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All-Pairs Shortest Paths. Find the distance between every pair of vertices in a weighted graph G. From A one may reach C faster by reaching by traveling from A to B and then from B to C. All-Pairs Shortest Paths. We can make n calls to Dijkstra’s algorithm O(n(n+m)log n) time. - PowerPoint PPT Presentation

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  • All-Pairs Shortest PathsFind the distance between every pair of vertices in a weighted graph G.

  • All-Pairs Shortest PathsWe can make n calls to Dijkstras algorithm O(n(n+m)log n) timeAlternatively ... Use dynamic programming: The Floyds AlgorithmO(n3) time.

  • Floyds Shortest Path AlgorithmIdea #1: Number the vertices 1, 2, , n.Idea #2: Shortest path between vertex i and vertex j without passing through any other vertex is weight(edge(i,j)).Let it be D0(i,j).Idea #3: If Dk(i,j) is the shortest path between vertex i and vertex j using vertices numbered 1, 2, , k, as intermediate vertices, then Dn(i,j) is the solution to the shortest path problem between vertex i and vertex j.

  • Floyds Shortest Path AlgorithmAssume that we have Dk-1(i,j) for all i, and j.We wish to compute Dk(i,j).What are the possibilities? Choose between a path through vertex k. Path Length: Dk(i,j) = Dk-1(i,k) + Dk-1(k,j). (2) Skip vertex k altogether. Path Length: Dk(i,j) = Dk-1(i,j).

  • Floyds Shortest Path AlgorithmkjiUses only verticesnumbered 1,,k-1Uses only verticesnumbered 1,,k-1Uses only vertices numbered 1,,k(compute weight of this edge)

  • Floyds Shortest Path AlgorithmExample:4135211182438D0

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  • Floyds Shortest Path AlgorithmExample:4135211182438D1

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  • Floyds Shortest Path AlgorithmExample:4135211182438D2

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  • Floyds Shortest Path AlgorithmExample:4135211182438D3

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  • Floyds Shortest Path AlgorithmExample:4135211182438D4

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  • Floyds Shortest Path AlgorithmExample:4135211182438D5

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  • Floyds Shortest Path AlgorithmAlgorithm AllPair(G) {assumes vertices 1,,n} for all vertex pairs (i,j) if i = jd0[i,i] 0else if (i,j) is an edge in Gd0[i,j] weight of edge (i,j)else d0[i,j] + for k 1 to n do for i 1 to n do for j 1 to n do dk[i,j] min(dk-1[i,j], dk-1[i,k]+dk-1[k,j])O(n+m)orO(n2)

    O(n3)

  • Floyds Shortest Path AlgorithmObservation: dk [i,k] = dk-1[i,k] dk[k,j] = dk-1[k,j]

  • Floyds Shortest Path AlgorithmAlgorithm AllPair(G) {assumes vertices 1,,n} for all vertex pairs (i,j) if i = jd[i,i] 0else if (i,j) is an edge in Gd[i,j] weight of edge (i,j)else d[i,j] + for k 1 to n do for i 1 to n do for j 1 to n do d[i,j] min(d[i,j], d[i,k]+d[k,j])

  • Floyds Shortest Path AlgorithmAlgorithm AllPair(G) {assumes vertices 1,,n} for all vertex pairs (i,j) path[i,j] nullif i = jd[i,i] 0else if (i,j) is an edge in Gd[i,j] weight of edge (i,j)else d[i,j] + for k 1 to n do for i 1 to n do for j 1 to n do if (d[i,k]+d[k,j] < d[i,j]) path[i,j] vertex k d[i,j] d[i,k]+d[k,j]

  • Floyds Shortest Path AlgorithmCan be applied to Directed GraphsCan accept ve edges as long as there are no ve cycles.

  • Transitive ClosureTransitive Relation: A relation R on a set A is called transitive if and only if for any a, b, and c in A, whenever R , and R , R

  • Transitive ClosureGiven a graph G, the transitive closure of G is the G* such thatG* has the same vertices as Gif G has a path from u to v (u v), G* has a edge from u to vBADCEGG*The transitive closure providesreachability information about a graph.

  • Computing the Transitive ClosureWe can perform BFS starting at each vertexO(n(n+m))Alternatively ... Use dynamic programming: Warshalls Algorithm

  • Floyd-Warshall Transitive ClosureIdea #1: Number the vertices 1, 2, , n.Idea #2: Consider paths that use only vertices numbered 1, 2, , k, as intermediate vertices:kjiUses only verticesnumbered 1,,k-1Uses only verticesnumbered 1,,k-1Uses only vertices numbered 1,,k(add this edge if its not already in)

  • Warshalls Transitive ClosureExample:4132A0

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  • Warshalls Transitive ClosureExample:4132A1

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  • Warshalls Transitive ClosureExample:4132A2

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  • Warshalls Transitive ClosureExample:4132A3

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  • Warshalls Transitive ClosureExample:4132A4

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  • Warshalls AlgorithmWarshalls algorithm numbers the vertices of G as v1 , , vn and computes a series of graphs G0, , GnG0=G Gk has a edge (vi, vj) if G has a path from vi to vj with intermediate vertices in the set {v1 , , vk} We have that Gn = G*In phase k, graph Gk is computed from Gk - 1Running time: O(n3), assuming areAdjacent is O(1) (e.g., adjacency matrix)Algorithm FloydWarshall(G)Input graph GOutput transitive closure G* of Gi 1for all v G.vertices()denote v as vii i + 1G0 Gfor k 1 to n doGk Gk - 1 for i 1 to n (i k) dofor j 1 to n (j i, k) doif Gk - 1.areAdjacent(vi, vk) Gk - 1.areAdjacent(vk, vj) if Gk.areAdjacent(vi, vj) Gk.insertEdge(vi, vj , k)return Gn

  • Shortest Path Algorithm for DAGsUse Dijkstras algorithmTime: O((n+m) log n)Use Topological sort based algorithmTime: O(n+m).

  • Directed Graph With ve weightORDPVDMCODFWSFOLAXLGAHNL$170-$100$277-$200-$100$210$224$246$68$511$28$2610

  • Directed Graph With ve weightORD4PVD3MCO1DFW5SFO8LAX6LGA2HNL7$170-$100$277-$200-$100$210$224$246$68$511$28$2610

  • Directed Graph With ve weightORD4PVD3MCO1DFW5SFO8LAX6LGA2HNL7$170-$100$277-$200-$100$210$224$246$68$511$28$2610$210$224$224

  • Directed Graph With ve weightORD4PVD3MCO1DFW5SFO8LAX6LGA2HNL7$170-$100$277-$200-$100$210$224$246$68$511$28$2610$210$224$261

  • Directed Graph With ve weightORD4PVD3MCO1DFW5SFO8LAX6LGA2HNL7$170-$100$277-$200-$100$210$224$246$68$511$28$2610$210$224$238

  • Directed Graph With ve weightORD4PVD3MCO1DFW5SFO8LAX6LGA2HNL7$170-$100$277-$200-$100$210$224$246$68$511$28$2610$210$224$238$408

  • Directed Graph With ve weightORD4PVD3MCO1DFW5SFO8LAX6LGA2HNL7$170-$100$277-$200-$100$210$224$246$68$511$28$2610$210$224$238$408$208$308

  • Directed Graph With ve weightORD4PVD3MCO1DFW5SFO8LAX6LGA2HNL7$170-$100$277-$200-$100$210$224$246$68$511$28$2610$210$224$238$408$208$308

  • Directed Graph With ve weightORD4PVD3MCO1DFW5SFO8LAX6LGA2HNL7$170-$100$277-$200-$100$210$224$246$68$511$28$2610$210$224$238$408$208$268$719

  • Directed Graph With ve weightORD4PVD3MCO1DFW5SFO8LAX6LGA2HNL7$170-$100$277-$200-$100$210$224$246$68$511$28$2610$210$224$238$408$208$268$719

  • Directed Graph With ve weightORD4PVD3MCO1DFW5SFO8LAX6LGA2HNL7$170-$100$277-$200-$100$210$224$246$68$511$28$2610$210$224$238$408$208$268$719

  • DAG-based AlgorithmWorks even with negative-weight edgesUses topological orderDoesnt use any fancy data structuresIs much faster than Dijkstras algorithmRunning time: O(n+m).Algorithm DagDistances(G, s)for all v G.vertices()v.parent nullif v = sv.distance 0else v.distance Perform a topological sort of the verticesfor u 1 to n do {in topological order}for each e G.outEdges(u)w G.opposite(u,e)r getDistance(u) + weight(e)if r < getDistance(w)w.distance r w.parent u

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