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Distance Queries over Dynamic Interval Graphs

Authors Jingbang Chen , Meng He , J. Ian Munro , Richard Peng, Kaiyu Wu , Daniel J. Zhang



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Author Details

Jingbang Chen
  • Cheriton School of Computer Science, University of Waterloo, Canada
Meng He
  • Faculty of Computer Science, Dalhousie University, Halifax, Canada
J. Ian Munro
  • Cheriton School of Computer Science, University of Waterloo, Canada
Richard Peng
  • Cheriton School of Computer Science, University of Waterloo, Canada
Kaiyu Wu
  • Cheriton School of Computer Science, University of Waterloo, Canada
Daniel J. Zhang
  • School of Computer Science, Georgia Tech, Atlanta, GA, USA

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Jingbang Chen, Meng He, J. Ian Munro, Richard Peng, Kaiyu Wu, and Daniel J. Zhang. Distance Queries over Dynamic Interval Graphs. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 18:1-18:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ISAAC.2023.18

Abstract

We design the first dynamic distance oracles for interval graphs, which are intersection graphs of a set of intervals on the real line, and for proper interval graphs, which are intersection graphs of a set of intervals in which no interval is properly contained in another. For proper interval graphs, we design a linear space data structure which supports distance queries (computing the distance between two query vertices) and vertex insertion or deletion in O(lg n) worst-case time, where n is the number of vertices currently in G. Under incremental (insertion only) or decremental (deletion only) settings, we design linear space data structures that support distance queries in O(lg n) worst-case time and vertex insertion or deletion in O(lg n) amortized time, where n is the maximum number of vertices in the graph. Under fully dynamic settings, we design a data structure that represents an interval graph G in O(n) words of space to support distance queries in O(n lg n/S(n)) worst-case time and vertex insertion or deletion in O(S(n)+lg n) worst-case time, where n is the number of vertices currently in G and S(n) is an arbitrary function that satisfies S(n) = Ω(1) and S(n) = O(n). This implies an O(n)-word solution with O(√{nlg n})-time support for both distance queries and updates. All four data structures can answer shortest path queries by reporting the vertices in the shortest path between two query vertices in O(lg n) worst-case time per vertex. We also study the hardness of supporting distance queries under updates over an intersection graph of 3D axis-aligned line segments, which generalizes our problem to 3D. Finally, we solve the problem of computing the diameter of a dynamic connected interval graph.

Subject Classification

ACM Subject Classification
  • Theory of computation → Data structures design and analysis
  • Information systems → Data structures
Keywords
  • interval graph
  • proper interval graph
  • intersection graph
  • geometric intersection graph
  • distance oracle
  • distance query
  • shortest path query
  • dynamic graph

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