All Pairs Shortest Paths

1
All Pairs Shortest Paths
2
outline

• application of single source shortest path
  algorithms
• direct methods to solve the problem
• matrix multiplication
• Floyds algorithm
• transitive closure (Warshalls algorithm)

3
motivation

• computer network
• aircraft network (e.g. flying time, fares)
• railroad network
• table of distances between all pairs of cities
  for a road atlas

4
single source shortest path algorithms

• if edges are non-negative
• running the Dijkstra algorithm n-times, once for
  each vertex as the source
• running time O(nm lg n)
• (using binary-heap)

5
single source shortest path algorithms

• negative-weight edges
• Bellman-Ford algorithm
• running time O(n2 m)

6
data structure

• adjacency matrix
• c E ? ? as n x n matrix C
• 0 if i j,
• cij c(i,j) if i ? j and (i,j) ? E,
• 8 if i ? j and (i,j) ? E

7
matrix multiplication (idea)

• dij(m) minimum weight of any path
• from i to j
• that contains at most m edges

8
matrix multiplication (idea)
dik(m-1)
k
p
ckj
i
j
dij(m-1)
v
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matrix multiplication (idea)

• dij(m) min (dij(m-1), min1kn dik(m-1)
  ckj)
• look at all possible predecessors k of j and
  compare

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matrix multiplication (structure)

• dij(1) cij
• dij(m) min (dij(m-1), min1kn dik(m-1)
  ckj)
• min1kn dik(m-1) ckj

11
matrix multiplication (structure)

• Compute a series of matrices
• D(1), D(2), , D(n-1)
• D(m) D(m-1) ? C
• final matrix D(n-1) contains the final
  shortest-path weights

12
matrix multiplication (pseudo-code)

• EXTEND-SHORTEST-PATHS (D,C )
• n ? rows D
• let D' (d'ij ) be an n x n matrix
• for i ? 1 to n
• do for j ? 1 to n
• do d'ij ? 8
• for k ? 1 to n
• do d'ij ? min (d'ij , dik ckj)
• return D'

13
matrix multiplication (example)
2
4
3
1
3
8
2
-4
1
-5
7
5
4
6
14
matrix multiplication (example)
2
4
3
1
3
8
2
-4
1
-5
7
5
4
6
15
matrix multiplication (example)
2
4
3
1
3
8
2
-4
1
-5
7
5
4
6
16
matrix multiplication (example)
2
4
3
1
3
8
2
-4
1
-5
7
5
4
6
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matrix multiplication (example)

• ?
• 1
• d14(2) (0 3 8 ? ?4) ? ?
• 0
• 6
• min (?, 4, ?, ?, 2)
• 2

18
matrix multiplication (example)
2
4
3
1
3
8
2
-4
1
-5
7
5
4
6
19
matrix multiplication (example)
2
4
3
1
3
8
2
-4
1
-5
7
5
4
6
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relation to matrix multiplication

• C A ? B
• cij Sk1n aik ? bkj
• compare
• D(m) D(m-1) ? C
• ? dij(m) min1kn dik(m-1) ckj

21
matrix multiplication

• compute the sequence of n-1 matrices
• D(1) D(0) ? C C,
• D(2) D(1) ? C C2,
• D(3) D(2) ? C C3,
• D(n-1) D(n-2) ? C Cn-1

22
matrix multiplication (pseudo-code)

• ALL-PAIRS-SHORTEST-PATHS (C )
• n ? rows C
• D (1) ? C
• for m ? 2 to n - 1
• do D (m) ? EXTEND-SHORTEST-PATHS
• (D (m-1),C )
• return D (n-1)
• negative cycles dii lt 0

23
matrix multiplication (example)
2
4
3
1
3
8
2
-4
1
-5
7
5
4
6
24
matrix multiplication (example)
2
4
3
1
3
8
2
-4
1
-5
7
5
4
6
25
matrix multiplication (example)
2
4
3
1
3
8
2
-4
1
-5
7
5
4
6
26
matrix multiplication (running time)

• O(n4)
• Improving the running time
• compute not all D(m) matrices
• interested only in D(n-1), which is D(m) for all
  integers m n-1

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improving the running time

• compute the sequence
• D(1) C,
• D(2) C2 C ? C,
• D(4) C4 C2 ? C2,
• D(8) C8 C4 ? C4
• ...
• D(2 ? ?lg(n-1)?) C2? ?lg(n-1)?
• C2 ? ( ?lg(n-1)? -1) ? C2 ?( ?lg(n-1)? -1)
• so we need only ?lg(n-1)? matrix products
• O(n3 lg n)

28
Floyds algorithm
1
2
.
i
j
k-1
k
k1
.
n
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Floyds algorithm

• dij(0) cij
• (no intermediate vertices at all)
• dij(k) min(dij(k-1), dik(k-1) dkj(k-1)) if
  k 1
• result D(n) (dij(n)) d(i,j)
• (because all intermediate vertices are in the
  set 1, 2, , n)

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Floyds algorithm (example)
2
4
3
1
3
8
2
-4
1
-5
7
5
4
6
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Floyds algorithm (example)
2
2
?
3
3
3
8
2
4
4
1
?4
?
5
5
4 ? 3 ? 5
32
Floyds algorithm (example)
2
4
3
1
3
8
2
-4
1
-5
7
5
4
6
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Floyds algorithm (example)
1
1
?
3
3
3
4
4
4
2
1
5
5
5
7
4 ? 5 ? 2 3 ? 4 ? 1 ? 5 ? 4 3 ? 5 ?
34
Floyds algorithm (example)
2
4
3
1
3
8
2
-4
1
-5
7
5
4
6
35
Floyds algorithm (pseudo-code)

• FLOYD (C )
• n ? rows C
• D (0) ? C
• for k ? 1 to n
• do for i ? 1 to n
• do for j ? 1 to n
• dij(k) ? min (dij(k-1), dik(k-1) dkj(k-1))
• return D (n)
• running time O(n3)

36
transitive closure (the problem)

• find out whether there is a path from i to j
  and compute
• G (V,E),
• where E (i,j) there is a path from i to
  j in G

a
c
b
i
j
37
transitive closure (the problem)

• find out whether there is a path from i to j
  and compute
• G (V,E),
• where E (i,j) there is a path from i to
  j in G

a
c
b
i
j
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transitive closure

• one way
• set cij 1 and
• run the Floyd algorithm
• running time O(n3)

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transitive closure

• another way
• substitute and min by AND and OR
• in Floyds algorithm
• running time O(n3)

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transitive closure (pseudo-code)

• TRANSITIVE-CLOSURE (G )
• n ? ?V G ?
• for i ? 1 to n
• do for j ? 1 to n
• do if i j OR (i,j ) ? E G
• then tij(0) ? 1
• else tij(0) ? 0
• for k ? 1 to n
• do for i ? 1 to n
• do for j ? 1 to n
• do tij(k) ? tij(k-1) OR (tik(k-1) AND
  tkj(k-1) )
• return T (n)

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table of running times
42
Floyds algorithm
k
dik(k-1)
dkj(k-1)
i
dij(k-1)
j