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Shortest Beer Path Queries in Digraphs with Bounded Treewidth

Authors Joachim Gudmundsson, Yuan Sha

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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

  1. Noga Alon and Baruch Schieber. Optimal preprocessing for answering on-line product queries. Technical Report No.71/87, Tel Aviv University, 1987. Google Scholar
  2. Joyce Bacic, Saeed Mehrabi, and Michiel Smid. Shortest beer path queries in outerplanar graphs. In Proceedings of the 32nd International Symposium on Algorithms and Computation (ISAAC), volume 212 of LIPIcs, pages 62:1-62:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. Google Scholar
  3. Michael A. Bender and Martin Farach-Colton. The LCA problem revisited. In Proceedings of the 4th Latin American Symposium on Theoretical Informatics (LATIN), pages 88-94, 2000. Google Scholar
  4. Hans L. Bodlaender. A linear time algorithm for finding tree-decompositions of small treewidth. In Proceedings of the 25th Annual ACM Symposium on Theory of Computing (STOC), pages 226-234, 1993. Google Scholar
  5. Panagiotis Charalampopoulos, Pawel Gawrychowski, Shay Mozes, and Oren Weimann. Almost optimal distance oracles for planar graphs. In Proceedings of the 51st Symposium on Theory of Computing (STOC), pages 138-151. ACM, 2019. Google Scholar
  6. Shiva Chaudhuri and Christos D. Zaroliagis. Shortest paths in digraphs of small treewidth. part I: sequential algorithms. Algorithmica, 27(3):212-226, 2000. Google Scholar
  7. Bernard Chazelle. Computing on a free tree via complexity-preserving mappings. Algorithmica, 2:337-361, 1987. Google Scholar
  8. Danny Z. Chen and Jinhui Xu. Shortest path queries in planar graphs. In Proceedings of the 32nd Annual ACM Symposium on Theory of Computing (STOC), pages 469-478. ACM, 2000. Google Scholar
  9. Vincent Cohen-Addad, Søren Dahlgaard, and Christian Wulff-Nilsen. Fast and compact exact distance oracle for planar graphs. In Proceedings of the 58th IEEE Annual Symposium on Foundations of Computer Science (FOCS), pages 962-973. IEEE Computer Society, 2017. Google Scholar
  10. Hristo N. Djidjev. On-line algorithms for shortest path problems on planar digraphs. In Proceedings of the 22nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG), volume 1197 of Lecture Notes in Computer Science, pages 151-165, 1996. Google Scholar
  11. Jittat Fakcharoenphol and Satish Rao. Planar graphs, negative weight edges, shortest paths, near linear time. In 42nd Annual Symposium on Foundations of Computer Science, (FOCS), pages 232-241, 2001. Google Scholar
  12. Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, Michal Pilipczuk, and Marcin Wrochna. Fully polynomial-time parameterized computations for graphs and matrices of low treewidth. ACM Trans. Algorithms, 14(3):34:1-34:45, 2018. Google Scholar
  13. Viktor Fredslund-Hansen, Shay Mozes, and Christian Wulff-Nilsen. Truly subquadratic exact distance oracles with constant query time for planar graphs. In Proceedings of the 32nd International Symposium on Algorithms and Computation (ISAAC), volume 212 of LIPIcs, pages 25:1-25:12. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. Google Scholar
  14. Leonidas J. Guibas, John Hershberger, Daniel Leven, Micha Sharir, and Robert Endre Tarjan. Linear-time algorithms for visibility and shortest path problems inside triangulated simple polygons. Algorithmica, 2:209-233, 1987. Google Scholar
  15. Tesshu Hanaka, Hirotaka Ono, Kunihiko Sadakane, and Kosuke Sugiyama. Shortest beer path queries based on graph decomposition, 2023. URL: https://arxiv.org/abs/2307.02787.
  16. Dov Harel and Robert Endre Tarjan. Fast algorithms for finding nearest common ancestors. SIAM J. Comput., 13(2):338-355, 1984. Google Scholar
  17. Tuukka Korhonen. A single-exponential time 2-approximation algorithm for treewidth. In Proceedings of the 62nd IEEE Annual Symposium on Foundations of Computer Science (FOCS), pages 184-192, 2021. Google Scholar
  18. Yaowei Long and Seth Pettie. Planar distance oracles with better time-space tradeoffs. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2517-2537. SIAM, 2021. Google Scholar
  19. Shay Mozes and Christian Sommer. Exact distance oracles for planar graphs. In Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 209-222. SIAM, 2012. Google Scholar
  20. Seth Pettie. An inverse-ackermann type lower bound for online minimum spanning tree verification. Combinatorica, 26(2):207-230, 2006. Google Scholar
  21. Liam Roditty and Uri Zwick. On dynamic shortest paths problems. In Proceedings of the 12th Annual European Symposium (ESA), volume 3221 of Lecture Notes in Computer Science, pages 580-591. Springer, 2004. Google Scholar
  22. Mikkel Thorup and Uri Zwick. Approximate distance oracles. J. ACM, 52(1):1-24, 2005. Google Scholar
  23. Virginia Vassilevska Williams and Ryan Williams. Subcubic equivalences between path, matrix and triangle problems. In 51th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 645-654. IEEE Computer Society, 2010. Google Scholar
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