Document Open Access Logo

Shortest Beer Path Queries in Interval Graphs

Authors Rathish Das, Meng He, Eitan Kondratovsky, J. Ian Munro, Anurag Murty Naredla, Kaiyu Wu

PDF
Thumbnail PDF

File

LIPIcs.ISAAC.2022.59.pdf
  • Filesize: 0.86 MB
  • 17 pages

Document Identifiers

Author Details

Rathish Das
  • Department of Computer Science, University of Liverpool, UK
Meng He
  • Faculty of Computer Science, Dalhousie University, Halifax, Canada
Eitan Kondratovsky
  • Cheriton School of Computer Science, University of Waterloo, Canada
J. Ian Munro
  • Cheriton School of Computer Science, University of Waterloo, Canada
Anurag Murty Naredla
  • Cheriton School of Computer Science, University of Waterloo, Canada
Kaiyu Wu
  • Cheriton School of Computer Science, University of Waterloo, Canada

Cite As Get BibTex

Rathish Das, Meng He, Eitan Kondratovsky, J. Ian Munro, Anurag Murty Naredla, and Kaiyu Wu. Shortest Beer Path Queries in Interval Graphs. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 59:1-59:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ISAAC.2022.59

Abstract

Our interest is in paths between pairs of vertices that go through at least one of a subset of the vertices known as beer vertices. Such a path is called a beer path, and the beer distance between two vertices is the length of the shortest beer path.
We show that we can represent unweighted interval graphs using 2n log n + O(n) + O(|B|log n) bits where |B| is the number of beer vertices. This data structure answers beer distance queries in O(log^ε n) time for any constant ε > 0 and shortest beer path queries in O(log^ε n + d) time, where d is the beer distance between the two nodes. We also show that proper interval graphs may be represented using 3n + o(n) bits to support beer distance queries in O(f(n)log n) time for any f(n) ∈ ω(1) and shortest beer path queries in O(d) time. All of these results also have time-space trade-offs.
Lastly we show that the information theoretic lower bound for beer proper interval graphs is very close to the space of our structure, namely log(4+2√3)n - o(n) (or about 2.9 n) bits.

Subject Classification

ACM Subject Classification
  • Theory of computation → Data compression
  • Mathematics of computing → Paths and connectivity problems
Keywords
  • Beer Path
  • Interval Graph

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Hüseyin Acan, Sankardeep Chakraborty, Seungbum Jo, and Srinivasa Rao Satti. Succinct encodings for families of interval graphs. Algorithmica, 83(3):776-794, 2021. URL: https://doi.org/10.1007/s00453-020-00710-w.
  2. Joyce Bacic, Saeed Mehrabi, and Michiel Smid. Shortest beer path queries in outerplanar graphs. In Hee-Kap Ahn and Kunihiko Sadakane, editors, 32nd International Symposium on Algorithms and Computation, ISAAC 2021, December 6-8, 2021, Fukuoka, Japan, volume 212 of LIPIcs, pages 62:1-62:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2021.62.
  3. Amotz Bar-Noy, Reuven Bar-Yehuda, Ari Freund, Joseph Naor, and Baruch Schieber. A unified approach to approximating resource allocation and scheduling. J. ACM, 48(5):1069-1090, 2001. URL: https://doi.org/10.1145/502102.502107.
  4. Kellogg S. Booth and George S. Lueker. Testing for the consecutive ones property, interval graphs, and graph planarity using pq-tree algorithms. J. Comput. Syst. Sci., 13(3):335-379, 1976. URL: https://doi.org/10.1016/S0022-0000(76)80045-1.
  5. Timothy M. Chan, Kasper Green Larsen, and Mihai Patrascu. Orthogonal range searching on the ram, revisited. In Ferran Hurtado and Marc J. van Kreveld, editors, Proceedings of the 27th ACM Symposium on Computational Geometry, Paris, France, June 13-15, 2011, pages 1-10. ACM, 2011. URL: https://doi.org/10.1145/1998196.1998198.
  6. Rathish Das, Meng He, Eitan Kondratovsky, J. Ian Munro, Anurag Murty Naredla, and Kaiyu Wu. Shortest beer path queries in interval graphs, 2022. URL: http://arxiv.org/abs/2209.14401.
  7. Martin Charles Golumbic. Algorithmic graph theory and perfect graphs. Elsevier, 2004. Google Scholar
  8. G Hajós. Über eine art von graphen. int. Math. Nachr, 11:1607-1620, 1957. Google Scholar
  9. Meng He, J. Ian Munro, Yakov Nekrich, Sebastian Wild, and Kaiyu Wu. Distance oracles for interval graphs via breadth-first rank/select in succinct trees. In Yixin Cao, Siu-Wing Cheng, and Minming Li, editors, 31st International Symposium on Algorithms and Computation, ISAAC 2020, December 14-18, 2020, Hong Kong, China (Virtual Conference), volume 181 of LIPIcs, pages 25:1-25:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2020.25.
  10. Pavel Klavík­ and Peter Zeman. Automorphism Groups of Geometrically Represented Graphs. In Ernst W. Mayr and Nicolas Ollinger, editors, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015), volume 30 of Leibniz International Proceedings in Informatics (LIPIcs), pages 540-553, Dagstuhl, Germany, 2015. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: https://doi.org/10.4230/LIPIcs.STACS.2015.540.
  11. C Lekkeikerker and Johan Boland. Representation of a finite graph by a set of intervals on the real line. Fundamenta Mathematicae, 51:45-64, 1962. Google Scholar
  12. J. Ian Munro, Venkatesh Raman, and S. Srinivasa Rao. Space efficient suffix trees. J. Algorithms, 39(2):205-222, 2001. URL: https://doi.org/10.1006/jagm.2000.1151.
  13. Yakov Nekrich. New data structures for orthogonal range reporting and range minima queries. In Dániel Marx, editor, Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10 - 13, 2021, pages 1191-1205. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.73.
  14. Mihai Patrascu. Succincter. In 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008, October 25-28, 2008, Philadelphia, PA, USA, pages 305-313. IEEE Computer Society, 2008. URL: https://doi.org/10.1109/FOCS.2008.83.
  15. G. Pólya. Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen. Acta Mathematica, 68(none):145-254, 1937. URL: https://doi.org/10.1007/BF02546665.
  16. N. Sloane. The on-line encyclopedia of integer sequences. Notices Amer. Math. Soc., 50:912-915, September 2003. Google Scholar
  17. Dan E. Willard. Log-logarithmic worst-case range queries are possible in space theta(n). Inf. Process. Lett., 17(2):81-84, 1983. URL: https://doi.org/10.1016/0020-0190(83)90075-3.
  18. Peisen Zhang, Eric A. Schon, Stuart G. Fischer, Eftihia Cayanis, Janie Weiss, Susan Kistler, and Philip E. Bourne. An algorithm based on graph theory for the assembly of contigs in physical mapping of DNA. Comput. Appl. Biosci., 10(3):309-317, 1994. URL: https://doi.org/10.1093/bioinformatics/10.3.309.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact