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Succinct Planar Encoding with Minor Operations

Authors Frank Kammer , Johannes Meintrup



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Author Details

Frank Kammer
  • THM, University of Applied Sciences Mittelhessen, Giessen, Germany
Johannes Meintrup
  • THM, University of Applied Sciences Mittelhessen, Giessen, Germany

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Frank Kammer and Johannes Meintrup. Succinct Planar Encoding with Minor Operations. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 44:1-44:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ISAAC.2023.44

Abstract

Let G be an unlabeled planar and simple n-vertex graph. Unlabeled graphs are graphs where the label-information is either not given or lost during the construction of data-structures. We present a succinct encoding of G that provides induced-minor operations, i.e., edge contractions and vertex deletions. Any sequence of such operations is processed in O(n) time in the word-RAM model. At all times the encoding provides constant time (per element output) neighborhood access and degree queries. Optional hash tables extend the encoding with constant expected time adjacency queries and edge-deletion (thus, all minor operations are supported) such that any number of edge deletions are computed in O(n) expected time. Constructing the encoding requires O(n) bits and O(n) time. The encoding requires ℋ(n) + o(n) bits of space with ℋ(n) being the entropy of encoding a planar graph with n vertices. Our data structure is based on the recent result of Holm et al. [ESA 2017] who presented a linear time contraction data structure that allows to maintain parallel edges and works for labeled graphs, but uses Θ(n log n) bits of space. We combine the techniques used by Holm et al. with novel ideas and the succinct encoding of Blelloch and Farzan [CPM 2010] for arbitrary separable graphs. Our result partially answers the question raised by Blelloch and Farzan whether their encoding can be modified to allow modifications of the graph.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
Keywords
  • planar graph
  • r-division
  • separator
  • succinct encoding
  • graph minors

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References

  1. Joyce Bacic, Saeed Mehrabi, and Michiel Smid. Shortest Beer Path Queries in Outerplanar Graphs. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021), volume 212 of Leibniz International Proceedings in Informatics (LIPIcs), pages 62:1-62:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2021.62.
  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, SODA '03, pages 679-688, USA, 2003. Society for Industrial and Applied Mathematics. URL: https://doi.org/10.5555/644108.644219.
  3. Guy E. Blelloch and Arash Farzan. Succinct representations of separable graphs. In Amihood Amir and Laxmi Parida, editors, Combinatorial Pattern Matching, pages 138-150. Springer Berlin Heidelberg, 2010. URL: https://doi.org/10.1007/978-3-642-13509-5_13.
  4. Gerth Stølting Brodal and Rolf Fagerberg. Dynamic representations of sparse graphs. In In Proc. 6th International Workshop on Algorithms and Data Structures (WADS 99), pages 342-351. Springer-Verlag, 1999. URL: https://doi.org/10.1007/3-540-48447-7_34.
  5. Yi-Ting Chiang, Ching-Chi Lin, and Hsueh-I Lu. Orderly spanning trees with applications to graph encoding and graph drawing. In Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '01, pages 506-515. Society for Industrial and Applied Mathematics, 2001. URL: https://doi.org/10.5555/365411.365518.
  6. Jack Edmonds. Paths, trees, and flowers. Canadian Journal of Mathematics, 17:449-467, 1965. URL: https://doi.org/10.4153/CJM-1965-045-4.
  7. Amr Elmasry, Torben Hagerup, and Frank Kammer. Space-efficient Basic Graph Algorithms. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015), volume 30 of Leibniz International Proceedings in Informatics (LIPIcs), pages 288-301. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015. URL: https://doi.org/10.4230/LIPIcs.STACS.2015.288.
  8. Amr Elmasry and Frank Kammer. Space-efficient plane-sweep algorithms. In 27th International Symposium on Algorithms and Computation, ISAAC 2016, volume 64 of LIPIcs, pages 30:1-30:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2016.30.
  9. Greg N. Federickson. Fast algorithms for shortest paths in planar graphs, with applications. SIAM Journal on Computing, 16(6):1004-1022, 1987. URL: https://doi.org/10.1137/0216064.
  10. Volodymyr Floreskul, Konstantin Tretyakov, and Marlon Dumas. Memory-efficient fast shortest path estimation in large social networks. Proceedings of the International AAAI Conference on Web and Social Media, 8:91-100, 2014. URL: https://doi.org/10.1609/icwsm.v8i1.14532.
  11. Michael T. Goodrich. Planar separators and parallel polygon triangulation. J. Comput. Syst. Sci., 51(3):374-389, 1995. URL: https://doi.org/10.1006/jcss.1995.1076.
  12. Torben Hagerup. Space-efficient DFS and applications to connectivity problems: Simpler, leaner, faster. Algorithmica, 82(4):1033-1056, 2020. URL: https://doi.org/10.1007/s00453-019-00629-x.
  13. Xin He, Ming-Yang Kao, and Hsueh-I Lu. A fast general methodology for information-theoretically optimal encodings of graphs. SIAM Journal on Computing, 30(3):838-846, 2000. URL: https://doi.org/10.1137/S0097539799359117.
  14. Klaus Heeger, Anne-Sophie Himmel, Frank Kammer, Rolf Niedermeier, Malte Renken, and Andrej Sajenko. Multistage graph problems on a global budget. Theoretical Computer Science, 868:46-64, 2021. URL: https://doi.org/10.1016/j.tcs.2021.04.002.
  15. Jacob Holm, Giuseppe F. Italiano, Adam Karczmarz, Jakub Lacki, Eva Rotenberg, and Piotr Sankowski. Contracting a Planar Graph Efficiently. In 25th Annual European Symposium on Algorithms (ESA 2017), volume 87 of Leibniz International Proceedings in Informatics (LIPIcs), pages 50:1-50:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.ESA.2017.50.
  16. Jacob Holm and Eva Rotenberg. Good r-Divisions Imply Optimal Amortized Decremental Biconnectivity. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021), volume 187 of Leibniz International Proceedings in Informatics (LIPIcs), pages 42:1-42:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.STACS.2021.42.
  17. Frank Kammer and Johannes Meintrup. Space-Efficient Graph Coarsening with Applications to Succinct Planar Encodings. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022), volume 248 of Leibniz International Proceedings in Informatics (LIPIcs), pages 62:1-62:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2022.62.
  18. Frank Kammer and Johannes Meintrup. Succinct planar encoding with minor operations, 2023. URL: https://arxiv.org/abs/2301.10564.
  19. Frank Kammer, Johannes Meintrup, and Andrej Sajenko. Space-efficient vertex separators for treewidth. Algorithmica, 84(9):2414-2461, 2022. URL: https://doi.org/10.1007/s00453-022-00967-3.
  20. David R. Karger. Global min-cuts in RNC, and other ramifications of a simple min-cut algorithm. In Proceedings of the Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '93, pages 21-30. Society for Industrial and Applied Mathematics, 1993. URL: https://doi.org/10.5555/313559.313605.
  21. Kenneth Keeler and Jeffery Westbrook. Short encodings of planar graphs and maps. Discrete Applied Mathematics, 58(3):239-252, 1995. URL: https://doi.org/10.1016/0166-218X(93)E0150-W.
  22. Philip N. Klein and Shay Mozes. Optimization algorithms for planar graphs. planarity.org, 2017. URL: http://planarity.org.
  23. Philip N. Klein, Shay Mozes, and Christian Sommer. Structured recursive separator decompositions for planar graphs in linear time. In STOC '13: Proceedings of the forty-fifth annual ACM symposium on Theory of Computing. Association for Computing Machinery, 2013. URL: https://doi.org/10.1145/2488608.2488672.
  24. J. Ian Munro and Venkatesh Raman. Succinct representation of balanced parentheses and static trees. SIAM Journal on Computing, 31(3):762-776, 2001. URL: https://doi.org/10.1137/S0097539799364092.
  25. Rajeev Raman, Venkatesh Raman, and Srinivasa Rao Satti. Succinct indexable dictionaries with applications to encoding k-ary trees, prefix sums and multisets. ACM Trans. Algorithms, 3(4):43-es, November 2007. URL: https://doi.org/10.1145/1290672.1290680.
  26. Ben Strasser, Dorothea Wagner, and Tim Zeitz. Space-Efficient, Fast and Exact Routing in Time-Dependent Road Networks. In 28th Annual European Symposium on Algorithms (ESA 2020), volume 173 of Leibniz International Proceedings in Informatics (LIPIcs), pages 81:1-81:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ESA.2020.81.
  27. Hisao Tamaki. Positive-instance driven dynamic programming for treewidth. J. Comb. Optim., 37(4):1283-1311, 2019. URL: https://doi.org/10.1007/s10878-018-0353-z.
  28. György Turán. On the succinct representation of graphs. Discret. Appl. Math., 8(3):289-294, 1984. URL: https://doi.org/10.1016/0166-218X(84)90126-4.
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