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Truly Subquadratic Exact Distance Oracles with Constant Query Time for Planar Graphs

Authors Viktor Fredslund-Hansen , Shay Mozes , Christian Wulff-Nilsen

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Author Details

Viktor Fredslund-Hansen
  • Department of Computer Science, University of Copenhagen, Denmark
Shay Mozes
  • The Interdisciplinary Center, Herzliya, Israel
Christian Wulff-Nilsen
  • Department of Computer Science, University of Copenhagen, Denmark

Funding

  • Mozes, Shay: Partially supported by Israel Science Foundation grant No. 592/17.
  • Wulff-Nilsen, Christian: This research is supported by the Starting Grant 7027-00050B from the Independent Research Fund Denmark under the Sapere Aude research career programme and by Basic Algorithms Research Copenhagen (BARC).

Cite AsGet BibTex

Viktor Fredslund-Hansen, Shay Mozes, and Christian Wulff-Nilsen. Truly Subquadratic Exact Distance Oracles with Constant Query Time for Planar Graphs. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 25:1-25:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ISAAC.2021.25

Abstract

We present a truly subquadratic size distance oracle for reporting, in constant time, the exact shortest-path distance between any pair of vertices of an undirected, unweighted planar graph G. For any ε > 0, our distance oracle requires O(n^{5/3+ε}) space and is capable of answering shortest-path distance queries exactly for any pair of vertices of G in worst-case time O(log (1/ε)). Previously no truly sub-quadratic size distance oracles with constant query time for answering exact shortest paths distance queries existed.

Subject Classification

ACM Subject Classification
  • Theory of computation → Data structures design and analysis
  • Theory of computation → Shortest paths
  • Mathematics of computing → Graph algorithms
Keywords
  • distance oracle
  • planar graph
  • shortest paths
  • subquadratic

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References

  1. Amir Abboud, Pawel Gawrychowski, Shay Mozes, and Oren Weimann. Near-Optimal Compression for the Planar Graph Metric. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 530-549. Society for Industrial and Applied Mathematics, Philadelphia, PA, January 2018. URL: https://doi.org/10.1137/1.9781611975031.35.
  2. Srinivasa Arikati, Danny Z. Chen, L. Paul Chew, Gautam Das, Michiel Smid, and Christos D. Zaroliagis. Planar Spanners and Approximate Shortest Path Queries Among Obstacles in the Plane. In Josep Diaz and Maria Serna, editors, Algorithms - ESA '96, pages 514-528, Berlin, Heidelberg, 1996. Springer Berlin Heidelberg. Google Scholar
  3. Sergio Cabello. Many Distances in Planar Graphs. Algorithmica, 62(1-2):361-381, February 2012. URL: https://doi.org/10.1007/s00453-010-9459-0.
  4. Sergio Cabello. Subquadratic Algorithms for the Diameter and the Sum of Pairwise Distances in Planar Graphs. ACM Transactions on Algorithms, 15(2):1-38, December 2018. URL: https://doi.org/10.1145/3218821.
  5. Timothy M. Chan. All-Pairs Shortest Paths for Unweighted Undirected Graphs in o(mn) Time. ACM Trans. Algorithms, 8(4):34:1-34:17, 2012. URL: https://doi.org/10.1145/2344422.2344424.
  6. Timothy M. Chan and Dimitrios Skrepetos. Faster Approximate Diameter and Distance Oracles in Planar Graphs. Algorithmica, 81(8):3075-3098, August 2019. URL: https://doi.org/10.1007/s00453-019-00570-z.
  7. Panagiotis Charalampopoulos, Paweł Gawrychowski, Shay Mozes, and Oren Weimann. Almost Optimal Distance Oracles for Planar Graphs. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, pages 138-151, New York, NY, USA, 2019. Association for Computing Machinery. URL: https://doi.org/10.1145/3313276.3316316.
  8. Danny Z. Chen and Jinhui Xu. Shortest Path Queries in Planar Graphs. In Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, STOC 2000, pages 469-478, New York, NY, USA, 2000. Association for Computing Machinery. URL: https://doi.org/10.1145/335305.335359.
  9. Vincent Cohen-Addad, Soren Dahlgaard, and Christian Wulff-Nilsen. Fast and Compact Exact Distance Oracle for Planar Graphs. In 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS), pages 962-973. IEEE, October 2017. URL: https://doi.org/10.1109/FOCS.2017.93.
  10. Hristo Djidjev. On-Line Algorithms for Shortest Path Problems on Planar Digraphs. In Proceedings of the 22nd International Workshop on Graph-Theoretic Concepts in Computer Science, WG 1996, pages 151-165, Berlin, Heidelberg, 1996. Springer-Verlag. Google Scholar
  11. Jittat Fakcharoenphol and Satish Rao. Planar Graphs, Negative Weight Edges, Shortest Paths, and Near Linear Time. Journal of Computer and System Sciences, 72(5):868-889, August 2006. URL: https://doi.org/10.1016/j.jcss.2005.05.007.
  12. Paweł Gawrychowski, Shay Mozes, Oren Weimann, and Christian Wulff-Nilsen. Better Tradeoffs for Exact Distance Oracles in Planar Graphs. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, pages 515-529, USA, 2018. Society for Industrial and Applied Mathematics. Google Scholar
  13. Qian-Ping Gu and Gengchun Xu. Constant Query Time (1+ε)-Approximate Distance Oracle for Planar Graphs. Theor. Comput. Sci., 761:78-88, 2019. URL: https://doi.org/10.1016/j.tcs.2018.08.024.
  14. Ken-ichi Kawarabayashi, Philip N. Klein, and Christian Sommer. Linear-space approximate distance oracles for planar, bounded-genus and minor-free graphs. In Luca Aceto, Monika Henzinger, and Jirí Sgall, editors, Automata, Languages and Programming - 38th International Colloquium, ICALP 2011, Zurich, Switzerland, July 4-8, 2011, Proceedings, Part I, volume 6755 of Lecture Notes in Computer Science, pages 135-146. Springer, 2011. URL: https://doi.org/10.1007/978-3-642-22006-7_12.
  15. Philip Klein. Preprocessing an Undirected Planar Network to Enable Fast Approximate Distance Queries. In Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '02, pages 820-827, USA, 2002. Society for Industrial and Applied Mathematics. Google Scholar
  16. Philip N. Klein, Shay Mozes, and Christian Sommer. Structured recursive separator decompositions for planar graphs in linear time. In Proceedings of the 45th annual ACM symposium on Symposium on theory of computing - STOC '13, page 505, New York, New York, USA, 2013. ACM Press. URL: https://doi.org/10.1145/2488608.2488672.
  17. Jason Li and Merav Parter. Planar Diameter via Metric Compression. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, pages 152-163, New York, NY, USA, 2019. Association for Computing Machinery. URL: https://doi.org/10.1145/3313276.3316358.
  18. Yaowei Long and Seth Pettie. Planar Distance Oracles with Better Time-Space Tradeoffs. In Proceedings of the Thirty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, pages 2517-2537. Society for Industrial and Applied Mathematics, 2021. URL: https://epubs.siam.org/doi/abs/10.1137/1.9781611976465.149.
  19. Shay Mozes and Christian Sommer. Exact Distance Oracles for Planar Graphs. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, pages 209-222, USA, 2012. Society for Industrial and Applied Mathematics. Google Scholar
  20. Yahav Nussbaum. Improved Distance Queries in Planar Graphs. In Proceedings of the 12th International Conference on Algorithms and Data Structures, WADS 2011, pages 642-653, Berlin, Heidelberg, 2011. Springer-Verlag. Google Scholar
  21. Mihai Pǎtraşcu and Liam Roditty. Distance Oracles beyond the Thorup-Zwick Bound. SIAM Journal on Computing, 43(1):300-311, January 2014. URL: https://doi.org/10.1137/11084128X.
  22. N Sauer. On the Density of Families of Sets. Journal of Combinatorial Theory, Series A, 13(1):145-147, July 1972. URL: https://doi.org/10.1016/0097-3165(72)90019-2.
  23. Christian Sommer. Shortest-Path Queries in Static Networks. ACM Computing Surveys, 46(4):1-31, April 2014. URL: https://doi.org/10.1145/2530531.
  24. Mikkel Thorup. Compact Oracles for Reachability and Approximate Distances in Planar Digraphs. Journal of the ACM (JACM), 51(6):993-1024, November 2004. URL: https://doi.org/10.1145/1039488.1039493.
  25. Christian Wulff-Nilsen. Algorithms for Planar Graphs and Graphs in Metric Spaces. PhD thesis, Department of Computer Science, University of Copenhagen, Denmark, 2010. URL: https://di.ku.dk/english/research/phd/phd-theses/2010/thesischristianwulff.pdf_kopi.
  26. Christian Wulff-Nilsen. Constant Time Distance Queries in Planar Unweighted Graphs with Subquadratic Preprocessing Time. Computational Geometry, 46(7):831-838, October 2013. URL: https://doi.org/10.1016/j.comgeo.2012.01.016.
  27. Christian Wulff-Nilsen. Approximate Distance Oracles for Planar Graphs with Improved Query Time-Space Tradeoff. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '16, pages 351-362, USA, 2016. Society for Industrial and Applied Mathematics. Google Scholar
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