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COSC 3101A - Design and Analysis of Algorithms 12

COSC 3101A - Design and Analysis of Algorithms 12. All-Pairs Shortest Paths Matrix Multiplication Floyd-Warshall Algorithm. 2. 4. 3. 8. 1. 3. 2. 1. -4. -5. 7. 5. 4. 6. All-Pairs Shortest Paths. Given: Directed graph G = (V, E) Weight function w : E → R Compute:

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COSC 3101A - Design and Analysis of Algorithms 12

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  1. COSC 3101A - Design and Analysis of Algorithms12 All-Pairs Shortest Paths Matrix Multiplication Floyd-Warshall Algorithm

  2. 2 4 3 8 1 3 2 1 -4 -5 7 5 4 6 All-Pairs Shortest Paths • Given: • Directed graph G = (V, E) • Weight function w : E → R • Compute: • The shortest paths between all pairs of vertices in a graph • Representation of the result: an n × n matrix of shortest-path distances δ(u, v) COSC3101A

  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 COSC3101A

  4. All-Pairs Shortest Paths - Solutions • Run BELLMAN-FORD once from each vertex: • O(V2E), which is O(V4) if the graph is dense(E = (V2)) • If no negative-weight edges, could run Dijkstra’s algorithm once from each vertex: • O(VElgV) with binary heap, O(V3lgV) if the graph is dense • We can solve the problem in O(V3) in all cases, with no elaborate data structures COSC3101A

  5. 2 4 3 8 1 3 2 1 -4 -5 7 5 4 6 All-Pairs Shortest Paths • Assume the graph (G) is given as adjacency matrix of weights • W = (wij), n x n matrix, |V| = n • Vertices numbered 1 to n if i = j wij = if i  j , (i, j)  E if i  j , (i, j)  E • Output the result in an n x n matrix D = (dij), where dij = δ(i, j) • Solve the problem using dynamic programming 0 weight of (i, j) ∞ COSC3101A

  6. at most m edges  11 j i p’ Optimal Substructure of a Shortest Path • All subpaths of a shortest path are shortest paths • Let p: a shortest path p from vertex i to j that contains at most m edges • If i = j • w(p) = 0 and p has no edges k at most m - 1 edges • If i  j:p = i k  j • p’ has at most m-1 edges • p’ is a shortest path δ(i, j) = δ(i, k) + wkj COSC3101A

  7. at most m edges  11 j i Recursive Solution • lij(m) = weight of shortest path i j that contains at most m edges • m = 0: lij(0) = if i = j if i  j • m  1: lij(m) = • Shortest path from i to j with at most m – 1 edges • Shortest path from i to j containing at most m edges, considering all possible predecessors (k) of j 0  k lij(m-1) min { , } min {lik(m-1) + wkj} 1  k  n = min {lik(m-1) + wkj} 1  k  n COSC3101A

  8. Computing the Shortest Paths • m = 1: lij(1) = • The path between i and j is restricted to 1 edge • Given W = (wij), compute: L(1), L(2), …, L(n-1), where L(m) = (lij(m)) • L(n-1) contains the actual shortest-path weights Given L(m-1) and W  compute L(m) • Extend the shortest paths computed so far by one more edge • If the graph has no negative cycles: all simple shortest paths contain at most n - 1 edges δ(i, j) = lij(n-1) and lij(n), lij(n+1). . . wij L(1) = W = lij(n-1) COSC3101A

  9. j i k k Extending the Shortest Path lij(m)= min {lik(m-1) + wkj} 1  k  n j i * = L(m-1) W L(m) n x n Replace: min  + +   Computing L(m) looks like matrix multiplication COSC3101A

  10. EXTEND(L, W, n) • create L’, an n × n matrix • for i ← 1to n • do for j ← 1to n • do lij’ ←∞ • for k ← 1to n • do lij’ ← min(lij’, lik + wkj) • return L’ Running time: (n3) lij(m)= min {lik(m-1) + wkj} 1  k  n COSC3101A

  11. SLOW-ALL-PAIRS-SHORTEST-PATHS(W, n) • L(1) ← W • for m ← 2 to n - 1 • do L(m) ←EXTEND (L(m - 1), W, n) • return L(n - 1) Running time: (n4) COSC3101A

  12. 2 4 3 8 1 3 2 1 -4 -5 7 5 4 6 Example W L(m-1) = L(1) 0 3 8 2 -4 3 0 -4 1 7 … and so on until L(4) L(m) = L(2) 4 0 5 11  2 -1 -5 0 -2 8 1 6 0  COSC3101A

  13. Example l14(2) = (0 3 8  -4)   1  0 6 = min (, 4, , , 2) = 2 COSC3101A

  14. Relation to Matrix Multiplication C = a  b • cij = Σk=1n aik bkj Compare: D(m) = d(m-1) C  dij(m) = min1≤k≤n {dik(m-1) + ckj} COSC3101A

  15. Improving Running Time • No need to compute all L(m) matrices • If no negative-weight cycles exist: L(m) = L(n - 1) for all m  n – 1 • We can compute L(n-1) by computing the sequence: L(1) = W L(2) = W2 = W  W L(4) = W4 = W2 W2 L(8) = W8 = W4 W4 … COSC3101A

  16. FASTER-APSP(W, n) • L(1) ← W • m ← 1 • while m < n - 1 • do L(2m) ← EXTEND(L(m), L(m), n) • m ← 2m • return L(m) • OK to overshoot: products don’t change after L(n - 1) • Running Time: (n3lg n) COSC3101A

  17. 2 4 3 8 1 3 2 1 -4 -5 7 5 4 6 The Floyd-Warshall Algorithm • Given: • Directed, weighted graph G = (V, E) • Negative-weight edges may be present • No negative-weight cycles could be present in the graph • Compute: • The shortest paths between all pairs of vertices in a graph COSC3101A

  18. From A one may reach C faster by traveling from A to B and then from B to C. The Idea COSC3101A

  19. 3 6 0.5 1 4 3 2 1 2 5 The Structure of a Shortest Path • Vertices in G are given by V = {1, 2, …, n} • Consider a path p = v1, v2, …, vl • An intermediate vertex of p is any vertex in the set {v2, v3, …, vl-1} • E.g.: p = 1, 2, 4, 5: {2, 4} p = 2, 4, 5: {4} 2 COSC3101A

  20. The Structure of a Shortest Path • For any pair of vertices i, j  V, consider all paths from i to j whose intermediate vertices are all drawn from a subset {1, 2, …, k} • Let p be a minimum-weight path from these paths p1 pu j i pt No vertex on these paths has index > k COSC3101A

  21. 3 6 0.5 1 2 4 3 2 1 Example dij(k) = the weight of a shortest path from vertex i to vertex j with all intermediary vertices drawn from {1, 2, …, k} • d13(0) = • d13(1) = • d13(2) = • d13(3) = • d13(4) = 6 6 5 5 4.5 COSC3101A

  22. k j i The Structure of a Shortest Path • k is not an intermediate vertex of path p • Shortest path from i to j with intermediate vertices from {1, 2, …, k} is a shortest path from i to j with intermediate vertices from {1, 2, …, k - 1} • k is an intermediate vertex of path p • p1 is a shortest path from i to k • p2 is a shortest path from k to j • k is not intermediary vertex of p1, p2 • p1 and p2 are shortest paths from i to k with vertices from {1, 2, …, k - 1} k j i p1 p2 COSC3101A

  23. A Recursive Solution (cont.) • k = 0 • dij(k) = dij(k) = the weight of a shortest path from vertex i to vertex j with all intermediary vertices drawn from {1, 2, …, k} wij COSC3101A

  24. A Recursive Solution (cont.) dij(k) = the weight of a shortest path from vertex i to vertex j with all intermediary vertices drawn from {1, 2, …, k} • k  1 • Case 1: k is not an intermediate vertex of path p • dij(k) = k j i dij(k-1) COSC3101A

  25. A Recursive Solution (cont.) dij(k) = the weight of a shortest path from vertex i to vertex j with all intermediary vertices drawn from {1, 2, …, k} • k  1 • Case 2: k is an intermediate vertex of path p • dij(k) =  k j i dik(k-1) + dkj(k-1) COSC3101A

  26. i j + Computing the Shortest Path Weights • dij(k) = wij if k = 0 min {dij(k-1) , dik(k-1) + dkj(k-1) } if k  1 • The final solution: D(n) = (dij(n)): dij(n) = (i, j)  i, j  V j (k, j) i (i, k) D(k-1) D(k) COSC3101A

  27. 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 dij(k) ← min (dij(k-1), dik(k-1) + dkj(k-1)) • return D(n) • Running time: (n3) COSC3101A

  28. 2 0 3 8 -4 4 3  8 0 1 7   1 3 2 4 0    1 -4 -5 7 2 0 5 4 6 6 0    1 2 3 4 5 D(4) D(3) 0 3 8 4 -4 0 3 8 -4 0 3 4 -4 1 0 1 7 0 1 7 0 1     2 4 0 5 11 4 0 4 0 5   3 2 -5 0 -2 2 5 -5 0 -2 2 -1 -5 0 -2 4 6 0 6 0 6 0       5 Example dij(k) = min {dij(k-1) , dik(k-1) + dkj(k-1) } D(1) D(0) = W 1 2 3 4 5 1 2 3 4 5 1 1 2 2 3 3 5 -5 -2 4 4 5 5 D(2) 4 -1 3 -4 -1 5 11 7 3 -1 8 5 1 COSC3101A

  29. D E B C A Transitive Closure • Given a graph G, the transitive closure of G is the G* such that • G* has the same vertices as G • if G has a path from u to v (u  v), G* has a edge from u to v D E B G C A The transitive closure provides reachability information about a graph. G* COSC3101A

  30. Transitive Closure of a Directed Graph Def: the transitive closure of G is defined as the graph G*= (V, E*) where E* = {(i,j): there is a path from i to j in G} Given a directed graph G = (V, E) we want to determine if there is a path from i to j for all vertex pairs COSC3101A

  31. Algorithms • The transitive closure can be computed using Floyd-Warshall (i.e. There is a path if ) • There is a more direct approach that can save time in practice and it is very similar to Floyd-Warshall: Let be a boolean value indicting whether there is a path from i to j or not with intermediary vertices in the set { 1,2,…, k}. (I,j) is placed in the set E* iff = 1 ={ 0 if i  j and (i, j)  E 1 if i = j or (i, j)  E COSC3101A

  32. Transitive-Closure Pseudo Code Transitive-Closure(G) (n3) 1 n  | V(G)| 2 for i 1 to n 3 do for j  1 to n 4 do if i = j or (i,j) E[G] 5 then tij(0) 1 6 else tij(0) 0 7 for k  1 to n 8 for i 1 to n 9 for j  1 to n 10 11 return T(n) COSC3101A

  33. Readings • Chapter 25 COSC3101A

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