APSP (All-Pairs Shortest Paths (APSP))

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Description

The shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized.

Related Problems

Subproblem: APSP on Dense Directed Graphs with Arbitrary Weights, APSP on Dense Undirected Graphs with Arbitrary Weights, APSP on Geometrically Weighted Graphs, APSP on Dense Undirected Graphs with Positive Integer Weights, APSP on Sparse Directed Graphs with Arbitrary Weights, APSP on Sparse Undirected Graphs with Positive Integer Weights, APSP on Sparse Undirected Graphs with Arbitrary Weights, APSP on Dense Directed Unweighted Graphs, APSP on Dense Undirected Unweighted Graphs, APSP on Sparse Directed Unweighted Graphs, APSP on Sparse Undirected Unweighted Graphs, (5/3)-approximate ap-shortest paths

Related: APSP on Dense Undirected Graphs with Arbitrary Weights, APSP on Geometrically Weighted Graphs, APSP on Dense Undirected Graphs with Positive Integer Weights, APSP on Sparse Directed Graphs with Arbitrary Weights, APSP on Sparse Undirected Graphs with Positive Integer Weights, APSP on Sparse Undirected Graphs with Arbitrary Weights, APSP on Dense Directed Unweighted Graphs, APSP on Dense Undirected Unweighted Graphs, APSP on Sparse Directed Unweighted Graphs, APSP on Sparse Undirected Unweighted Graphs, (5/3)-approximate ap-shortest paths

Parameters

$n$: number of vertices

$m$: number of edges

Table of Algorithms

Name Year Time Space Approximation Factor Model Reference
Shimbel Algorithm 1953 $O(n^{4})$ $O(n^{2})$ Exact Deterministic Time
Floyd–Warshall algorithm 1962 $O(n^{3})$ $O(n^{2})$ Exact Deterministic Time
Seidel's algorithm 1995 $O (n^{2.{37}3} \log n)$ $O(n^{2})$ Exact Deterministic Time
Williams 2014 $O(n^{3} /{2}^{(\log n)^{0.5}})$ $O(n^{2})$ Exact Deterministic Time
Pettie & Ramachandran 2002 $O(mn \log \alpha(m,n))$ Exact Deterministic Time
Thorup 1999 $O(mn)$ $O(mn)$ Exact Deterministic Time & Space
Chan (Geometrically Weighted) 2009 $O(n^{2.{84}4})$ $O(l n^{2})$ Exact Deterministic Time
Chan 2009 $O(n^{3} \log^{3} \log n / \log^{2} n)$ $O(n^{2})$ Exact Deterministic Time

Time Complexity Graph

All-Pairs Shortest Paths (APSP) - APSP - Time.png