Closing the Gap: Minimum Space Optimal Time Distance Labeling Scheme for Interval Graphs

Authors Meng He , Kaiyu Wu



PDF
Thumbnail PDF

File

LIPIcs.CPM.2024.17.pdf
  • Filesize: 1.07 MB
  • 18 pages

Document Identifiers

Author Details

Meng He
  • Faculty of Computer Science, Dalhousie University, Halifax, Canada
Kaiyu Wu
  • Faculty of Computer Science, Dalhousie University, Halifax, Canada

Cite AsGet BibTex

Meng He and Kaiyu Wu. Closing the Gap: Minimum Space Optimal Time Distance Labeling Scheme for Interval Graphs. In 35th Annual Symposium on Combinatorial Pattern Matching (CPM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 296, pp. 17:1-17:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.CPM.2024.17

Abstract

We present a distance labeling scheme for an interval graph on n vertices that uses at most 3lg n + lg lg n + O(1) bits per vertex to answer distance queries, which ask for the distance between two given vertices, in constant time. Our labeling scheme improves the distance labeling scheme of Gavoille and Paul for connected interval graphs which uses at most 5lg n + O(1) bits per vertex to achieve constant query time. Our improved space cost matches a lower bound proven by Gavoille and Paul within additive lower order terms and is thus optimal. Based on this scheme, we further design a 6lg n + 2lg lg n +O(1) bit distance labeling scheme for circular-arc graphs, with constant distance query time, which improves the 10lg n + O(1) bit distance labeling scheme of Gavoille and Paul. We give a n/2 + O(lg^ 2n) bit labeling scheme for chordal graphs which answers distance queries in O(1) time. The best known lower bound is n/4 - o(n) bits.

Subject Classification

ACM Subject Classification
  • Theory of computation → Data structures design and analysis
  • Theory of computation → Data compression
Keywords
  • Distance Labeling
  • Interval Graph
  • Circular-Arc Graph
  • Chordal Graph

Metrics

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

References

  1. Ittai Abraham and Cyril Gavoille. On approximate distance labels and routing schemes with affine stretch. In David Peleg, editor, Distributed Computing - 25th International Symposium, DISC 2011, Rome, Italy, September 20-22, 2011. Proceedings, volume 6950 of Lecture Notes in Computer Science, pages 404-415. Springer, 2011. URL: https://doi.org/10.1007/978-3-642-24100-0_39.
  2. Hüseyin Acan, Sankardeep Chakraborty, Seungbum Jo, and Srinivasa Rao Satti. Succinct data structures for families of interval graphs. In Algorithms and Data Structures - 16th International Symposium, WADS 2019, Edmonton, AB, Canada, August 5-7, 2019, Proceedings, pages 1-13, 2019. URL: https://doi.org/10.1007/978-3-030-24766-9_1.
  3. Stephen Alstrup, Philip Bille, and Theis Rauhe. Labeling schemes for small distances in trees. SIAM Journal on Discrete Mathematics, 19(2):448-462, 2005. URL: https://doi.org/10.1137/S0895480103433409.
  4. Stephen Alstrup, Søren Dahlgaard, and Mathias Bæk Tejs Knudsen. Optimal induced universal graphs and adjacency labeling for trees. In 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, pages 1311-1326, 2015. URL: https://doi.org/10.1109/FOCS.2015.84.
  5. Stephen Alstrup, Cyril Gavoille, Esben Bistrup Halvorsen, and Holger Petersen. Simpler, faster and shorter labels for distances in graphs. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '16, pages 338-350, USA, 2016. Society for Industrial and Applied Mathematics. Google Scholar
  6. Stephen Alstrup, Inge Li Gørtz, Esben Bistrup Halvorsen, and Ely Porat. Distance Labeling Schemes for Trees. In Ioannis Chatzigiannakis, Michael Mitzenmacher, Yuval Rabani, and Davide Sangiorgi, editors, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016), volume 55 of Leibniz International Proceedings in Informatics (LIPIcs), pages 132:1-132:16, Dagstuhl, Germany, 2016. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ICALP.2016.132.
  7. Stephen Alstrup, Esben Bistrup Halvorsen, and Kasper Green Larsen. Near-optimal labeling schemes for nearest common ancestors. In Proceedings of the 2014 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 972-982, 2014. URL: https://doi.org/10.1137/1.9781611973402.72.
  8. Stephen Alstrup, Haim Kaplan, Mikkel Thorup, and Uri Zwick. Adjacency labeling schemes and induced-universal graphs. SIAM Journal on Discrete Mathematics, 33(1):116-137, 2019. URL: https://doi.org/10.1137/16M1105967.
  9. Amotz Bar-Noy, Reuven Bar-Yehuda, Ari Freund, Joseph Naor, and Baruch Schieber. A unified approach to approximating resource allocation and scheduling. In F. Frances Yao and Eugene M. Luks, editors, Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, May 21-23, 2000, Portland, OR, USA, pages 735-744. ACM, 2000. URL: https://doi.org/10.1145/335305.335410.
  10. Marthe Bonamy, Louis Esperet, Carla Groenland, and Alex Scott. Optimal labelling schemes for adjacency, comparability, and reachability. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021, pages 1109-1117, New York, NY, USA, 2021. Association for Computing Machinery. URL: https://doi.org/10.1145/3406325.3451102.
  11. Kellogg S. Booth and George S. Lueker. Testing for the consecutive ones property, interval graphs, and graph planarity using pq-tree algorithms. Journal of Computer and System Sciences, 13(3):335-379, 1976. URL: https://doi.org/10.1016/S0022-0000(76)80045-1.
  12. Danny Z. Chen, D. T. Lee, R. Sridhar, and Chandra N. Sekharan. Solving the all-pair shortest path query problem on interval and circular-arc graphs. Networks, 31(4):249-258, 1998. URL: https://onlinelibrary.wiley.com/doi/abs/10.1002/%28SICI%291097-0037%28199807%2931%3A4%3C249%3A%3AAID-NET5%3E3.0.CO%3B2-D.
  13. Lenore J Cowen. Compact routing with minimum stretch. Journal of Algorithms, 38(1):170-183, 2001. URL: https://doi.org/10.1006/jagm.2000.1134.
  14. Tamar Eilam, Cyril Gavoille, and David Peleg. Compact routing schemes with low stretch factor. Journal of Algorithms, 46(2):97-114, 2003. URL: https://doi.org/10.1016/S0196-6774(03)00002-6.
  15. Pierre Fraigniaud and Amos Korman. An optimal ancestry labeling scheme with applications to XML trees and universal posets. J. ACM, 63(1), February 2016. URL: https://doi.org/10.1145/2794076.
  16. Ofer Freedman, Paweł Gawrychowski, Patrick K. Nicholson, and Oren Weimann. Optimal distance labeling schemes for trees. In Proceedings of the ACM Symposium on Principles of Distributed Computing, PODC '17, pages 185-194, New York, NY, USA, 2017. Association for Computing Machinery. URL: https://doi.org/10.1145/3087801.3087804.
  17. Cyril Gavoille and Christophe Paul. Optimal distance labeling for interval graphs and related graph families. SIAM J. Discret. Math., 22(3):1239-1258, July 2008. URL: https://doi.org/10.1137/050635006.
  18. Cyril Gavoille and David Peleg. Compact and localized distributed data structures. Distrib. Comput., 16(2–3):111-120, September 2003. URL: https://doi.org/10.1007/s00446-002-0073-5.
  19. Cyril Gavoille, David Peleg, Stéphane Pérennes, and Ran Raz. Distance labeling in graphs. Journal of Algorithms, 53(1):85-112, 2004. URL: https://doi.org/10.1016/j.jalgor.2004.05.002.
  20. Fǎnicǎ Gavril. The intersection graphs of subtrees in trees are exactly the chordal graphs. Journal of Combinatorial Theory, Series B, 16(1):47-56, 1974. URL: https://doi.org/10.1016/0095-8956(74)90094-X.
  21. Paweł Gawrychowski, Fabian Kuhn, Jakub Łopuszański, Konstantinos Panagiotou, and Pascal Su. Labeling schemes for nearest common ancestors through minor-universal trees. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '18, pages 2604-2619, USA, 2018. Society for Industrial and Applied Mathematics. Google Scholar
  22. Pawel Gawrychowski and Przemyslaw Uznanski. Better distance labeling for unweighted planar graphs. Algorithmica, 85(6):1805-1823, 2023. URL: https://doi.org/10.1007/S00453-023-01133-Z.
  23. 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.
  24. Michal Katz, Nir A. Katz, and David Peleg. Distance labeling schemes for well-separated graph classes. Discrete Applied Mathematics, 145(3):384-402, 2005. URL: https://doi.org/10.1016/j.dam.2004.03.005.
  25. Hung Le and Christian Wulff-Nilsen. Optimal approximate distance oracle for planar graphs. In 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO, USA, February 7-10, 2022, pages 363-374. IEEE, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00044.
  26. Yaowei Long and Seth Pettie. Planar distance oracles with better time-space tradeoffs. In Dániel Marx, editor, Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10 - 13, 2021, pages 2517-2537. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.149.
  27. J. Ian Munro and Kaiyu Wu. Succinct data structures for chordal graphs. In 29th International Symposium on Algorithms and Computation, ISAAC 2018, December 16-19, 2018, Jiaoxi, Yilan, Taiwan, pages 67:1-67:12, 2018. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2018.67.
  28. Mihai Patrascu and Liam Roditty. Distance oracles beyond the thorup-zwick bound. SIAM J. Comput., 43(1):300-311, 2014. URL: https://doi.org/10.1137/11084128X.
  29. David Peleg. Informative labeling schemes for graphs. Theoretical Computer Science, 340(3): 577-593, 2005. Mathematical Foundations of Computer Science 2000. URL: https://doi.org/10.1016/j.tcs.2005.03.015.
  30. Fernando Magno Quintão Pereira and Jens Palsberg. Register allocation via coloring of chordal graphs. In Kwangkeun Yi, editor, Programming Languages and Systems, pages 315-329, Berlin, Heidelberg, 2005. Springer Berlin Heidelberg. Google Scholar
  31. Gaurav Singh, N. S. Narayanaswamy, and G. Ramakrishna. Approximate distance oracle in o(n 2) time and o(n) space for chordal graphs. In M. Sohel Rahman and Etsuji Tomita, editors, WALCOM: Algorithms and Computation - 9th International Workshop, WALCOM 2015, Dhaka, Bangladesh, February 26-28, 2015. Proceedings, volume 8973 of Lecture Notes in Computer Science, pages 89-100. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-15612-5_9.
  32. Virginia Vassilevska Williams, Yinzhan Xu, Zixuan Xu, and Renfei Zhou. New bounds for matrix multiplication: from alpha to omega. In David P. Woodruff, editor, Proceedings of the 2024 ACM-SIAM Symposium on Discrete Algorithms, SODA 2024, Alexandria, VA, USA, January 7-10, 2024, pages 3792-3835. SIAM, 2024. URL: https://doi.org/10.1137/1.9781611977912.134.
  33. Nicholas C. Wormald. Counting labelled chordal graphs. Graphs and Combinatorics, 1(1):193-200, 1985. URL: https://doi.org/10.1007/BF02582944.
  34. 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. Computer Applications in the Biosciences, 10(3):309-317, 1994. URL: https://doi.org/10.1093/bioinformatics/10.3.309.