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All Pairs Shortest Paths in Sparse Graphs

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  • First Online:
Encyclopedia of Algorithms

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Years and Authors of Summarized Original Work

  • 2004; Pettie

Problem Definition

Given a communications network or road network, one of the most natural algorithmic questions is how to determine the shortest path from one point to another. The all pairs shortest path problem (APSP) is, given a directed graph G = (V, E, l), to determine the distance and shortest path between every pair of vertices, where \(\vert V \vert = n,\vert E\vert = m\), and l : E → ℝ is the edge length (or weight) function. The output is in the form of two n × n matrices: D(u, v) is the distance from u to v and S(u, v) = w if (u, w) is the first edge on a shortest path from u to v. The APSP problem is often contrasted with the point-to-point and single source (SSSP) shortest path problems. They ask for, respectively, the shortest path from a given source vertex to a given target vertex and all shortest paths from a given source vertex.

Definition of Distance

If ℓassigns only non-negative edge lengths then the...

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Authors and Affiliations

  1. Electrical Engineering and Computer Science (EECS) Department, University of Michigan, Ann Arbor, MI, USA

    Seth Pettie

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  1. Seth Pettie
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Correspondence to Seth Pettie .

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Editors and Affiliations

  1. Department of Electrical Engineering and Computer Science, Northwestern University, Evanston, IL, USA

    Ming-Yang Kao

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Pettie, S. (2016). All Pairs Shortest Paths in Sparse Graphs. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2864-4_11

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