Chapter 25: All-Pairs Shortest-Paths

Transcription

1 Chapter : All-Pairs Shortest-Paths

2 When no negative edges Some Algorithms Using Dijkstra s algorithm: O(V ) Using Binary heap implementation: O(VE lg V) Using Fibonacci heap: O(VE + V log V) When no negative cycles Floyd-Warshall [96]: O(V ) time When negative cycles Using Bellman-Ford algorithm: O(V E) = O(V ) Johnson [977]: O(VE + V log V) time based on a clever combination of Bellman-Ford and Dijkstra

3 A dynamic programming approach. characterize the structure of an optimal solution,. recursively define the value of an optimal solution,. compute the value of an optimal solution in a bottom-up fashion.

4 The structure of an 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 n- is finite.

5 The structure of an optimal solution (cont.) If i = j, then p has weight and no edge If i j, we decompose p into i ~ k -> j where p contains at most m- edges. Moreover, p is a shortest path from i to k and δ(i,j) = δ(i,k) + w kj, where δ(i,j) dente the shortest weight path from i to j p'

6 Recursive solution to the allpairs shortest-path problem Define: l (m) ij = minimum weight of any path from i to j that contains at most m edges. if i = j l () = ij if i j Then l (m) ij = min{ l (m-) ij, min k n {l (m-) ik + w kj }} = min k n {l (m-) ik + w kj } (why?) 6

7 Recursive solution to the allpairs shortest-path problem Since shortest path from i to j contains at most n- edges, δ(i,j) = l ij (n-) = l ij (n) = l ij (n+) = Computing the shortest-path weight bottom up: Compute L (),L (),., L (n-) where L (m) =(l ij (m) ) for all i and j ote that L () = W. 7

8 () L () L Example: W = L () = L () x W

9 () L () L L () = L () x W L () = L () x W

10 EXTEDED-SHORTEST- PATHS(L, W) Given matrices L (m-) and W return L (m) n <- L.row Let L = (l ij ) be a new n x n matrix for i = to n for j = to n l ij = 6 for k = to n 7 l ij = min(l ij, l ik + w kj ) 8 return L

11 Matrix Multiplication Let l (m-) -> a w -> b l (m) -> c min -> + + -> C ij = k= to n a ik b kj l ij (m) = min k n {l ik (m-) + w kj } time complexity : O(n )

12 SLOW-ALL-PAIRS- SHORTEST-PATHS(W) L () = L () W = W L () = L () W = W L () = L () W = W L (n-) = L (n-) W = W n-

13 SLOW-ALL-PAIRS- SHORTEST-PATHS(W) n = W.rows L () = W for m = to n- let L (m) be a new n x n matrix L (m) = EXTEDED-SHORTEST- PATHS( L (m-), W ) 6 return L (n-) Time complexity : O(n )

14 Improving the running time L () = W L () = W =W.W L () =W = W.W. L (n) = W n =W n/.w n/ i.e., using repeating squaring! Time complexity: O(n lg n)

15 FASTER-ALL-PAIRS-SHORTEST- PATHS FASTER-ALL-PAIRS-SHORTEST-PATHS(W). n =W.row. L () =W. m =. while m < n-. let L (m) be a new n x n matrix 6. L (m) = Extend-Shorest-Path(L (m), L (m) ) 7. m = m 8. return L (m)

16 The Floyd-Warshall algorithm A different dynamic programming formulation The structure of a shortest path: Let V(G)={,,,n}. For any pair of vertices i, j єv, consider all paths from i to j whose intermediate vertices are drawn from {,,,k}, and let p be a minimum weight path among them. 6

17 The structure of a shortest path If k is not in p, then all intermediate vertices are in {,,,k-}. If k is an intermediate vertex of p, then p p can be decomposed into i ~ k p ~ j where p is a shortest path from i to k with all the intermediate vertices in {,,,k-} and p is a shortest path from k to j with all the intermediate vertices in {,,,k-}. 7

18 A recursive solution to the allpairs shortest path Let d ij (k) =the weight of a shortest path from vertex i to vertex j with all intermediate vertices in the set {,,,k}. d ij (k) = w ij if k = = min(d ij (k-), d ik (k-) + d kj (k-) ) if k D (n) = (d ij (n) ) if the final solution! 8

19 FLOYD-WARSHALL(W). n = W.rows. D () = W. for k = to n. Let D (k) = (d ij (k) ) be a new n x n matrix. for i = to n 6. for j = to n 7. d ij (k) = min (d ij (k-), d ik (k-) + d kj (k-) ) 8. return D (n) Complexity: O(n ) 9

20 Example () D () D () ()

21 6 7 8 () D () () D ()

22 6 8 7 () D () D () ()

23 Constructing a shortest path π (), π (),, π (n) π (k) ij : is the predecessor of the vertex j on a shortest path from vertex i with all intermediate vertices in the set {,,,k}. π () ij = IL if i=j or w ij = = i if i j and w ij < π ij (k) = π ij (k-) if d ij (k-) d ik (k-) + d kj (k-) = π kj (k-) if d ij (k-) > d ik (k-) + d kj (k-)

24 Transitive closure of a directed graph Given a directed graph G = (V, E) with V = {,,, 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: t ij () = if i j and (i,j) E for k if i=j or (i,j) є E t ij (k) = t ij (k-) (t ik (k-) t kj (k-) )

25 Transitive-Closure of a Directed Graph Given a directed graph G=(V, E) with V= {,, n} We might wish to determine whether G contains a path from I to j for all vertex pairs i, j єv. We define the transitive closure of G as the graph G* = (V, E*), where E* = {(i, j): there is a path from vertex i to vertex j in G}

26 TRASITIVE-CLOSURE(G) n = G.V Let T () = (t () ij ) be a new n x n matrix for i = to n for j = to n if i == j or (i, j) G.E 6 t () ij = 7 else t () ij = Time complexity: O(n ) 8 for k = to n 9 Let T (k) = (t (k) ij ) be a new n x n matrix for i = to n for j = to n t (k) ij = t (k-) ij (t (k-) ik t (k-) kj ) return T (n) 6

27 7 Example () T () T () T () T () T

28 When no negative edges Some Algorithms Using Dijkstra s algorithm: O(V ) Using Binary heap implementation: O(VE lg V) Using Fibonacci heap: O(VE + V log V) When no negative cycles Floyd-Warshall [96]: O(V ) time When negative cycles Using Bellman-Ford algorithm: O(V E) = O(V ) Johnson [977]: O(VE + V log V) time based on a clever combination of Bellman-Ford and Dijkstra 8

29 Johnson s algorithm for sparse graphs If all edge weights in G are nonnegative, we can find all shortest paths in O(V lg V + VE) by using Dijkstra s algorithm with Fibonacci heap Bellman-Ford algorithm takes O(V E) Using reweighting technique 9

30 Reweighting technique 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 ŵ satisfies:. For all pairs of vertices u, v єv, a shortest path from u to v using weight function w is also a shortest path from u to v using the weight function ŵ. (u,v) є E(G), ŵ(u,v) is nonnegative

31 Lemma: (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) є E, ŵ(u,v) = w(u,v) + h(u) h(v) Let P=<v,v,,v k > be a path from vertex v to v k Then w(p) = δ(v,v k ) if and only if ŵ(p) = ˆ (v,v k ) Also, G has a negative-weight cycle using weight function w iff G has a negative weight cycle using weight function ŵ. Question: how to setting the value of h(v i ) for all i?

32 Example:

33 h(v) = δ(s, v) ŵ(u, v) = w(u, v) + h(u) h(v)

34 Proof ŵ(p) = w(p) + h(v ) h(v k ) ŵ(p) = ŵ(v i-,v i ) k k i = (w(v i-,v i ) + h(v i- )-h(v i )) i k = w(v i-,v i ) + h(v )-h(v k ) i = w(p) + h(v ) h(v k )

35 Proof Because h(v ) and h(v k ) do not depend on the path, if one path from v to v k is shorter than another using weight function w, then it is also shorter using ŵ. Thus, w(p) = δ(v,v k ) if and only if ŵ(p) = (v,v k )

36 Proof G has a negative-weight cycle using w iff G has a negative-weight cycle using ŵ. Consider any cycle C=<v,v,,v k > with v =v k. Then ŵ(c) = w(c) + h(v ) h(v k ) = w(c). Question: how to setting the value of h(v i ) for all i? 6

37 Producing nonnegative weight by reweighting Given a weighted directed graph G = (V, E) We make a new graph G = (V,E ), V = V {s}, E = E {(s,v): v є V} and w(s,v) =, for all v in V Let h(v) = δ(s, v) for all v V We have h(v) h(u) + w(u, v) (if there is no negative weight cycle) (why?) ŵ(u, v) = w(u, v) + h(u) h(v) 7

38 Example:

39 S

40 S

41 h(v) = δ(s, v) ŵ(u, v) = w(u, v) + h(u) h(v) S - - -

42 JOHSO algorithm Computing G, where G.V = G.V {s} and G.E= G.E {(s, v): v є G.V} and w(s, v) =. if BELLMA-FORD(G, w, s)= FALSE print the input graph contains negative weight cycle else for each vertex v є G.V set h(v) to be the value of δ(s, v) computed by the BF algorithm 6 for each edge (u, v) є G.E, ŵ(u, v) = w(u, v) + h(u) h(v)

43 JOHSO algorithm 7 Let D = (d uv ) be a new n x n matrix 8 for each vertex u є G.V run DIJKSTRA (G, ŵ, u) to compute ˆ (u, v) for all v є V[G]. for each vertex v є G.V d uv = (u, v) + h(v) h(u) ˆ return D Complexity: O(V lgv + VE)

44 / /- / /- / / / /- / /- / / /7 / /

45 / /- /- /8 / / /- / / /6

46 Homework Practice at home:.-,.-,.-6 Exercises:.-,.-6,.-8 (Due: Jan. ) Practice at home :.-, Exercises:.- (Due: Jan. ) Bonus: Write a program to find all pairs shortest path of a graph G. There may exist negative weight cycle on graph G. 6