License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ESA.2023.72
URN: urn:nbn:de:0030-drops-187257
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/18725/
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Kociumaka, Tomasz ; Polak, Adam

Bellman-Ford Is Optimal for Shortest Hop-Bounded Paths

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LIPIcs-ESA-2023-72.pdf (0.7 MB)

Abstract

This paper is about the problem of finding a shortest s-t path using at most h edges in edge-weighted graphs. The Bellman-Ford algorithm solves this problem in O(hm) time, where m is the number of edges. We show that this running time is optimal, up to subpolynomial factors, under popular fine-grained complexity assumptions.
More specifically, we show that under the APSP Hypothesis the problem cannot be solved faster already in undirected graphs with nonnegative edge weights. This lower bound holds even restricted to graphs of arbitrary density and for arbitrary h ∈ O(√m). Moreover, under a stronger assumption, namely the Min-Plus Convolution Hypothesis, we can eliminate the restriction h ∈ O(√m). In other words, the O(hm) bound is tight for the entire space of parameters h, m, and n, where n is the number of nodes.
Our lower bounds can be contrasted with the recent near-linear time algorithm for the negative-weight Single-Source Shortest Paths problem, which is the textbook application of the Bellman-Ford algorithm.

BibTeX - Entry

@InProceedings{kociumaka_et_al:LIPIcs.ESA.2023.72,
  author =	{Kociumaka, Tomasz and Polak, Adam},
  title =	{{Bellman-Ford Is Optimal for Shortest Hop-Bounded Paths}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{72:1--72:10},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-295-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{274},
  editor =	{G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2023/18725},
  URN =		{urn:nbn:de:0030-drops-187257},
  doi =		{10.4230/LIPIcs.ESA.2023.72},
  annote =	{Keywords: Fine-grained complexity, graph algorithms, lower bounds, shortest paths}
}

Keywords: Fine-grained complexity, graph algorithms, lower bounds, shortest paths
Collection: 31st Annual European Symposium on Algorithms (ESA 2023)
Issue Date: 2023
Date of publication: 30.08.2023

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