# APSP (All-Pairs Shortest Paths (APSP))

## 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 |