � i � � d � f � g j q c � g k � e � c � � � � � � � d � � c � � � p k c d l � � � � g d e e p f � k � � c � g g � d i � � g � � od k k f � f � g d p � � g f g � d g k � � � g f g � q f p p � d l � � c f � � d k k � � f � g m � � � \ � � � � ] � a � � _ } | � � � � � \ � } b d g � � f i � � � � � � � i � � e q f � c d g f k � d � c d � � � � i � e � � i f � �f g � j � g p � � � � g k � � � p d � � � � c d � f � k � � � g � � q � � � m � � � � � e � � l � � c d � � � � � d p � � � f � c i f �� � � � � � � m � o � � � � d g g � � � � � q c � f � f � � � � � � � � j � c d g � � � d � � � c d � f � f � g � � � � � � � � � d � d p p m

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� c � � � f � � � f p p i � � c i � � � � � � � p � d � g � k d � � � � d p � � � f � c i k � � f � g j � � � q � c d l � � � l � � � k d � � � d �k � d p � o � c � � d � f � i d � � � f d p m � g � k f � � � � f � g f � � � � e � � f d p f � � f g � � i � e d � � f � � p d � d � � d j � m � m � � � f g �e d � � � � g i d � � c f g � j � � i e � � d � f � g d p � � � i � � � � j e d � d p p � p d p � � � f � c i � j � d g k � i f � � k d p � � � f � c i � j � �d e e � � � f i d � f � g d p � � � f � c i � m � � q � � p k � � � d � � � � k � l � � � d g � g � f � � � � i � � � � � � � d g � � g � � o � c � � �� � e f � � m � g � � c � � k f � � � � f � g f � � � � d f g d � � � � � � � g k � � � � d g k f g � � o d l � � d � � � � d � � d g d p � � f � j q c f � cq � c d l � p d � � � p � f � g � � � k m � � f p p d g � � c � � k f � � � � f � g i f � c � � � � � � � � k � g � i � � f � d p d p � � � f � c i � � d �

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� z � � � p � � f g � � � � p f ¡ � � � � l � � m� z � � � � � f g k � e � g k � g � � � � m� � g k � e � g k � g � � � � � � l � � � � � � � l � � d g k � p f ¡ � � m� � � � � � � � � l � � � � k � i f g d � f g � � � � m� z � � � � � k f � � � � � k � d i f p � � g f d g e d � c m

� � f � d p � � d � � � k f k � d � � � g k � � � � d g k d p p � c � � � k � � � f � g � � c d � q � � � � � � k f g � c � c � i � q � � �� � p � � f � g � j � f g � � i � k f � � d � f � g � � o � c � � � i d � d e e � d � � g � c � � g d p m� n � � � i e p � � � � � k � � � f � g � � d g � � � c d p p � g � f g � m � o � � � � d g g � � � � � c � q � � � � p l � � c � e � � � p � i jc � � � d � � � � i � � � � � � � � f � g � o � � i d � f i f � f g � e d � � f d p � � � k f � m� p p � n � � � i e p � � � e � � � o � c d l � d l � � � � e � � f � � o � � i m � � e p d f g � c d � � � � � g � q � c � � � i e p d � � jd g k � � � � � � p p f g d � i d g � d � e � � � � d � e � � � f � p � m � � e e � � � � c d � � � � q d g � � � e � � l � � c d � � � i �e � � � p � i � f � � n � � � i e p � � � m

� � � � n m � c f � d p i � � � d p q d � � � d � � j � � k � g � � � p � q f � m h � i � i � � � � c d � � c � � � � � f � � d � � f �d p i � � � d p q d � � � c � � c f g � � c d � � c � e � � � p � i f � d � � f g � � � � � � � g k m

� � � � � � i � � g � q g � n � � � i e p � � � e � � � p � i � j � � � � m � c f � i � d g � � c d � � � � q d g � � �� g k d e � p � g � i f d p � f i � o � g � � f � g � � c d � i d e � d g f g � � d g � � � o � � � d g f g � � d g � � � o � m� d � � � � � � � � � � � � c � k f � � � � f � g � � � � � � � � �

� � c � q � c � � � � � � � � g � � � � o � � � � � � k � � � f � g j � � � c � q f g � � c d � � � � f o d g k � g p � f o � � � � � � m� f � � � � � e e � � � � c d � � � � c d l � d � � p � � f � g � � � d g k � c � q c � q � � i d e � c f � � � d � � p � � f � go � � � � � � m � c � g � � e e � � � � c d � � � � c d l � d � � p � � f � g � � � � � � d g k � c � q c � q � � i d e � c f �� � d � � p � � f � g o � � � m

� o � � � � d g g � � � � � � � � � � q c d � � f � j d � p � d � � � � p p i � q c d � � � � q � � p k p f � � � � � k � m � � e p d f gq c f � c � p � i � g � � � o e � � � p � i � q f p p p f � � p � i d e � � q c f � c � p � i � g � � � o e � � � p � i � m h � i � i � � �� c d � � � � d � � � � � f g � � � � � d g � p d � � � c � � p � i � g � � � o � g � e � � � p � i f g � � � c � � � i i � g � p � i � g � �� o � c � � � c � � e � � � p � i m� � � � � � i d � � d � p � d � � � g � � � k � � � f � g � g � c � � � d i � f i f p d � � � � g � � c d � � � � c d l � � � � g � � o � � � j� � i d � � � � � � � c d � � g k � � � � d g k � c � � g � � � c d � q � c d l � k � g � � f � c � � f g � p d � � � � � g c � i � q � � �e � � � p � i � m

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� � � � � � _ � ] } _ � ` � � a � � _ } | � ~ b � c d e � m z � j � e � c � � � � c z � m � � m� \ � } \ � � � � \ � b h d � f � � � � g k � o � m� � � _ } | } � _ ] ` a � \ _ ` \ � � ] � _ } � b h d � f � � � � g k � o � m� \ } � � � \ � b h d � f � � � � g k � o p g � j q c � � � � � �� � m� _ ` � ] � � _ ` a b h d � f � � � � g k � o � m� � � \ ` } \ � b h d � f � � � � g k � o � m d g � d e e � � � f i d � f � g d p � � � f � c i � d � � � f i e p � m � � � � d � � � d � � k � g � f i e p � � � � � k � c � � � f � � f � � m �� c � � � � � � e � � l f g � i d g � � d � f � � � � g k � f � � � � � � � i f g � � e q f � c d p � q � � � � � g k � g � c �� e � f i d p � � p � � f � g � � m � m j � � n � � �

� � � � m � � � � j e � � l f k � d g � e e � � � � � g k � g � c � � � � � � o� � � � c � � � f � � f � � � p d � f l � � � � c f � � d i � ¡ � d g � f � � � � m � m j � c � � c � � � � � � � q f � � � d � � � g k � � � � o � � � c � � � f � d � i � � � � q f � � � c � � � � � � � � m

; < = > ? @ < � � A B � < = ? @ @ < G = < E w G H � < G < @ w > I G � � ? G = > I D G EJ T M V V � U � U � � Q N R Z[ \ ] ^ b � c f � i d � � � f d p f � g � � � � l � � � k f g � h m � c � � � d � � � k k � � � � f e � f � g � o � � g � � d � f g � o � g � � f � g � f g � m� m � g � � c j � � � � � � � � � � � � � � � � � � � � � � � m ` ^ � � } _ � ` ] ` ^ � � ` ~ } � � � } _ � \ ` ^ � � } _ � ` b � d � � � f i � q � f g � � � k � � � k � � � � � � � g � � � d � d i � � c � k o � �

d g d p � � f g � � � � � � � f l � d p � � � f � c i � j q c � � � � c � � f � � � � o � c � � � � � � � f l � � � � e � � � p � i � f � e � � k f � � d � p � m� g � � c � � � � � c g f ¡ � � o � � � � p l f g � � � � � � � � g � � � � d g k � c f � q � � � � o � � � � i i d � f � g � d � q � p p � f � � � � � � � �� c � � � p � � f � g j � � � c � � � g � � d p o � � i � o � c � � � p � � f � g j d g k � c � g d � � � i e � � � l � � f o � f � � � � � � � � � g � � �� c � � � � c f g k � � � f � g m � � i � � f i � � � c � � � d � � e d � d i � � � � � q c � � � l d p � � � � � � k � g � � � g � q m � c f � f �� g � m � g � c � � � � � � � � o � c � f g k � � � f � g e � � � o j � � � q f p p � � � d p p � � g k � � � q c d � � c � � � l d p � � � i � � � � � m¥ � q f p p � � g � f k � � d o d i � � � � � d i e p � j � c d � � o � c � � � � � � � � � � � � � � � � � m

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� c � � f � � g d � � f g � i � � � � d � f � � f g k d � d � � � � � � � � � k � � f � g m � o � � � � � � k � � � � j � � c � f � c � � d p d g � � k j� � � � � f g � � � � j � � � q f p p p � d � g � c d � � c � i f g f i � i � � f � � k � � � � � � � � d � � e � � k � � � k � � � c �� � � � � � f l � � � g � � � � � � f � g � f l � g � � p � q m � � � � � � � k � g � � � � c � g � i � � � � o p � d l � � f g � c � i f g f i � i �� f � � k � � � � � � � � o c � f � c � � m � � � � g � � � � � � d i f g f i � i � � f � � k � � � � � � � � o c � f � c � � j � � � � � � d � � d� � � � g � k � q c � � � � c f p k � � g � � g � f � � � o d i f g f i � i � � f � � k � � � � � � � � o c � f � c � � � � � d g k � � � m � c � �� c � g � i � � � � o p � d l � � � � � � � � � � � � � � � � � � j � � � � � � � � � � � � � � � � � � m � � f � � d � � � � � � � � c d �� � � � � � � � � m

L(0) = 1 L(1)=1 L(2)=2 L(3)=3 L(4)=5

� f � � � � ¤ ¤ f g f i � i � � f � � k � � � � � � � � m� o � � � � � e d g k � c � � f � � g d � � f � � � f � � o � � d g � i � � � � o � � � i � j � � � q f p p � � � � � l � � c d � � � d e e � d � �� � � � � q � � e � g � g � f d p p � j � � � g � � d � o d � � d � � � m � � f � � � i e � f g � � � � � g   � � � � � � � c d � � � � � � � � j o � �

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� m � � f g � � c f � � � � � � � � � � � � � � � � � � � � q � q f p p � c � q � c d � � c � p � i i d c � p k � o � � f � � � p o j q c � g � l � � � � m� f g � � � � j q � c d l � � � � � � � � � � � � � m � � q j � f g � � � � d g k � � d � � � � � c � � � f � � p �p � � � � c d g j q � � d g d e e p � � c � f g k � � � f � g c � e � � c � � f � j o � � i q c f � c q � c d l �

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� � } _ � � � ] } _ � ` � _ ` } ~ ] ` ^ � � � b � g � � k � � � � k � � � � i f g � � c � d � � f � � p d � f � g e � f g � � � o d g � g k f � � � � � k� � d e c j q � q f p p � d p p k � e � c � � � � � � � d � � c j d g k � � � � c � � � � � � � � � � � � � � e � � l f k � k � � � c � � � d � � c � � d f k� � m � g e d � � f � � p d � j p � � � � d � � � � � � � p l � � f o d l � � � � � � f � d g d � � f � � p d � f � g e � f g � j c � q q � � p k q � � g � qf � � � f � � � � � � � � � � � f g � c � � � � � � � � �¥ � d � � � i � � c d � � f � � � g g � � � � k � f o g � � j q � � d g d e e p � � c f � d p � � � f � c i � � � d � c f g k f l f k � d p � � g g � � � � k� � i e � g � g � � m � � q � d � � � i � f � � g p � � g � � � � � f g � c � � � � o � � � � � m � � � d � � � � f � � g k f � � � � � k j � c �� � � � � � � c d � d � f i e p � � � � � � � � � � � m � f � � � � ¢ j q � � d g g � � k f � � f g � � f � c � � � q � � g o � � q d � k � k � � � d g k� d � � � k � � � j d g k q �   � � � � d p p � c � i � d � � � k � � � m � p � � j � c � � � d � � g � � � � � � � k � � � m � � � � � c � � p k � d � �d i � i � g � � � � � g l f g � � � � � � � � p o q c � � c f � f � � � � � m �� � � g � q j p � � � � � � g � f k � � � c � � � e f � d p � d � � � o d l � � � � � � j q c � � � � f � g � � d p � d o d g k � f � g � � � c �� � � � m � � � � � p � � � � � � � � � � � � � � � � c � � c f p k � � g � o � m � � � � d � c � c f p k � c � � � f � d � � � � � � � � o � c � � � �� � � � � � � � � k d � � c f � � c f p k m � o o � � � � i � � c f p k j � c � � � f � g � � d � � � k � � � � f g � � � d e � � e � � d g � � � � � � � o

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e � f g � f o d g k � g p � � c � � � � � f � � � d � � � � � � � � � � � � k d � d � c f p k � o � � � � c � c d � � c � � � f � g � � d � �� k � � o � � i d g � l � � � � � f g � c f � � � � � � � � � � d e � � e � � d g � � � � � � � o � m

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� ` ] � � ~ _ ~ � � } | \ � � � ^ � � � � � \ � ~ � ` � \ } | � ^ b � c � e � � � p � i q f � c � c � � � � k � � � p � � � � � g d p � � � f � c i f �� c d � k � e � g k f g � � g c � q f � e f � � � d � � i � g � f g � e d � c � j f � i d � � e � g k d g f g � � k f g d � � d i � � g � � o � f i �d � � f l f g � d � c � � g d p i d � f i � i � � q m � g � f k � � � c � o � p p � q f g � � � d i e p � � o � � i e d � � � ¤ f g � h � m� o � c � d p � � � f � c i q � � � � i d � � � g � � � c � � � � g k � � q d p � g � � c � � k � � � � o q � f � c � � j � � � j � � � j � c �d p � � � f � c i q � � p k � � � i f g d � � f g � q � d � � i � g � f g � � � � e � m � � q � l � � j f o � c � d p � � � f � c i q � � � � � � � � � �d � � i � g � � � f g � � c � i f k k p � � k � � j f � q f p p � � g � f g � � � � p � f i e � � l � � c � � � q � � � g p � d � f g � p � � g f � m� j � � � j � � � d � � i � g � f g � q f p p � � g � � k � k � � o � � � q � � � � � c � � g d p � � q m � g � � g � � d p j � � � k � � � p � � � � � g� d g � d � � � f i � � � � � � � �� � � � q c � � � � � f � � c � i d � f i � i � � q m

� ` � � � � � \ � \ ` } b ¥ � c d l � � c � q g � c d � f o � c � d � � i � g � f g � e d � c q d � � c � � � g f g d � d k q d � � c �d p � � � f � c i � � � p k � � g o � � d l � � � p � g � � f i � � � o � � � � � g l � � � f g � � g � c � � g d p � � q m � � � � � i � � o � � i� c � � � d i e p � q � � c � q � k � � c d � d i � � � p � � f � d p q d � � � e � � c � � q f � � � � � p � � � � c � d � � i � g � f g � e d � cq c f � c c � p k � � c � i d � f i � i d i � � g � � o � � q m � i e � � f g � � c f � e d � c f � � ¡ � f l d p � g � � � k � � � � i f g f g �� c � e d � c � o i d � f i � i � d e d � f � � o � � i � � � � f g � c � � � � f k � d p g � � q � � � m � � c f � f � � � d � � p � � c � � d i �d � � c � � � � � � � d g � e � � � e � � � p � i � f l � g � g � c � p d � � � � d i m � � � f � g � � � g � q g c � q o d � � � c f � i � � c � kq � � � � f g � c � q � � � � � d � � j � � � � c � � � f � d g � � c � � � f i e p � � � � d � � � � � c d � f � � � d � d g � � � k � � � f l � � � � k� � � g k � � f g � � � i � � o d g k � � m� ^ � � ` ^ ~ � � ] � � � � a � � _ } | � b � c � � k i � g k � � � d � e d p � � � f � c i f � � � � k � � � p � � � � � g j q f � c � g � p f � � p �� c d g � � m ¥ c � g � g k f g � � c � d � � i � g � f g � e d � c j q � � � � � � � d k � c � � f � � � � � d � � c f g � c � � � � f k � d pg � � q � � � j � � d � � f g � d � � c � � � � � � � � j d g k � c � � q � � g k � c � � c � � � � � � d � � i � g � f g � e d � c � q c � � � � c �p � g � � c � o � c � e d � c f � � c � g � i � � � � o � k � � � � g � c � e d � c � m ¥ � � p d f i � c d � � c f � � c � f � � f � e d � � f � � p d � p �g f � � f g � c d � j f o q � k � � � j � c � g � i � � � � o � � q d � � i � g � d � f � g � g � � k � k q f p p � � d � i � � � � � � � m� f g � � � d � c d � � i � g � d � f � g � d � � � � � � � � f i � � � � � i e � � � � � f g � � � � j � c � � l � � d p p � � g g f g � � f i �q f p p � � � � � � � � � � � � �

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Page 122: Design and analysis of computer algorithms (dave mount, 1999)

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Page 123: Design and analysis of computer algorithms (dave mount, 1999)

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Page 125: Design and analysis of computer algorithms (dave mount, 1999)

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Page 126: Design and analysis of computer algorithms (dave mount, 1999)

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Page 131: Design and analysis of computer algorithms (dave mount, 1999)

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