I t d ti t Al ith Introduction to Algorithms
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It d ti t Al ith Introduction to Algorithms All-Pairs Shortest Paths My T. Thai @ UF
Transcript of I t d ti t Al ith Introduction to Algorithms
I t d ti t Al ithIntroduction to Algorithms
All-Pairs Shortest Paths
My T. Thai @ UF
Single-Source Shortest Paths ProblemSingle-Source Shortest Paths Problem
Input: A weighted, directed graph G = (V, E) Output: An n × n matrix of shortest-path
distances δ. δ(i, j) is the weight of a shortest path ( j) g pfrom i to j.
1 2 3 4 5
1 0 1 ‐3 2 ‐4
2 3 0 ‐4 1 ‐1
3 7 4 0 5 3
4 2 ‐1 ‐5 0 ‐2
5 8 5 1 6 0
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5 8 5 1 6 0
Can use algorithms for Single-Source Shortest Paths
Run BELLMAN-FORD once from each vertex Time:
If there are no negative-weight edges, could run Dijkstra’s algorithm once from each vertexDijkstra s algorithm once from each vertex Time:
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OutlineOutline Shortest paths and matrix multiplication
Floyd-Warshall algorithm Floyd Warshall algorithm
Johnson’s algorithm Johnson’s algorithm
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Recursive solutionRecursive solution
Optimal substructure: subpaths of shortest paths are shortest paths
Recursive solution: Let = weight of shortestgpath from i to j that contains ≤ m edges.
WhWhere wij:My T. Thai
Computing the shortest-path weights bottom upbottom up
All simple shortest paths contain ≤ n − 1 edges
Compute from bottom up: L(1), L(2), . . . , L(n-1). Compute from bottom up: L , L , . . . , L . Compute L(i+1) from L(i) by extending oneL(i) by extending one more edge
iTime:
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Shortest paths and matrix multiplicationShortest paths and matrix multiplication Extending shortest paths by one more edge
lik t i d t L(i+1) L(i) Wlikes matrix product: L(i+1)= L(i).W Compute L(1), L(2), L(4) . . . , L(r) with
Time:Time:
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Floyd-Warshall algorithmFloyd Warshall algorithm For path p = <v1, v2, . . . , vl> , v2 … vl-1 are
intermediate vertices from v1 to vl
Define = shortest-path weight of any path from i to j with all intermediate vertices in {1, 2, . . . , k}
Consider a shortest path with all intermediate i i {1 2 k}vertices in {1, 2, . . . , k}:
If k is not an intermediate vertex, all intermediate vertices in {1 2 k -1}in {1, 2, . . . , k -1}
If k is an intermediate vertex:
Floyd-Warshall algorithmFloyd Warshall algorithm Recursive formula:
Time: N t i h t t Time: Note: since we have at most n vertices, return
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Constructing a shortest pathConstructing a shortest path is the predecessor of vertex j on a shortest
path from vertex i with all intermediate vertices in the set {1, 2, . . . , k}
(Use vertex k)
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Johnson’s algorithmJohnson s algorithm Reweighting edges to get non-negative weight
edges: For all u, v V, p is a shortest path using
w if and only if p is a shortest path using
For all (u, v) E, Run Dijkstra’s algorithm once from each vertexj g
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Proof of lemma 25 1Proof of lemma 25.1 First, we prove
i h l With cycle ,
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Producing nonnegative weightsProducing nonnegative weights Construct
Since no edges enter s, has the same set of cycles Gas G
has a negative-weight cycle if and only if G does
Define: Define: Claim: Proof: Triangle inequality of shortest paths Proof: Triangle inequality of shortest paths
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SummarySummary Dynamic-programming algorithm based on matrix
multiplication Define sub-optimal solutions based on the length of
thpaths
Use the technique of “repeated squaring” Time: Time:
Floyd-Warshall algorithm Define sub optimal solutions based on the set of Define sub-optimal solutions based on the set of
allowed intermediate vertices Time:
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Johnson’s algorithm Reweight edges to non-negative weight edges Run Dijkstra’s algorithm once from each vertex Time: Faster than Floyd-Warshall algorithm when the graph
is dense E = o(V2)
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