Design Analysis and Algorithm

7/25/2019 Design Analysis and Algorithm

1/83

UNIT 2Algorithms Analysis

Techniques & DesignTechniqueEfciency o Algorithms,

Analysis o Recursive Programs,

Solving Recurrence Equation

ivi!e " #onquer Algorithms

ynamic $rogramming

%ree!y Algorithm

&ac'trac'ing(o$e #haitali

)rushnaRoll no* +


2/83

Efciency o Algorithms,

e-nition

A!vance! .IS

/un!amental techniques 0hich are use! to !esign analgorithm efciently*

+1 ivi!ean!#onquer

21 %ree!y metho!

31 ynamic Programming

41 &ac'trac'ing

1 &ranchan!&oun!


3/83

Analysis o Recursive Programs

e-nition

A!vance! .IS

Recurrence relations oten arise in calculating thetime an! s$ace com$le5ity o algorithms1

Any $ro6lem can 6e solve! either 6y 0riting

recursive algorithm or 6y 0riting nonrecursive

algorithm1

A recursive algorithm is one 0hich ma'es a

recursive call to itsel 0ith smaller in$uts1


4/83

7e oten use a recurrence relation to !escri6e the

running time o a recursive algorithm1

A recurrence relation is an equation or

inequality that !escri6es a unction in terms o its

value on smaller in$uts or as a unction o

$rece!ing 8or lo0er9 terms1

A!vance! .IS

O O O ' '


5/83

METHOD !OR O"#$% RE'(RRE%'E

RE"AT$O%

e-nition

A!vance! .IS

7e 0ill intro!uce three metho!s o solving therecurrence equation*

+1 The Su6stitution .etho!

21 Iteration .etho!

31 The Recursiontree metho!

41 .aster metho!


6/83

In su6stitution metho!, 0e guess a 6oun! an!then use mathematical in!uction to $rove ourguess correct1

The iteration metho! converts the recurrence intoa summation an! then relies on techniques or6oun!ing summations to solve the recurrence1

The .aster metho! $rovi!es 6oun!s or therecurrence o the orm

A!vance! .IS


7/83

Divi)e & conquer technique

e-nition

A!vance! .IS

ivi!e " conquer technique is a to*+)ona**roach to solve a $ro6lem1

The algorithm 0hich ollo0s !ivi!e an! conquer

technique involves 3 ste$s*

-. Divi)e the original $ro6lem into a set o su6

$ro6lems1

/. 'onquer 8or Solve9 every su6$ro6lem

in!ivi!ually, recursive1

0. 'om1ine the solutions o these su6 $ro6lems to

et the solution o ori inal ro6lem1


8/83

&inary search

e.g. 2 4 5 6 7 8 9

search :* nee!s 3 com$arisons

time* ;8log n9 The 6inary search can 6e use! only i the

elements are sorte! an! store! in an array1


9/83

Algorithm 6inarysearch

$n*ut* A sorte! sequence o n elements store! in an array

Out*ut2The $osition o 5 8to 6e searche!91

te* -2 I only one element remains in the array, solve it!irectly1

te* /2 #om$are 5 0ith the mi!!le element o the array1

Ste$ 21+* I 5 < mi!!le element, then out$ut it an! sto$1

Ste$ 212* I 5 = mi!!le element, then recursively solve the$ro6lem 0ith 5 an! the let hal array1

Ste$ 213* I 5 > mi!!le element, then recursively solve the

$ro6lem 0ith 5 an! the right hal array1


10/83

Algorithm &inSearch8a, lo0, high, 59

?? a@* sorte! sequence in non!ecreasing or!er

?? lo0, high* the 6oun!s or searching in a @

?? 5* the element to 6e searche!

?? I 5 < a@B, or some B, then return B else return C+

i 8lo0 > high9 then return C+ ?? invali! range i 8lo0 < high9 then ?? i small P

i 85


11/83

4 ++

.erge t0o sorte! sequences into a singleone1

time com$le5ity* ;8mDn9,

m an! n* lengths o the t0o sorte! lists

T0o0ay merge

[25 37 48 57][12 33 86 92]

[12 25 33 37 48 57 86 92]

merge


12/83

4 +2

Sort into non!ecreasing or!er

log2n $asses are require!1

time com$le5ity* ;8nlogn9

.erge sort

[25][57][48][37][12][92][86][33]

[25 57][37 48][12 92][33 86]

[25 37 48 57][12 33 86 92]

[12 25 33 37 48 57 86 92]

pass 1

pass 2

pass 3


13/83

Algorithm .ergeSort

$n*ut2 A set S o n elements1

Out*ut2The sorte! sequence o the in$uts in

non!ecreasing or!er1

te* -2 I S2, solve it !irectly1

te* /2 Recursively a$$ly this algorithm to solve the let

hal $art an! right hal $art o S, an! the results are

store! in S+ an! S2, res$ectively1te* 02 Perorm the t0o0ay merge scheme on S+ an! S21


14/83

P < 8A++D A2298&++D &229F < 8A2+D A229&++R < A++8&+2 &229

S < A228&2+ &++9

T < 8A++D A+29&22U < 8A2+ A++98&++D &+29

G < 8A+2 A2298&2+D &2291

#++< P D S T D G

#+2< R D T

#2+< F D S

#22< P D R F D U

Strassens matri5 multi$licaiton

#++

< A++

&++

D A+2

&2+

#+2

< A++

&+2

D A+2

&22#

2+ < A

2+&

++D A

22

&2+

#22

< A2+

&+2

D A22

&22


15/83

Time com$le5ity

: multi$lications an! +H a!!itions or

su6tractions Time com$le5ity*

T(n) =

b

7T(n/2)+an2, n 2

, n > 2

)(O)(O

)(

constantais,7)(

)1(7))()(1(

)4/(7

)4/(7)((7

)2/(7)(

81.27log

7log4log7log4log7loglog

4

72

loglog

4

72

1

4

72

4

7

4

72

22

4

72

2

2

2

2

2

222222

22

nn

ncnncn

ccn

Tan

nTanan

nTaan

nTannT

n

nn

kk

n

=

+=+=

+

+++++=

=

++=

++=

+=

+


16/83

4 +

.a5imum num6ers

-n!ing the ma5imum o a set S o nnum6ers


17/83

ree)y Algorithm

A greedy algorithmal0ays ma'es the choice that

loo's 6est at the moment

.y every!ay e5am$les*

7al'ing to the #orner

Playing a 6ri!ge han!

The ho$e* a locally o$timal choice 0ill lea! to aglo6ally o$timal solution

/or some $ro6lems, it 0or's

Inclu!es

som6o!yJgmail1com


18/83

A Generic Greedy Algorithm:

(1) Initialize Cto be the set o can!i!ate sol"tions (2)Initialize a set S= the e#$t% set (the set is to be theo$ti#al sol"tion &e a'e const'"cting). () hile Can! Sis (still) not a sol"tion !o (.1) selectx'o#set C"sing a g'ee!% st'ateg% (.2) !eletex'o# C

(.) i *x Sis afeasiblesol"tion, thenS= S*x (i.e., a!!xto set S) (4)i Sis a sol"tion then 'et"'n S

()else 'et"'nfailure

In gene'al, a g'ee!% algo'ith# is eicient beca"se it #a-es ase"ence o (local) !ecisions an! ne/e' bac-t'ac-s. The

sol"tion is not al&a%s o$ti#al, ho&e/e'.


19/83

e-nition

A .inimum S$anning Tree 8.ST9 is asu6gra$h o an un!irecte! gra$hsuch that the su6gra$h s$ans

8inclu!es9 all no!es, is connecte!, isacyclic, an! has minimum total e!ge0eight


20/83

Algorithm #haracteristics

&oth PrimKs an! )rus'alKs Algorithms0or' 0ith un!irecte! gra$hs

&oth 0or' 0ith 0eighte! an!un0eighte! gra$hs 6ut are moreinteresting 0hen e!ges are 0eighte!

&oth are gree!y algorithms that$ro!uce o$timal solutions


21/83

PrimKs Algorithm

Similar to iB'straKs Algorithm e5ce$tthat d

v

recor!s e!ge 0eights, not

$ath lengths


22/83

3al4+Through

InitialiLearray

K dv pv

A 0

B 0

C 0 D 0

E 0

F 0

G 0

H 0

5

/6

A

H

7

!

E

D

'

8

/

-9

-:

05

0

8

:

;

0

-9

/


23/83

5

/6

A

H

7

!

E

D

'

8

/

-9

-:

05

0

8

:

;

0

-9

Start 0ith any no!e,say

K dv pv

A

B

C

D T

E

F

G

H

/


24/83

5

/6

A

H

7

!

E

D

'

8

/

-9

-:

05

0

8

:

;

0

-9

U$!ate !istances o

a!Bacent, unselecte!no!es

K dv pv

A

B

C

D T

E 2

F 18

G 2

H

/


25/83

5

/6

A

H

7

!

E

D

'

8

/

-9

-:

05

0

8

:

;

0

-9

Select no!e 0ith

minimum !istance

K dv pv

A

B

C

D T

E 2

F 18

G T 2

H

/


26/83

5

/6

A

H

7

!

E

D

'

8

/

-9

-:

05

0

8

:

;

0

-9

U$!ate !istances o


K dv pv

A

B

C

D T

E 7 3

F 18

G T 2

H 3

/


27/83

5

/6

A

H

7

!

E

D

'

8

/

-9

-:

05

0

8

:

;

0

-9

Select no!e 0ith

minimum !istance

K dv pv

A

B

C T

D T

E 7 3

F 18

G T 2

H 3

/


28/83

5

/6

A

H

7

!

E

D

'

8

/

-9

-:

05

0

8

:

;

0

-9

U$!ate !istances o


K dv pv

A

B 4

C T

D T

E 7 3

F

G T 2

H 3

/


29/83

5

/6

A

H

7

!

E

D

'

8

/

-9

-:

05

0

8

:

;

0

-9

Select no!e 0ith

minimum !istance

K dv pv

A

B 4

C T

D T

E 7 3

F T

G T 2

H 3

/


30/83

5

/6

A

H

7

!

E

D

'

8

/

-9

-:

05

0

8

:

;

0

-9

U$!ate !istances o


K dv pv

A 1 0

B 4

C T

D T

E 2 0

F T

G T 2

H 3

/


31/83

5

/6

A

H

7

!

E

D

'

8

/

-9

-:

05

0

8

:

;

0

-9

Select no!e 0ith

minimum !istance

K dv pv

A 1 0

B 4

C T

D T

E T 2 0

F T

G T 2

H 3

/


32/83

5

/6

A

H

7

!

E

D

'

8

/

-9

-:

05

0

8

:

;

0

-9

U$!ate !istances o


K dv pv

A 1 0

B 4

C T

D T

E T 2 0

F T

G T 2

H 3

/

Ta6le entriesunchange!


33/83

5

/6

A

H

7

!

E

D

'

8

/

-9

-:

05

0

8

:

;

0

-9

Select no!e 0ith

minimum !istance

K dv pv

A 1 0

B 4

C T

D T

E T 2 0

F T

G T 2

H T 3

/


34/83

5

/6

A

H

7

!

E

D

'

8

/

-9

-:

05

0

8

:

;

0

-9

U$!ate !istances o


K dv pv

A 4 5

B 4

C T

D T

E T 2 0

F T

G T 2

H T 3

/


35/83

5

/6

A

H

7

!

E

D

'

8

/

-9

-:

05

0

8

:

;

0

-9

Select no!e 0ith

minimum !istance

K dv pv

A T 4 5

B 4

C T

D T

E T 2 0

F T

G T 2

H T 3

/


36/83

5

/6

A

H

7

!

E

D

'

8

/

-9

-:

05

0

8

:

;

0

-9

U$!ate !istances o


K dv pv

A T 4 5

B 4

C T

D T

E T 2 0

F T

G T 2

H T 3

/

Ta6le entriesunchange!


37/83

5

/6

A

H

7

!

E

D

'

8

/

-9

-:

05

0

8

:

;

0

-9

Select no!e 0ith

minimum !istance

K dv pv

A T 4 5

B T 4

C T

D T

E T 2 0

F T

G T 2

H T 3

/


38/83

5

A

H

7

!

E

D

'

/

05

0

0

#ost o .inimum

S$anning Tree < dv=/-

K dv pv

A T 4 5

B T 4

C T

D T

E T 2 0

F T

G T 2

H T 3

/

Done


39/83

6'"s-als lgo'ith#

o'- &ith e!ges, 'athe' than no!es

T&o ste$s 9o't e!ges b% inc'easing e!ge &eight 9elect the i'st :;: < 1 e!ges that !o not

gene'ate a c%cle


40/83

al-Th'o"gh#onsi!er an un!irecte!, 0eightgra$h

6

-

A

H

7

!

E

D

'

0

/

5

) 1

(,3) 2

(>,3)

(,)

(3,5)

(,0)

(?,) 4

6

-

A

H

7

!

E

D

'

0

/

5

) 4

(?,0) 4

(?,5) 4(,5)

(,0) @

(,?) 8

(,0) 1


42/83

Select -rst GC+ e!ges 0hich !o

not generate a cycle

edge d v

(,>) 1

(,3) 2

(>,3)

(,)

(3,5)

(,0)

(?,) 4

6

-

A

H

7

!

E

D

'

0

/

5

) 4

(?,0) 4

(?,5) 4

(,5)

(,0) @

(,?) 8

(,0) 1


43/83



edge d v

(,>) 1

(,3) 2

(>,3)

(,)

(3,5)

(,0)

(?,) 4

6

-

A

H

7

!

E

D

'

0

/

5

) 4

(?,0) 4

(?,5) 4

(,5)

(,0) @

(,?) 8

(,0) 1


44/83



edge d v

(,>) 1

(,3) 2

(>,3)

(,)

(3,5)

(,0)

(?,) 4

6

-

A

H

7

!

E

D

'

0

/

5

) 4

(?,0) 4

(?,5) 4

(,5)

(,0) @

(,?) 8

(,0) 1

Acce$ting e!ge 8E,%9 0oul! create acycle


45/83



edge d v

(,>) 1

(,3) 2

(>,3)

(,)

(3,5)

(,0)

(?,) 4

6

-

A

H

7

!

E

D

'

0

/

5

) 4

(?,0) 4

(?,5) 4

(,5)

(,0) @

(,?) 8

(,0) 1


46/83



edge d v

(,>) 1

(,3) 2

(>,3)

(,)

(3,5)

(,0)

(?,) 4

6

-

A

H

7

!

E

D

'

0

/

5

) 4

(?,0) 4

(?,5) 4

(,5)

(,0) @

(,?) 8

(,0) 1


47/83



edge d v

(,>) 1

(,3) 2

(>,3)

(,)

(3,5)

(,0)

(?,) 4

6

-

A

H

7

!

E

D

'

0

/

5

) 4

(?,0) 4

(?,5) 4

(,5)

(,0) @

(,?) 8

(,0) 1


48/83



edge d v

(,>) 1

(,3) 2

(>,3)

(,)

(3,5)

(,0)

(?,) 4

6

-

A

H

7

!

E

D

'

0

/

5

) 4

(?,0) 4

(?,5) 4

(,5)

(,0) @

(,?) 8

(,0) 1


49/83



edge d v

(,>) 1

(,3) 2

(>,3)

(,)

(3,5)

(,0)

(?,) 4

6

-

A

H

7

!

E

D

'

0

/

5

) 4

(?,0) 4

(?,5) 4

(,5)

(,0) @

(,?) 8

(,0) 1


50/83



edge d v

(,>) 1

(,3) 2

(>,3)

(,)

(3,5)

(,0)

(?,) 4

6

-

A

H

7

!

E

D

'

0

/

5

) 4

(?,0) 4

(?,5) 4

(,5)

(,0) @

(,?) 8

(,0) 1


51/83



edge d v

(,>) 1

(,3) 2

(>,3)

(,)

(3,5)

(,0)

(?,) 4

6

-

A

H

7

!

E

D

'

0

/

5

) 4

(?,0) 4

(?,5) 4

(,5)

(,0) @

(,?) 8

(,0) 1


52/83



edge d v

(,>) 1

(,3) 2

(>,3)

(,)

(3,5)

(,0)

(?,) 4

6

-

A

H

7

!

E

D

'

0

/

5

) 4

(?,0) 4

(?,5) 4

(,5)

(,0) @

(,?) 8

(,0) 1


53/83



edge d v

(,>) 1

(,3) 2

(>,3)

(,)

(3,5)

(,0)

(?,) 4

6

-

A

H

7

!

E

D

'

/

0

0

0

edge d v

(?,>) 4

(?,0) 4

(?,5) 4

(,5)

(,0) @

(,?) 8

(,0) 1DoneTotal 'ost = dv=21

5

Mnotconsi!ere!


54/83


55/83

>na*sac4 *ro1lem


56/83

>na*sac4 *ro1lem

%iven a 'na$sac' 0ith ma5imum ca$acity W, an!

a set Sconsisting o nitems

Each item ihas some 0eight wian! 6ene-t value

bi 8all wi an! Ware integer values9

Pro6lem* o0 to $ac' the 'na$sac' to achieve

ma5imum total value o $ac'e! itemsO

Inclu!es

A!vance! .IS


57/83

Pro6lem, in other 0or!s, is to -n!

9+- >na*sac4 *ro1lem

The $'oble# is calle! a A0-1B $'oble#,

beca"se each ite# #"st be enti'el% acce$te!

o' 'eCecte!.

Ti

i

Ti

i Wwb s"bCect to#aD

ynamic Programming


58/83

ynamic Programming

ynamic Programming is an algorithm !esign technique oro$timiLation $ro6lems* oten minimiLing or ma5imiLing1

i'e !ivi!e an! conquer, P solves $ro6lems 6y com6iningsolutions to su6$ro6lems1

Unli'e !ivi!e an! conquer, su6$ro6lems are notin!e$en!ent1 Su6$ro6lems may share su6su6$ro6lems,

o0ever, solution to one su6$ro6lem may not aQect thesolutions to other su6$ro6lems o the same $ro6lem1 8.ore onthis later19

P re!uces com$utation 6y Solving su6$ro6lems in a 6ottomu$ ashion1

Storing solution to a su6$ro6lem the -rst time it is solve!1

oo'ing u$ the solution 0hen su6$ro6lem is encountere! again1

)ey* !etermine structure o o$timal solutions


59/83

#om$ +22, S$ring 24

Ste$s in ynamic Programming

1. Characterize structure of an optimal solution.

2. Dene value of optimal solution recursively.

. Compute optimal solution values either top!

down with caching or bottom!up in a table.

". Construct an optimal solution from computed

values.

!loy)?s Algorithm2 All *airs shortest *aths


60/83

!loy) s Algorithm2 All *airs shortest *aths

Pro6lem* In a 0eighte! 8!i9gra$h, -n! shortest $aths

6et0een every $air o vertices

Same i!ea* construct solution through series o

matrices 89, , 8n9 using increasing su6sets o thevertices allo0e! as interme!iate

E5am$le*

3

42

+

4

+

+

3

4 + 4 3

+

!loy)?s Algorithm @matri generationB


61/83

!loy) s Algorithm @matri generationB

On the -th ite'ation, the algo'ith# !ete'#ines sho'test $aths

bet&een e/e'% $ai' o /e'tices i, C that "se onl% /e'tices a#ong 1,E,- as inte'#e!iate

(-)Fi,CG = #in *(-1)Fi,CG, (-1)Fi,-G + (-1)F-,CG

i

#

$

D@k+-BCi,j

D@k+-BCi,k

D@k+-BCk,j

Initialcon!itionO

!loy)?s Algorithm @eam*leB


62/83

!loy) s Algorithm @eam*leB

3 2 : +

D89 FO(

Documents

Design Analysis and Algorithm