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Shortest Paths

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Abstract

Computing shortest paths in a given graph is one of the fundamental problems in computer science. The input is given by a connected directed graph G=(V,E) where V={v 1,…,v n } is a set of n vertices and E is a set of m edges. In addition, there is a weight function w:E→ℕ which assigns positive integer weights to the edges. We denote by w max =max  eE w(e) the maximum of the weights of all edges and distinguish between two problems. In the single-source shortest-path (SSSP) problem, there is one designated vertex sV and the task is to compute a shortest path from s to every other vertex v i V∖{s}. W. l. o. g., we assume s=v 1 throughout this chapter. The length of a path is measured by the sum of the weights of the edges that are used in this path. A generalization of the SSSP problem is the all-pairs shortest-path (APSP) problem, where the task is to compute from each vertex v i V a shortest path to every other vertex v j V∖{v i }. The SSSP and APSP problems can be solved by using Dijkstra’s algorithm and the Floyd-Warshall algorithm, respectively. Using appropriate data structures, single-source shortest paths and all-pairs shortest paths can be computed in time O(m+nlog n) and O(nm+n 2log n), respectively (Mehlhorn and Sanders, 2008).

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  • DOI: 10.1007/978-3-642-16544-3_8
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References

  • Mehlhorn K, Sanders P (2008) Algorithms and Data Structures: The Basic Toolbox. Springer

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Correspondence to Carsten Witt .

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© 2010 Springer-Verlag Berlin Heidelberg

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Neumann, F., Witt, C. (2010). Shortest Paths. In: Bioinspired Computation in Combinatorial Optimization. Natural Computing Series. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16544-3_8

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  • DOI: https://doi.org/10.1007/978-3-642-16544-3_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-16543-6

  • Online ISBN: 978-3-642-16544-3

  • eBook Packages: Computer ScienceComputer Science (R0)