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Shortest Beer Path Queries in Interval Graphs

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

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

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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.

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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

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

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