Document Open Access Logo

Succinct Data Structures for Chordal Graph with Bounded Leafage or Vertex Leafage

Authors Meng He , Kaiyu Wu

PDF

File

Thumbnail PDF
PDF
LIPIcs.WADS.2025.35.pdf
  • Filesize: 1.09 MB
  • 23 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Data compression
  • Theory of computation → Data structures design and analysis
Keywords
  • Chordal Graph
  • Leafage
  • Vertex Leafage
  • Succinct Data Structure

Metrics

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

Abstract

We improve the recent succinct data structure result of Balakrishnan et al. for chordal graphs with bounded vertex leafage (SWAT 2024). A chordal graph is a widely studied graph class which can be characterized as the intersection graph of subtrees of a host tree, denoted as a tree representation of the chordal graph. The vertex leafage and leafage parameters of a chordal graph deal with the existence of a tree representation with a bounded number of leaves in either the subtrees representing the vertices or the host tree itself.
We simplify the lower bound proof of Balakrishnan et al. which applied to only chordal graphs with bounded vertex leafage, and extend it to a lower bound proof for chordal graphs with bounded leafage as well. For both classes of graphs, the information-theoretic lower bound we (re-)obtain for k = o(n) is (k-1)nlog n - knlog k - o(knlog n) bits, where the leafage or vertex leafage of the graph is at most k = o(n). We further extend the range of the parameter k to Θ(n) as well.
Then we give a succinct data structure using (k-1)nlog (n/k) + o(knlog n) bits to answer adjacent queries, which test the adjacency between pairs of vertices, in O((log k)/(log log n) + 1) time compared to the O(klog n) time of the data structure of Balakrishnan et al. For the neighborhood query which lists the neighbours of a given vertex, our query time is O((log n)/(log log n)) per neighbour compared to O(k²log n) per neighbour.
We also extend the data structure ideas to obtain a succinct data structure for chordal graphs with bounded leafage k, answering an open question of Balakrishnan et al. Our succinct data structure, which uses (k-1)nlog (n/k) + o(knlog n) bits, has query time O(1) for the adjacent query and O(1) per neighbour for the neighborhood query. Using slightly more space (an additional (1+ε)nlog n bits for any ε > 0) allows distance queries, which compute the number of edges in the shortest path between two given vertices, to be answered in O(1) time as well.

Cite As Get BibTex

Meng He and Kaiyu Wu. Succinct Data Structures for Chordal Graph with Bounded Leafage or Vertex Leafage. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 35:1-35:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.WADS.2025.35

Author Details

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

Funding

This work is supported by NSERC.

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. Girish Balakrishnan, Sankardeep Chakraborty, Seungbum Jo, N. S. Narayanaswamy, and Kunihiko Sadakane. Succinct data structure for graphs with d-dimensional t-representation. In Ali Bilgin, James E. Fowler, Joan Serra-Sagristà, Yan Ye, and James A. Storer, editors, Data Compression Conference, DCC 2024, Snowbird, UT, USA, March 19-22, 2024, page 546. IEEE, 2024. URL: https://doi.org/10.1109/DCC58796.2024.00063.
  3. Girish Balakrishnan, Sankardeep Chakraborty, N. S. Narayanaswamy, and Kunihiko Sadakane. Succinct data structure for chordal graphs with bounded vertex leafage. In Hans L. Bodlaender, editor, 19th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2024, June 12-14, 2024, Helsinki, Finland, volume 294 of LIPIcs, pages 4:1-4:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.SWAT.2024.4.
  4. Girish Balakrishnan, Sankardeep Chakraborty, N.S. Narayanaswamy, and Kunihiko Sadakane. Succinct data structure for path graphs. Information and Computation, 296:105124, 2024. URL: https://doi.org/10.1016/j.ic.2023.105124.
  5. E. A. Bender, L. B. Richmond, and N. C. Wormald. Almost all chordal graphs split. Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics, 38(2):214-221, 1985. URL: https://doi.org/10.1017/S1446788700023077.
  6. C. Berge. Some classes of perfect graphs. In Graph Theory and Theoretical Physics, pages 155-165. Academic Press, London-New York, 1967. Google Scholar
  7. 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.
  8. Prosenjit Bose, Meng He, Anil Maheshwari, and Pat Morin. Succinct orthogonal range search structures on a grid with applications to text indexing. In Frank K. H. A. Dehne, Marina L. Gavrilova, Jörg-Rüdiger Sack, and Csaba D. Tóth, editors, Algorithms and Data Structures, 11th International Symposium, WADS 2009, Banff, Canada, August 21-23, 2009. Proceedings, volume 5664 of Lecture Notes in Computer Science, pages 98-109. Springer, 2009. URL: https://doi.org/10.1007/978-3-642-03367-4_9.
  9. Boris Bukh and R. Amzi Jeffs. Enumeration of interval graphs and d-representable complexes, 2023. URL: https://doi.org/10.48550/arXiv.2203.12063.
  10. Peter Buneman. A characterisation of rigid circuit graphs. Discret. Math., 9(3):205-212, 1974. URL: https://doi.org/10.1016/0012-365X(74)90002-8.
  11. Steven Chaplick and Juraj Stacho. The vertex leafage of chordal graphs. Discret. Appl. Math., 168:14-25, 2014. Fifth Workshop on Graph Classes, Optimization, and Width Parameters, Daejeon, Korea, October 2011. URL: https://doi.org/10.1016/j.dam.2012.12.006.
  12. Clark, David. Compact PAT trees. PhD thesis, University of Waterloo, 1997. URL: http://hdl.handle.net/10012/64.
  13. G. A. Dirac. On rigid circuit graphs. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 25:71-76, 1961. URL: https://api.semanticscholar.org/CorpusID:120608513.
  14. Arash Farzan and Shahin Kamali. Compact navigation and distance oracles for graphs with small treewidth. Algorithmica, 69(1):92-116, 2014. URL: https://doi.org/10.1007/s00453-012-9712-9.
  15. Johannes Fischer and Volker Heun. A new succinct representation of rmq-information and improvements in the enhanced suffix array. In Bo Chen, Mike Paterson, and Guochuan Zhang, editors, Combinatorics, Algorithms, Probabilistic and Experimental Methodologies, pages 459-470, Berlin, Heidelberg, 2007. Springer Berlin Heidelberg. URL: https://doi.org/10.1007/978-3-540-74450-4_41.
  16. Michael L. Fredman and Dan E. Willard. Surpassing the information theoretic bound with fusion trees. J. Comput. Syst. Sci., 47(3):424-436, 1993. URL: https://doi.org/10.1016/0022-0000(93)90040-4.
  17. Fănică Gavril. A recognition algorithm for the intersection graphs of paths in trees. Discrete Mathematics, 23(3):211-227, 1978. URL: https://doi.org/10.1016/0012-365X(78)90003-1.
  18. 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.
  19. Isaac Goldstein, Tsvi Kopelowitz, Moshe Lewenstein, and Ely Porat. Conditional lower bounds for space/time tradeoffs. In Faith Ellen, Antonina Kolokolova, and Jörg-Rüdiger Sack, editors, Algorithms and Data Structures, pages 421-436, Cham, 2017. Springer International Publishing. URL: https://doi.org/10.1007/978-3-319-62127-2_36.
  20. Martin Charles Golumbic. Algorithmic graph theory and perfect graphs. Elsevier, 1980. Google Scholar
  21. 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.
  22. Meng He, J. Ian Munro, and Kaiyu Wu. Succinct data structures for path graphs and chordal graphs revisited. In Ali Bilgin, James E. Fowler, Joan Serra-Sagristà, Yan Ye, and James A. Storer, editors, Data Compression Conference, DCC 2024, Snowbird, UT, USA, March 19-22, 2024, pages 492-501. IEEE, 2024. URL: https://doi.org/10.1109/DCC58796.2024.00057.
  23. Douglas B. West In-Jen Lin, Terry A. McKee. The leafage of a chordal graph. Discussiones Mathematicae Graph Theory, 18(1):23-48, 1998. URL: https://doi.org/10.7151/dmgt.1061.
  24. Clyde L. Monma and Victor K.-W. Wei. Intersection graphs of paths in a tree. Journal of Combinatorial Theory, Series B, 41(2):141-181, 1986. URL: https://doi.org/10.1016/0095-8956(86)90042-0.
  25. 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.
  26. 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.
  27. Mihai Patrascu and Mikkel Thorup. Dynamic integer sets with optimal rank, select, and predecessor search. In 2014 IEEE 55th Annual Symposium on Foundations of Computer Science, pages 166-175, 2014. URL: https://doi.org/10.1109/FOCS.2014.26.
  28. Rajeev Raman, Venkatesh Raman, and Srinivasa Rao Satti. Succinct indexable dictionaries with applications to encoding k-ary trees, prefix sums and multisets. ACM Transactions on Algorithms, 3(4):43, November 2007. URL: https://doi.org/10.1145/1290672.1290680.
  29. Donald J Rose. Triangulated graphs and the elimination process. Journal of Mathematical Analysis and Applications, 32(3):597-609, 1970. URL: https://doi.org/10.1016/0022-247X(70)90282-9.
  30. Konstantinos Tsakalidis, Sebastian Wild, and Viktor Zamaraev. Succinct permutation graphs. Algorithmica, 85(2):509-543, 2023. URL: https://doi.org/10.1007/s00453-022-01039-2.
  31. Nicholas C. Wormald. Counting labelled chordal graphs. Graphs and Combinatorics, 1(1):193-200, 1985. URL: https://doi.org/10.1007/BF02582944.
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
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact