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Shortest Beer Path Queries in Digraphs with Bounded Treewidth

Authors Joachim Gudmundsson, Yuan Sha

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LIPIcs.ISAAC.2023.35.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Graph algorithms
  • Shortest Path
  • Data structures
  • Bounded treewidth

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Abstract

A beer digraph G is a real-valued weighted directed graph where some of the vertices have beer stores. A beer path from a vertex u to a vertex v in G is a path in G from u to v that visits at least one beer store.
In this paper we consider the online shortest beer path query in beer digraphs with bounded treewidth t. Assume that a tree decomposition of treewidth t on a beer digraph with n vertices is given. We show that after O(t³n) time preprocessing on the beer digraph, (i) a beer distance query can be answered in O(t³α(n)) time, where α(n) is the inverse Ackermann function, and (ii) a shortest beer path can be reported in O(t³α(n)L) time, where L is the number of edges on the path. In the process we show an improved O(t³α(n)L) time shortest path query algorithm, compared with the currently best O(t⁴α(n)L) time algorithm [Chaudhuri & Zaroliagis, 2000].
We also consider queries in a dynamic setting where the weight of an edge in G can change over time. We show two data structures. Assume t is constant and let β be any constant in (0,1). The first data structure uses O(n) preprocessing time, answers a beer distance query in O(α(n)) time and reports a shortest beer path in O(α(n) L) time. It can be updated in O(n^β) time after an edge weight change. The second data structure has O(n) preprocessing time, answers a beer distance query in O(log n) time, reports a shortest beer path in O(log n + L) time, and can be updated in O(log n) time after an edge weight change.

Cite As Get BibTex

Joachim Gudmundsson and Yuan Sha. Shortest Beer Path Queries in Digraphs with Bounded Treewidth. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 35:1-35:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ISAAC.2023.35

Author Details

Joachim Gudmundsson
  • The University of Sydney, Australia
Yuan Sha
  • The University of Sydney, Australia

References

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