Algorithms. All-Pairs Shortest Paths. Dong Kyue Kim Hanyang University

Transcription

1 Algorithms All-Pairs Shortest Paths Dong Kyue Kim Hanyang University

2 Contents Using single source shortest path algorithms Presents O(V 4 )-time algorithm, O(V 3 log V)-time algorithm, O(V 3 )-time algorithm O(V 4 )-time algorithm O(V 3 log V)-time algorithm Floyd-Warshall algorithm Transitive closure of a directed graph 2

3 Using single source shortest path algorithms Positive edges Negative edges 3

4 Using single source shortest path algorithms Using single source shortest path algorithms We can solve an all-pairs shortest-paths problem by running a single-source shortest-paths algorithm V times, once for each vertex as the source. 4

5 Using single source shortest path algorithms Positive edges Using dijkstra algorithm: The linear-array implementation O(V 3 + V E) = O(V 3 ). The binary min-heap implementation O(V E lg V), Fibonacci heap O(V 2 lg V + V E). 5

6 Using single source shortest path algorithms Negative edges Using Bellman-Ford algorithm: O(V 2 E) Dense graph O(V 4 ) 6

7 Using single source shortest path algorithms Use adjacency matrix Predecessor matrix 7

8 Using single source shortest path algorithms Use adjacency matrix We assume that the vertices are numbered 1, 2,..., V The input is an n n matrix W representing the edge weights of an n-vertex directed graph G = (V, E) w ij the weight of directed edge Negative-weight edges are allowed, but we assume for the time being that the input graph contains no negative-weight cycles. (i, j) if if if i i i j, jand (i, j) E, jand (i, j) E 8

9 Using single source shortest path algorithms Predecessor matrix Say Π = (π ij ) π ij = NIL if either i = j or there is not path from i to j. π ij is the predecessor of j on some shortest path from i to j. 9

10 Using single source shortest path algorithms For each vertex i V, we define predecessor subgraph of G for i as G π,i = (V π,i, E π,i ), where V π,i = {j V : π ij = NIL} {i} E π,i = {(π ij, j) j V π,i {i}} If G π,i is a shortest-paths tree, then PRINT-ALL- PAIRS-SHORTEST-PATH will print a shortest path from i to j. 1

11 Using single source shortest path algorithms PRINT-ALL-PAIRS-SHORTEST-PATH(Π, i, j) 1 if i = j 2 then print i 3 else if π ij = NIL 4 then print no path from i to j exists 5 else PRINT-ALL-PAIRS-SHORTEST-PATH(Π,i, π ij ) 6 print j 11

12 Shortest paths and matrix multiplication O(V 4 )-time algorithm Use dynamic programming Use matrix multiplication Associative (short explanation) Computing predecessor matrix 12

13 Shortest paths and matrix multiplication Use dynamic programming 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. 13

14 Shortest paths and matrix multiplication The structure of a shortest path On a graph G = (V, E), all subpaths of a shortest path are shortest paths. (Lemma 24.1) k P 1 P 2 i j P` : {1,2,,k-1} p is a shortest path from i to k, and so δ(i, j) = δ(i, k) + w kj. 14

15 Shortest paths and matrix multiplication A recursive solution to the all-pairs shortest-paths problem (m) l ij Let be the minimum weight of any path from vertex i to vertex j that contains at most m edges When m =, then () l ij When m 1, then if if i j i j l ( m) ij min( l min{ l 1kn ( m1) ij ( m1) ij, min{ l 1kn w kj } ( m1) ij w kj }) 15

16 Shortest paths and matrix multiplication If the graph contains no negative-weight cycles, For every pair of vertices i and j for which δ(i, j) < There is a shortest path from i to j that is simple and thus contains at most n - 1 edges. A path from vertex i to vertex j with more than n - 1 edges cannot have lower weight than a shortest path from i to j. Therefore, ( i, j) l l ( n1) ( n) ( n1) ij ij ij l... 16

17 Shortest paths and matrix multiplication Computing the shortest-path weights bottom up Input the matrix W = (w ij ), we now compute a series of matrices L (1), L (2),..., L (n-1), ( m) ( m) Where for m = 1, 2,..., n - 1, we have l ( l ). The final matrix L (n-1) contains the actual shortest-path weights. Observe that for all vertices i, j V, and so L (1) = W ij. ij 17

18 Shortest paths and matrix multiplication Let L (m 1) = L, L (m) = L. We have EXTEND-SHORTEST-PATHS(L, W) 1 n rows[l] 2 let L = (l ij ) be an n n matrix 3 for i 1 to n 4 do for j to n 5 do 6 for k 1 to n 7 do 8 return L Costs Θ(n 3 ) time. 18

19 Shortest paths and matrix multiplication EXTEND-SHORTEST-PATHS(L, W) 1 n rows[l] 2 let C be an n n matrix 3 for i 1 to n 4 do for j 1 to n 5 do c ij 6 for k 1 to n 7 do c ij c ij + a ik b kj 8 return L 19

20 Shortest paths and matrix multiplication SLOW-ALL-PAIRS-SHORTEST-PATHS(W) 1 n rows[w] 2 L (1) W 3 for m 2 to n 1 4 do L (m) EXTEND-SHORTEST-PATHS(L (m 1),W) 5 return L (n 1) Costs Θ(n 4 ) time. 2

21 Shortest paths and matrix multiplication Improving the running time We are interested only in matrix L (n-1). Recall that in the absence of negative-weight cycles, implies L (m) = L (n-1) for all integers m n - 1. Therefore, we can compute L (n-1) with only lg(n - 1) matrix products by computing the sequence 21

22 22 Shortest paths and matrix multiplication (4) (3) (2) (1) L L L L

23 Shortest paths and matrix multiplication L (2 L L L L (1) (2) (4) (8) W W W W W W W W W 2 4 W W 2 4 W W lg( n1) 2lg( n1) 2lg( n1) 1 2lg( n1) 1 23

24 Shortest paths and matrix multiplication

25 Shortest paths and matrix multiplication FASTER-ALL-PAIRS-SHORTEST-PATHS(W) 1 n rows[w] 2 L (1) W 3 m 1 4 while m < n do L (2m) EXTEND-SHORTEST-PATHS(L (m), L (m) ) 6 m 2m 7 return L (m) Costs Θ(n 3 ) time. 25

26 The Floyd-Warshall algorithm Intermediate Vertex An intermediate vertex of a simple path p = <v 1, v 2,, v l > is any vertex of p other than v 1 and v l. 26

27 The Floyd-Warshall algorithm The structure of a shortest path Floyd-Warshall algorithm is based on the observation of the intermediate vertices, which costs Θ( V 3 ) time. Let V = {1, 2,, n}. For any pair of vertices i, j V, consider all paths from i to j whose intermediate vertices are all drawn from {1, 2,, k}, and let p be a minimum weight path from among them. 27

28 The Floyd-Warshall algorithm If k is not an intermediate vertex of path p, then all intermediate vertices of p are in {1, 2,, k 1}. If k is an intermediate vertex of path p, then we break p down into all intermediate vertices in {1,2..,k-1} all intermediate vertices in {1,2..,k-1} P 1 P 2 k i j P: all intermediate vertices in {1,2,,k-1} 28

29 The Floyd-Warshall algorithm A recursive solution to the all-pairs shortest-paths problem (k ) d ij Let be the weight of a shortest path from vertex i to vertex j for which all intermediate vertices are in the set {1, 2,, k}. We have the following recurrence: d ( k ) ij wij min(d (k -1) ij,d (k -1) ik d (k -1) kj ) if k, if k 1. (25.5) Because for any path, all intermediate vertices are in the set {1, 2,, n}, the matrix D (n) (n) dij gives the final answer: ( ) d n ( i, j) for all i, j V. ij 29

30 The Floyd-Warshall algorithm FLOYD-WARSHALL(W) 1 n rows[w] 2 D () W 3 for k 1 to n 4 do for i 1 to n 5 do for j 1 to n ( k) ( k1) ( k1) ( k1) 6 do dij min d ij, dik dkj 7 return D (n) costs Θ(n 3 ) time. 3

31 The Floyd-Warshall algorithm 31

32 The Floyd-Warshall algorithm 32

33 The Floyd-Warshall algorithm 33

34 The Floyd-Warshall algorithm Constructing A Shortest Path k ij Let be the predecessor of vertex j on a shortest path from vertex i with all intermediate vertices in {1, 2,, k}. () ij NIL if i if i i jor w ij jand w,. ij ( k ) ij ( ( k1) ij k1) ik if if d d ( k1) ij ( k1) ij d d ( k1) ik ( k1) ik d d ( k1) kj ( k1) kj,. 34

35 The Floyd-Warshall algorithm Transitive Closure of Graph Given a directed graph G = (V, E) with vertex set V = {1, 2,, n}. The transitive closure of G is defined as the graph G = (V, E ), where E = {(i, j) : there is a path from vertex i to vertex j in G}. 35