25.All-Pairs Shortest Paths Hsu, Lih-Hsing. Computer Theory Lab. Chapter 25P.2.

25.All-Pairs Shortest Paths

Hsu, Lih-Hsing

Chapter 25 P.2

Computer Theory Lab.

G V E( , ) with w E R:

Suppose that w e( )0 for all e E .

Using Dijkstra’s algorithm

O v O v

O v v eO v v Ev

O v

( ) ( )

( log )( log )

( )

2 3

2

3

Suppose negative weight are allowed

Using Bellman-Ford algorithm

O VE O v E O v( ) ( ) ( ) 2 4

Can be improved.

Chapter 25 P.3


Input form: matrix W

w

i=j

(i,j)i j (i,j) Eij

0 if the weight of the

directed edge if and

otherwise

Output:

D d d i jij ij ( ), ( , ) .

Predecessor matrix ( ) ij

ij

ij

NILi=j

i jj

i

if either or there is

no path from to is some predecessor of

on the shortest path from

.

.

Chapter 25 P.4


F o r e a c h v e r t e x i V , w e d e f i n e d t h e p r e d e c e s s o r s u b g r a p h o f G f o r i

a s ),(,,, iii

EVG w h e r e }}{|),{(

}{}|{

,,

,

iVjjE

iNILjV

iiji

iji

P R I N T - A L L - S H O R T E S T - P A T H ( , ,i j )

1 i f i j

2 t h e n p r i n t i

3 e l s e i f i j N I L

4 t h e n p r i n t “ n o p a t h f r o m ” i t o j “ e x i s t s ”

5 e l s e P R I N T - A L L - S H O R T E S T - P A T H ( , , )i i j

6 P r i n t j

Chapter 25 P.5

Computer Theory Lab.25.1 Shortest paths and matrix

multiplication

A dynamic programming approach:

1. characterize the structure of the optimal solution,

2. recursively define the value of optimal solution,

3. compute the value of an optimal solution in a bottom-up fashion.

The structure of a optimal solution:

Consider a shortest path p from vertex i to vertex j, and suppose that p

contains at most m edges. Assume that there are no negative-weight

cycles. Hence m is finite.

Chapter 25 P.6


If i = j, then p has weight 0 and no edge.

If i j , we decompose p into ip

k j'

~ where p' contains at most

m-1 edges.

Moreover, p' is a shortest path from i to k and ( , ) ( , )i j i k wkj .

Chapter 25 P.7


A recursive solution to the all-pairs shortest-path problem.

Define: )(m

ijl = minimum weight of any path from i to j that contains at

most m edges.

jiif

jiiflij

0)0(

Then

}{min

}}{min,min{

)1(1

)1(1

)1()(

kjm

iknk

kjm

iknkm

ijm

ij

wl

wlll

Chapter 25 P.8


Since shortest path from i to j contains at most m-1 edges,

)1()()1(),( nij

nij

nij lllji

Computing the shortest-path weight bottom up:

Compute )1()2()1( ,,, nLLL where )( )()( m

ijm lL .

Note that WL )1(.

Chapter 25 P.9


EXTENDED-SHORTEST-PATHS(L, w)

1 ][Lrown

2 Let )( ijlL be an n n matrix

3 for i1 to n

4 do for j1 to n

5 do ijl

6 for k1 to n

7 do ),min(

kjikijijwlll

8 return L

Chapter 25 P.10


MATRIX-MULTIPLY(A,B)

1 n row A [ ]

2 Let C be an n n matrix

3 for i 1 to n

4 do for j 1 to n

5 do cij 0

6 for k 1 to n

7 do c c a bij ij ik kj

8 return C

Chapter 25 P.11


min

)(

)1(

cl

bw

al

m

m

(Time complexity: O n( )3 .)

Chapter 25 P.12


1)2()1(

3)2()3(

2)1()2(

)0()1(

nnn WWLL

WWLL

WWLL

WWLL

Chapter 25 P.13


SLOW-ALL-PAIRS-SHORTEST-PATHS(W)

1 n row W [ ]

2 WL )1(

3 for m 2 to n-1

4 do )(mL EXTENDED-SHORTEST-PATHS( ),)1( WL m

5 return )1( nL

Time complexity: O n( )4 .

Chapter 25 P.14


Improving the running time:

)1log()1log( 2)2(

224)4(

2)2(

)1(

nn

WL

WWWL

WWWL

WL

i.e., using repeating squaring!

Time complexity: O n n( log )3 .

Chapter 25 P.15


2

1

5 4

3

3

6

-4

7

2

8

1

4

-5

06

052

04

710

4830

)1(L

0618

20512

11504

71403

42830

)2(L

Chapter 25 P.16


06158

20512

115047

11403

42330

)3(L

06158

20512

35047

11403

42310

)4(L

Chapter 25 P.17


FASTER-ALL-PAIRS-SHORTEST-PATHS(W)

1 n row W [ ]

2 WL )1(

3 m 1

4 while n m 1

5 do )2( mL EXTENDED-SHORTEST-PATHS( ), )()( mm LL

6 m m 2

7 return )(mL

Chapter 25 P.18


25.2 The Floyd-Warshall algorithmA different dynamic programming formulation

The structure of a shortest path:

Let V(G)={1,2,…,n} and consider a subset {1,2,…,k} of vertices for some

k. For any pair of vertices i j V, , consider all paths from i to j whose

intermediate vertices are drawn from {1,2,…,k} and p be a minimum

weight path among them.

Chapter 25 P.19


If k is not an intermediate vertex of p, then all intermediate vertices are in

{1,2,…,k-1}.

If k is an intermediate vertices of p, then p can be decomposed into

ip

kp

j1 2~ ~

where p1 is a shortest path from i to k with all the

intermediate vertex in {1,2,…,k-1} and p2 is a shortest path from k to j

with all the intermediate vertex in {1,2,…,k-1}.

Chapter 25 P.20


A recursive solution to the all-pairs shortest path problem

Definition: dijk the weight of a shortest path from vertex i to vertex

j with all intermediate vertices in the set {1,2,…,k}.

1),min(

0)1()1()1(

)(

kifddd

kifwd k

kjk

ikk

ij

ijkij

D dnij

n( ) ( )( ) is the final solution!

Chapter 25 P.21


Computing the Shortest-path weight bottom-up.

FLOYD-WARSHALL(W)

1 n row W [ ]

2 D W( )0

3 for k 1 to n

4 do for i 1 to n

5 do for j 1 to n

6 d d d dijk

ijk

ikk

kjk( ) ( ) ( ) ( )min( , ) 1 1 1

7 return D n( )

Complexity: O n( )3 .

Chapter 25 P.22


C o n s t r u c t i n g a s h o r t e s t p a t h

( ) ( ) ( ), , . . . ,0 1 n

i jk( ) : i s t h e p r e d e c e s s o r o f t h e v e r t e x j o n a s h o r t e s t p a t h f r o m v e r t e x i

w i t h a l l i n t e r m e d i a t e v e r t i c e s i n t h e s e t { 1 , 2 , … , k } .

ij

ij

ij wjii

=wi=jNIL

and if

or if)0(

i j

k i jk

i jk

i kk

k jk

k jk

i jk

i kk

k jk

d d d

d d d

( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

1 1 1 1

1 1 1 1

i f

i f

Chapter 25 P.23


D( )0

0 3 8 4

0 1 7

4 0

2 5 0

6 0

( )0

1 1 1

2 2

3

4 4

5

N N

N N N

N N N N

N N N

N N N N

D( )1

0 3 8 4

0 1 7

4 0

2 5 5 0 2

6 0

( )1

1 1 1

2 2

3

4 1 4 1

5

N N

N N N

N N N N

N

N N N N

Chapter 25 P.24


D( )2

0 3 8 4 4

0 1 7

4 0 5 11

2 5 5 0 2

6 0

( )2

1 1 2 1

2 2

3 2 2

4 1 4 1

5

N

N N N

N N

N

N N N N

D( )3

0 3 8 4 4

0 1 7

4 0 5 11

2 1 5 0 2

6 0

( )3

1 1 2 1

2 2

3 2 2

4 3 4 1

5

N

N N N

N N

N

N N N N

Chapter 25 P.25


D( )4

0 3 1 4 4

3 0 4 1 1

7 4 0 5 3

2 1 5 0 2

8 5 1 6 0

( )4

1 4 2 1

4 4 2 1

4 3 2 1

4 3 4 1

4 3 4 5

N

N

N

N

N

D( )5

0 1 3 2 4

3 0 4 1 1

7 4 0 5 3

2 1 5 0 2

8 5 1 6 0

( )5

1 4 5 1

4 4 2 1

4 3 2 1

4 3 4 1

4 3 4 5

N

N

N

N

N

Chapter 25 P.26


Transitive closure of a directed graph: Given a directed graph G = (V, E)

with V = {1, 2,…, n}. The transitive closure of G is G V E* ( . *)

where E i j* {( , )| there is a path from i to j in G}.

Modify FLOYD-WARSHALL algorithm:

ti j (i,j) E

i j (i,j) Eij( )0 0

1

if and

if or

for k 1,

t t t tijk

ijk

ikk

kjk( ) ( ) ( ) ( )( ) 1 1 1

Chapter 25 P.27


T R A N S I T I V E - C L O S U R E ( G )

1 n G | |

2 f o r i 1 t o n

3 d o f o r 1j t o n

4 d o i f i j o r ( , ) ( )i j E G

5 t h e n t i j( )0 1

6 e l s e t i j( )0 0

7 f o r k 1 t o n

8 d o f o r i 1 t o n

9 d o f o r j 1 t o n

1 0 d o t t t ti jk

i jk

i kk

k jk( ) ( ) ( ) ( )( ) 1 1 1

1 1 r e t u r n T n( )

T i m e c o m p l e x i t y : O n( )3 .

Chapter 25 P.28


1

4 3

2T( )0

1 0 0 0

0 1 1 1

0 1 1 0

1 0 1 1

T ( )1

1 0 0 0

0 1 1 1

0 1 1 0

1 0 1 1

T ( )2

1 0 0 0

0 1 1 1

0 1 1 1

1 0 1 1

T ( )3

1 0 0 0

0 1 1 1

0 1 1 1

1 1 1 1

T ( )4

1 0 0 0

1 1 1 1

1 1 1 1

1 1 1 1

Chapter 25 P.29


25.3 Johnson’s algorithm for sparse graphs

t i m e c o m p l e x i t y O V V V E( l g )2 b y a s s u m i n g D i j k s t r a a l g o r i t h m t a k e s

O V V V E( l g )2 u s i n g F i b o n a c c i h e a p .

C o m b i n e w i t h B e l l m a n - F o r d a l g o r i t h m t a k e s O V E( ) .

u s i n g r e w e i g h t i n g t e c h n i q u e

I f a l l e d g e w e i g h t s i n G a r e n o n n e g a t i v e , w e c a n f i n d a l l s h o r t e s t p a t h s i n

O V V V E( l g )2 u s i n g D i j k s t r a a l g o r i t h m .

Chapter 25 P.30


If G has negative-weighted edge, we compute a new set of nonnegative

weight that allows us to use the same method. The new set of edge weight

w satisfies:

1. For all pairs of vertices u v V, , a shortest path from u to v using

weight function w is also a shortest path from u to v using the weight

function w .

2. ( , ) ( ), ( , )u v E G w u v is nonnegative.

Chapter 25 P.31


Preserving shortest paths by reweighting

Lemma 25.1 (Reweighting doesn’t change shortest paths)

Given a weighted directed graph G = (V, E) with weight function

w E R: , let h V R: be any function mapping vertices to real

numbers. For each edge ( , )u v E , define

( , ) ( , ) ( ) ( )w u v w u v h u h v .

Let P v v vk 0 1, ,..., be a path from vertex v0 to vk . Then

w P v vk( ) ( , ) 0 if and only if ( ) ( , )w P v vk 0 . Also, G has a

negative-weight cycle using weight function w iff G has a negative

weight cycle using weight function w .

Chapter 25 P.32


Proof.

( ) ( ) ( ) ( )w P w P h v h vk 0

( ) ( , )

( ( , ) ( ) ( ))

( , ) ( ) ( )

( ) ( ) ( )

w P w v v

w v v h v h v

w v v h v h v

w P h v h v

i ii

k

i ii

ki i

i ii

kk

k

11

11

1

11

0

0

Chapter 25 P.33


w P v v k( ) ( , ) 0 i m p l i e s ( ) ( , )w P v v k 0 .

S u p p o s e t h e r e i s a s h o r t e r p a t h P ' f r o m v 0 t o v k u s i n g t h e

w e i g h t f u n c t i o n w . T h e n )(ˆ)'(ˆ PwPw .

T h e n

).()'(

)()()()(ˆ

)'(ˆ)()()'(

0

0

PwPw

vhvhPwPw

PwvhvhPw

k

k

W e g e t a c o n t r a d i c t i o n !

Chapter 25 P.34


G has a negative-weight cycle using w iff G has a negative-weight

cycle using w .

Consider any cycle C v v vk 0 1, ,..., with v vk0 . Then

( ) ( ) ( ) ( ) ( )w C w C h v h v w Ck 0 .

Producing nonnegative weight by reweight

Chapter 25 P.35


2

3

45

1

-4

3

7

6

21

4

8

-5

Chapter 25 P.36


0

-4 0

-5

2

3

45

1

-4

3

7

6

21

4-1

8

-5

0

0

0

0

0

0s

Chapter 25 P.37


0

-4 0

-5

2

3

45

1

0

4

10

2

20

0-1

13

0

5

1

0

4

0

0s

( , ) ( , ) ( ) ( )w u v w u v h u h v

h v s v( ) ( , )

Chapter 25 P.38


D e f i n e h v s v( ) ( , ) f o r a l l v V '

'),(,0)()(),(),(ˆ

'),(),,()()(

Evuvhuhvuwvuw

Evuvuwuhvh

Chapter 25 P.39


J O H N S O N ( G )

1 C o m p u t i n g G ' , w h e r e }{}[]'[ sGVGV

a n d E G E G s v v V G[ ' ] [ ] { ( , ) | [ ] } .

2 i f B E L L M A N - F O R D ( G w s' , , ) = F A L S E

3 t h e n p r i n t “ t h e i n p u t g r a p h c o n t a i n s n e g a t i v e w e i g h t c y c l e ”

4 e l s e f o r e a c h v e r t e x v V G [ ' ]

5 d o s e t h ( v ) t o b e t h e v a l u e o f ( , )s v

6 f o r e a c h e d g e [ , ] [ ' ]u v E G

Chapter 25 P.40


7 do ( , ) ( , ) ( ) ( )w u v w u v h u h v

8 for each vertex u V G [ ]

9 do run DIJKSTRA(G w u, , ) to compute ( , ) u v for all v V G [ ] .

10 for each vertex v V G [ ]

11 do d u v h v h uuv ( , ) ( ) ( )

12 return D

Complexity: O V V VE( lg )2 .

Chapter 25 P.41


2/7

2/3 0/5

0/0

2

3

45

1

0

4

10

2

20

00/4

13

0

0/0

0/-4 2/2

2/-3

2

3

45

1

0

4

10

2

20

02/1

13

0

2/3

2/-1 0/1

0/-4

2

3

45

1

0

4

10

2

20

00/0

13

0

4/8

0/0 2/6

2/1

2

3

45

1

0

4

10

2

20

02/5

13

0

2/2

2/-2 0/0

0/-5

2

3

45

1

0

4

10

2

20

00/-1

13

0

( , ) / ( , ) u v u v

25.All-Pairs Shortest Paths Hsu, Lih-Hsing. Computer Theory Lab. Chapter 25P.2.

Documents

computer theory

lihhsing slide

matrix multiplication

floydwarshall algorithm

allpairs shortest paths

johnson s algorithm

sparse graphs