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8/14/2019 Algoritmo Floyd Warshall
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t h a t t h e r e i s n o p a t h . ) S o t h e l e a s t c o s t p a t h f r o m v t o w i s t h e l e a s t c o s t p a t h f r o m v t o k ( w h e r e
k = P v , w ] ) , f o l l o w e d b y w :
p a t h ( v ; w ) = p a t h ( v ; P v ; w ] ) ! w
w h e r e p a t h ( i ; j ) = i ! : : : ! j
a n d p a t h ( i ; i ) = i
0 . 3 A n E x a m p l e ( m o d i e d f r o m C L R )
4
5
6
3 4
1
5
2
3
4
7
2
8
1
F i g u r e 1 : E x a m p l e G r a p h
C o n s i d e r t h e g r a p h i n F i g u r e 1 . F o r t h i s g r a p h , w e w o u l d i n i t i a l i z e D a n d P t o b e :
D =
0
B
B
B
B
B
@
0 3 8 1 4
1 0 1 1 7
1 4 0 1 1
2 1 5 0 1
1 1 1 6 0
1
C
C
C
C
C
A
P =
0
B
B
B
B
B
@
N I L 1 1 N I L 1
N I L N I L N I L 2 2
N I L 3 N I L N I L N I L
4 N I L 4 N I L N I L
N I L N I L N I L 5 N I L
1
C
C
C
C
C
A
a n d o u r n a l v a l u e s f o r D a n d P a r e :
D =
0
B
B
B
B
B
@
0 3 8 4 4
3 0 6 1 7
7 4 0 5 1 1
2 5 5 0 6
8 1 1 1 1 6 0
1
C
C
C
C
C
A
P =
0
B
B
B
B
B
@
N I L 1 1 2 1
4 N I L 4 2 2
4 3 N I L 2 2
4 1 4 N I L 1
4 1 4 5 N I L
1
C
C
C
C
C
A
2
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F r o m t h e n a l m a t r i c e s , w e c a n s e e t h a t t h e s h o r t e s t p a t h f r o m v e r t e x 5 t o v e r t e x 2 h a s a c o s t
o f 1 1 ( D 5 , 2 ] ) a n d m o v e s f r o m 5 t o 2 v i a 1 ( P 5 , 2 ] ) , a n d f r o m 5 t o 1 v i a 4 ( P 5 , 1 ] ) , f o r a p a t h o f
5 ! 4 ! 1 ! 2 .
3
