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Succinct Data Structures for Chordal Graphs

Authors J. Ian Munro , Kaiyu Wu



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Author Details

J. Ian Munro
  • Cheriton School of Computer Science, University of Waterloo, Waterloo, Canada
Kaiyu Wu
  • Cheriton School of Computer Science, University of Waterloo, Waterloo, Canada

Cite AsGet BibTex

J. Ian Munro and Kaiyu Wu. Succinct Data Structures for Chordal Graphs. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 67:1-67:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.ISAAC.2018.67

Abstract

We study the problem of approximate shortest path queries in chordal graphs and give a n log n + o(n log n) bit data structure to answer the approximate distance query to within an additive constant of 1 in O(1) time. We study the problem of succinctly storing a static chordal graph to answer adjacency, degree, neighbourhood and shortest path queries. Let G be a chordal graph with n vertices. We design a data structure using the information theoretic minimal n^2/4 + o(n^2) bits of space to support the queries: - whether two vertices u,v are adjacent in time f(n) for any f(n) in omega(1). - the degree of a vertex in O(1) time. - the vertices adjacent to u in (f(n))^2 time per neighbour - the length of the shortest path from u to v in O(nf(n)) time

Subject Classification

ACM Subject Classification
  • Theory of computation → Shortest paths
  • Theory of computation → Data compression
Keywords
  • Succinct Data Structure
  • Chordal Graph

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References

  1. Niranka Banerjee, Venkatesh Raman, and Srinivasa Rao Satti. Maintaining Chordal Graphs Dynamically: Improved Upper and Lower Bounds. In Fedor V. Fomin and Vladimir V. Podolskii, editors, Computer Science - Theory and Applications - 13th International Computer Science Symposium in Russia, CSR 2018, Moscow, Russia, June 6-10, 2018, Proceedings, volume 10846 of Lecture Notes in Computer Science, pages 29-40. Springer, 2018. URL: http://dx.doi.org/10.1007/978-3-319-90530-3_4.
  2. Daniel K. Blandford, Guy E. Blelloch, and Ian A. Kash. Compact representations of separable graphs. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, January 12-14, 2003, Baltimore, Maryland, USA., pages 679-688. ACM/SIAM, 2003. URL: http://dl.acm.org/citation.cfm?id=644108.644219.
  3. Bostjan Bresar, Frantisek Kardos, Ján Katrenic, and Gabriel Semanisin. Minimum k-path vertex cover. Discrete Applied Mathematics, 159(12):1189-1195, 2011. URL: http://dx.doi.org/10.1016/j.dam.2011.04.008.
  4. Peter Buneman. A characterisation of rigid circuit graphs. Discrete Mathematics, 9(3):205-212, 1974. URL: http://dx.doi.org/10.1016/0012-365X(74)90002-8.
  5. Hagai Cohen and Ely Porat. Fast set intersection and two-patterns matching. Theoretical Computer Science, 411(40):3795-3800, 2010. URL: http://dx.doi.org/10.1016/j.tcs.2010.06.002.
  6. Ronald Fagin. Degrees of Acyclicity for Hypergraphs and Relational Database Schemes. J. ACM, 30(3):514-550, 1983. URL: http://dx.doi.org/10.1145/2402.322390.
  7. Arash Farzan and Shahin Kamali. Compact Navigation and Distance Oracles for Graphs with Small Treewidth. Algorithmica, 69(1):92-116, May 2014. URL: http://dx.doi.org/10.1007/s00453-012-9712-9.
  8. Arash Farzan and J. Ian Munro. Succinct encoding of arbitrary graphs. Theoretical Computer Science, 513:38-52, 2013. URL: http://dx.doi.org/10.1016/j.tcs.2013.09.031.
  9. Louis Ibarra. Fully dynamic algorithms for chordal graphs and split graphs. ACM Trans. Algorithms, 4(4):40:1-40:20, 2008. URL: http://dx.doi.org/10.1145/1383369.1383371.
  10. Guy Jacobson. Space-efficient Static Trees and Graphs. In 30th Annual Symposium on Foundations of Computer Science, Research Triangle Park, North Carolina, USA, 30 October - 1 November 1989, pages 549-554. IEEE Computer Society, 1989. URL: http://dx.doi.org/10.1109/SFCS.1989.63533.
  11. Gonzalo Navarro and Kunihiko Sadakane. Fully Functional Static and Dynamic Succinct Trees. ACM Trans. Algorithms, 10(3):16:1-16:39, 2014. URL: http://dx.doi.org/10.1145/2601073.
  12. M. Patrascu and L. Roditty. Distance Oracles beyond the Thorup-Zwick Bound. In 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, pages 815-823, October 2010. URL: http://dx.doi.org/10.1109/FOCS.2010.83.
  13. 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: http://dx.doi.org/10.1109/FOCS.2008.83.
  14. 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
  15. Donald J Rose. Triangulated graphs and the elimination process. Journal of Mathematical Analysis and Applications, 32(3):597-609, 1970. URL: http://dx.doi.org/10.1016/0022-247X(70)90282-9.
  16. Donald J. Rose, Robert Endre Tarjan, and George S. Lueker. Algorithmic Aspects of Vertex Elimination on Graphs. SIAM J. Comput., 5(2):266-283, 1976. URL: http://dx.doi.org/10.1137/0205021.
  17. 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: http://dx.doi.org/10.1007/978-3-319-15612-5_9.
  18. James R. Walter. Representations of chordal graphs as subtrees of a tree. Journal of Graph Theory, 2(3):265-267, 1978. URL: http://dx.doi.org/10.1002/jgt.3190020311.
  19. Nicholas C. Wormald. Counting labelled chordal graphs. Graphs and Combinatorics, 1(1):193-200, 1985. URL: http://dx.doi.org/10.1007/BF02582944.
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