MA 252: Data Structures and Algorithms Lecture 36. Partha Sarathi Mandal. Dept. of Mathematics, IIT Guwahati

Transcription

1 MA 252: Data Structures and Algorithms Lecture 36 Partha Sarathi Mandal Dept. of Mathematics, IIT Guwahati

2 The All-Pairs Shortest Paths Problem Given a weighted digraph G = (V, E) with weight function w: E R, (R, is the set of real numbers) determine the length of the shortest path (i.e., distance between all pairs of vertices in G). Here we assume that there are no cycles with zero or negative cost.

3 Dynamic Programming Solution (based on matrix multiplication) To simplify the notation, we assume that V = {1,2,3,...,n} Assume that the graph is represented by an n n matrix with the weights of the edges : Output Format: an n n matrix D = [d ij ] where d ij is the length of the shortest path from vertex i to j.

4 How to Decompose the Original Problem Subproblems with smaller sizes should be easier to solve. An optimal solution to a subproblem should be expressed in terms of the optimal solutions to subproblems with smaller sizes.

5 Decompose in a Natural Way Define d (m) ij to be the length of the shortest path from i to j that contains at most m edges. Let D (m) be the matrix [d ij (m) ]. d ij (n-1) is the true distance from i to j. Subproblems: compute D (m) for m = 1,2,, n-1.

6 A Recursive Formula d ij (0) = 0 if i = j d ij (0) = i j Consider a shortest path from i to j of length d ij (m) Case 1: at most m-1 edges shortest path It has at most m-1 edges, then d ij (m) = d ij (m-1) = d ij (m-1) + w jj (=0) Case 2: exactly m edges shortest path It has m edges. Let k be the vertex before j then d ij (m) = d ik (m-1) + w kj where 1 k n.

7 A Recursive Formula Combining the two cases d ij (m) = min(d ij (m-1), min k {d ik (m-1) + w kj }) where 1 k n. = min (m-1) k {d ik + w kj }

8 Bottom-up Computation of D (n-1) Bottom: D (1) =[w ij ], the weight matrix W. Computing D (m) from D (m-1), for m = 2,, n-2 using d ij (m) = min k {d ik (m-1) + w kj } 1 k n

9 Extend Shortest Paths Extend-Shortest-Paths(D,W) n rows[d] let D =(d ij ) n n for i 1 to n do for j 1 to n do d ij for k 1 to n do d ij min(d ij, d ik +w kj ) return D D for D (m-1) D for D (m)

10 Relation with matrix multiplication Extend-Shortest-Paths(D,W) n rows[d] let D =(d ij ) n n for i 1 to n do for j 1 to n do d ij for k 1 to n do d ij min(d ij, d ik +w kj ) return D Matrix-Multiplication(A,B) n rows[a] let C=(c ij ) n n for i 1 to n do for j 1 to n do c ij 0 for k 1 to n do c ij c ij + a ik.b kj return C d (m-1) a w b d (m) c min + +.

11 shortest-path weights calculation Returning to the all-pairs shortest-paths problem computing the shortest-path weights by extending shortest paths edge by edge By returning n-1 times the Extend-Shortest-Paths(D,W) D (1) = D (0).W = W D (2) = D (1).W = W 2.. D (n-1) = D (n-2).w = W (n-1)

12 Example

13 Example: Computing D (2) from D (1)

14 Example: Computing D (3) from D (2) D (3) gives the distances between any pair of vertices.

15 The Algorithm for Computing D (n-1) All-Pairs-Shortest-Paths(W) n rows[w] D (1) W for m 2 to n -1 do D (m) Extend-Shortest Shortest-Paths( Paths(D (m-1),w) return D (n-1) O(n 4 ) Extend-Shortest-Paths( Paths(D,W) n rows[d] let D =(d ij ) n n O(n 3 ) for i 1 to n do for j 1 to n do d ij for k 1 to n do d ij min(d ij, d ik +w kj ) return D

16 Improving the running time D (1) D (2) = W = W 2 = W.W D (4) = W 4 = W 2.W 2.. D (2^ lg (n-1) ) = W (2^ lg (n-1) ) = W (2^ lg (n-1) ).W(2^ lg (n-1) )

17 Faster all Pairs Shortest Paths Faster-All-Pairs-Shortest-Paths(W) n rows[w] D (1) W m 1 while n -1 > m do D (2m) Extend-Shortest Shortest-Paths( Paths(D (m ), D (m ) ) m 2m return D (m) O(n 3 lg n) Extend-Shortest-Paths( Paths(D,W) n rows[d] let D =(d ij ) n n O(n 3 ) for i 1 to n do for j 1 to n do d ij for k 1 to n do d ij min(d ij, d ik +w kj ) return D

18 All-Pairs Shortest Paths Problem The Floyd-Warshall Algorithm Given a weighted digraph G = (V, E) with weight function w: E R, (R, is the set of real numbers) determine the length of the shortest path (i.e., distance between all pairs of vertices in G). Here we assume that there are no cycles with zero or negative cost.

19 Decomposition Definition: The vertices v 2, v 2,,v l-1 are called the intermediate vertices of the path p = {v 1, v 2, v 2,,v l-1, v l }. 2 2 l-1 l Let c (k) ij be the length of the shortest path from i to j such that all intermediate vertices on the path (in any) are in set {1, 2,, k}. c (0) ij is set to be w ij, i.e., no intermediate vertex. Let D (k) be the n n matrix [c (k) ij]

20 Decomposition c (k) ij is the distance from i to j. our aim is to compute D (n). Subproblems: compute D (k) for k =0, 1,, n.

21 Structure of shortest paths Observation 1: A shortest path does not contain the same vertex twice. Observation 2: For a shortest path from i to j such that any intermediate vertices on the path are chosen from the set {1,2,,k} there are two possibilities: k is not a vertex on the path, the shortest such path has length c (k-1) ij k is a vertex on the path, the shortest such path has length c (k-1) ik + c (k-1) kj.

22 Floyd-Warshall recurrence Consider a shortest path from i to j containing the vertex k. It consists of a subpath from i to k and k to j. Each subpath can only contain intermediate vertices in {1,2,, k-1} and must be as short as possible, namely they have lengths c (k-1) ik and c (k-1) kj. Hence the path has length c (k-1) ik + c (k-1) kj. intermediate vertices in {1, 2,, k}

23 Floyd-Warshall recurrence Combining the two cases we get c (k) ij =min k {c (k-1) ij,c (k-1) ik + c (k-1) kj} intermediate vertices in {1, 2,, k}

24 Bottom-up Computation Bottom: D (0) = [w ij ] the weight matrix. Compute D (k) from D (k-1) using c (k) ij =min{c (k-1) ij, c (k-1) ik + c (k-1) kj} for k = 1,2,,n

25 Pseudocode for Floyd-Warshall Floyd-Warshall(W) n rows[w] D (0) W for k 1 to n do for i 1 to n do for j 1 to n do c (k) ij min{c (k-1) ij, c (k-1) ik + c (k-1) kj} return D (n)

26 Extracting the Shortest Paths Construct a predecessor matrix Π from the D matrix. Given the predecessor matrix Π, print-all-pairsshortest-path procedure can use to print vertices on a given shortest path. The predecessor pointers π ij can be used to extract the final path. The idea is as follows. Whenever we discover that the shortest path from i to j passes through an intermediate vertex k, we set π ij = k If the shortest path does not pass through any intermediate vertex, then π ij = NIL.

27 Pseudocode for Floyd-Warshall (Extracting the Shortest Paths) Floyd-Warshall(W) n rows[w] π (0) ij =NIL if i=j or w ij = = i if i j and w ij < C (0) W =[w ij ] Π (0) [π [ (0) ij] for k 1 to n do for i 1 to n do for j 1 to n do c (k) ij min{c (k-1) ij, c (k-1) ik + c (k-1) kj} π ij k return Π (n)

28 Example Floyd-Warshall(W) n rows[w] D (0) W for k 1 to n for i 1 to n for j 1 to n c (k) ij min{c (k-1) ij, c (k-1) ik + c (k-1) kj} π ij k return D (n)

29 Run time comparison with Floyd-Warshall Algorithm O(V 2 E) Bellman-Ford (sparse graph) O(V 4 ) Bellman-Ford (dense graph) O(VElg V) Dijkstra (binary-heap) (sparse graph) O(V 3 lg V) Dijkstra (binary-heap) (dense graph) O(V 2 lg V+VE) Dijkstra (Fibonacci-heap) (sparse graph) O(V 2 lg V+V 3 ) Dijkstra (Fibonacci-heap) (dense graph) ve weight edges are not allowed in case of Dijkstra s algorithm Dynamic Programming based algorithms: O(V 4 ) matrix multiplication methods O(V 3 lg V) repeated squaring O(V 3 ) Floyd-Warshall Algorithm

30 Transitive closure of a directed graph Given a digraph G=(V,E) we may wish to find weather there is a path in G from i to j for all vertex pairs i,j. the transitive closure of G is defined as the graph G* =(V,E*) E* ={(i,j): there is a path from i to j in G}

31 Transitive closure of a directed graph