Document Open Access Logo

Space-Efficient Depth-First Search via Augmented Succinct Graph Encodings

Authors Michael Elberfeld , Frank Kammer , Johannes Meintrup

PDF

File

Thumbnail PDF
PDF
LIPIcs.ISAAC.2025.29.pdf
  • Filesize: 0.76 MB
  • 16 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Data structures design and analysis
  • Theory of computation → Graph algorithms analysis
Keywords
  • Depth-First Search
  • Succinct
  • Space Efficient
  • Separable Graphs
  • Planar Graphs
  • Table Lookup
  • r-Division

Metrics

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

Abstract

We call a graph G separable if a balanced separator can be computed for G of size O(n^ε) with ε < 1. Many real-world graphs are separable such as graphs of bounded genus, graphs of constant treewidth, and graphs excluding a fixed minor. In particular, the well-known planar graphs are separable. We present a succinct encoding of separable graphs G such that, after the encoding is computed, any number of depth-first searches (DFS) can be performed from any given start vertex, each in o(n) time and o(n) bits in the word RAM model. After the execution of a DFS, the succinct encoding of G is augmented such that the DFS tree is encoded inside the encoding while maintaining succinctness. Afterward, the encoding provides common DFS-related queries in constant time. These queries include queries such as lowest-common ancestor of two given vertices in the DFS tree or queries that output the lowpoint of a given vertex in the DFS tree. Furthermore, for planar graphs, we show that the succinct encoding can be computed in O(n) bits and expected linear time, and a compact variant can be constructed in O(n) time and bits. For other separable graph classes 𝒢 the runtime and space usage depends on the specific algorithms used to find balanced separators in graphs of 𝒢.

Cite As Get BibTex

Michael Elberfeld, Frank Kammer, and Johannes Meintrup. Space-Efficient Depth-First Search via Augmented Succinct Graph Encodings. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 29:1-29:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ISAAC.2025.29

Author Details

Michael Elberfeld
  • THM, University of Applied Sciences Mittelhessen, Gießen, Germany
Frank Kammer
  • THM, University of Applied Sciences Mittelhessen, Gießen, Germany
Johannes Meintrup
  • THM, University of Applied Sciences Mittelhessen, Gießen, Germany

References

  1. Hüseyin Acan, Sankardeep Chakraborty, Seungbum Jo, and Srinivasa Rao Satti. Succinct data structures for families of interval graphs. In 16th International Symposium on Algorithms and Data Structures (WADS 2019), pages 1-13. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-24766-9_1.
  2. Luca Castelli Aleardi, Olivier Devillers, and Gilles Schaeffer. Optimal succinct representations of planar maps. In Proceedings of the Twenty-Second Annual Symposium on Computational Geometry (SCG 2006), pages 309-318. ACM, 2006. URL: https://doi.org/10.1145/1137856.1137902.
  3. Tetsuo Asano, Taisuke Izumi, Masashi Kiyomi, Matsuo Konagaya, Hirotaka Ono, Yota Otachi, Pascal Schweitzer, Jun Tarui, and Ryuhei Uehara. Depth-first search using O(n) bits. In Hee-Kap Ahn and Chan-Su Shin, editors, 25h International Symposium on Algorithms and Computation (ISAAC 2014), pages 553-564. Springer, 2014. URL: https://doi.org/10.1007/978-3-319-13075-0_44.
  4. Niranka Banerjee, Sankardeep Chakraborty, Venkatesh Raman, and Srinivasa Rao Satti. Space efficient linear time algorithms for bfs, DFS and applications. Theory Comput. Syst., 62(8):1736-1762, 2018. URL: https://doi.org/10.1007/S00224-017-9841-2.
  5. Tim Baumann and Torben Hagerup. Rank-select indices without tears. In 16th International Symposium on Algorithms and Data Structures (WADS 2019), pages 85-98. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-24766-9_7.
  6. Guy E. Blelloch and Arash Farzan. Succinct representations of separable graphs. In 21st Annual Symposium on Combinatorial Pattern Matching (CPM 2010), pages 138-150. Springer, 2010. URL: https://doi.org/10.1007/978-3-642-13509-5_13.
  7. Sankardeep Chakraborty, Venkatesh Raman, and Srinivasa Rao Satti. Biconnectivity, Chain Decomposition and st-Numbering Using O(n) Bits. In 27th International Symposium on Algorithms and Computation (ISAAC 2016), volume 64 of LIPIcs, pages 22:1-22:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2016.22.
  8. Yi-Ting Chiang, Ching-Chi Lin, and Hsueh-I Lu. Orderly spanning trees with applications to graph encoding and graph drawing. In 12th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2001), pages 506-515. SIAM, 2001. URL: http://dl.acm.org/citation.cfm?id=365411.365518.
  9. 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 LIPIcs, pages 288-301. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015. URL: https://doi.org/10.4230/LIPIcs.STACS.2015.288.
  10. 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.
  11. Richard F. Geary, Rajeev Raman, and Venkatesh Raman. Succinct ordinal trees with level-ancestor queries. ACM Trans. Algorithms, 2(4):510-534, 2006. URL: https://doi.org/10.1145/1198513.1198516.
  12. M.T. Goodrich. Planar separators and parallel polygon triangulation. Journal of Computer and System Sciences, 51(3):374-389, 1995. URL: https://doi.org/10.1006/jcss.1995.1076.
  13. 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.
  14. Dov Harel and Robert Endre Tarjan. Fast algorithms for finding nearest common ancestors. SIAM Journal on Computing, 13(2):338-355, 1984. URL: https://doi.org/10.1137/0213024.
  15. Jacob Holm, Giuseppe F. Italiano, Adam Karczmarz, Jakub Lacki, and Eva Rotenberg. Decremental SPQR-trees for Planar Graphs. In 26th Annual European Symposium on Algorithms (ESA 2018), volume 112 of LIPIcs, pages 46:1-46:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.ESA.2018.46.
  16. Jacob Holm and Eva Rotenberg. Good r-divisions imply optimal amortized decremental biconnectivity. Theory Comput. Syst., 68(4):1014-1048, 2024. URL: https://doi.org/10.1007/S00224-024-10181-Z.
  17. John Hopcroft and Robert Tarjan. Algorithm 447: efficient algorithms for graph manipulation. Commun. ACM, 16(6):372-378, June 1973. URL: https://doi.org/10.1145/362248.362272.
  18. Taisuke Izumi and Yota Otachi. Sublinear-Space Lexicographic Depth-First Search for Bounded Treewidth Graphs and Planar Graphs. In Artur Czumaj, Anuj Dawar, and Emanuela Merelli, editors, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020), volume 168 of LIPIcs, pages 67:1-67:17, Dagstuhl, Germany, 2020. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ICALP.2020.67.
  19. G. Jacobson. Space-efficient static trees and graphs. In 30th Annual Symposium on Foundations of Computer Science (FOCS 1989), pages 549-554, 1989. URL: https://doi.org/10.1109/SFCS.1989.63533.
  20. Frank Kammer, Dieter Kratsch, and Moritz Laudahn. Space-efficient biconnected components and recognition of outerplanar graphs. Algorithmica, 81(3):1180-1204, 2019. URL: https://doi.org/10.1007/S00453-018-0464-Z.
  21. 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 LIPIcs, pages 62:1-62:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2022.62.
  22. Kenneth Keeler and Jeffery Westbrook. Short encodings of planar graphs and maps. Discrete Appl. Math., 58(3):239-252, 1995. URL: https://doi.org/10.1016/0166-218X(93)E0150-W.
  23. Philip N. Klein, Shay Mozes, and Christian Sommer. Structured recursive separator decompositions for planar graphs in linear time. In 45th Annual ACM Symposium on Theory of Computing (STOC 2013), pages 505-514. Association for Computing Machinery, 2013. URL: https://doi.org/10.1145/2488608.2488672.
  24. Richard J. Lipton and Robert Endre Tarjan. A separator theorem for planar graphs. SIAM Journal on Applied Mathematics, 36(2):177-189, 1979. URL: https://doi.org/10.1137/0136016.
  25. J. Ian Munro, Patrick K. Nicholson, Louisa Seelbach Benkner, and Sebastian Wild. Hypersuccinct Trees - New Universal Tree Source Codes for Optimal Compressed Tree Data Structures and Range Minima. In 29th Annual European Symposium on Algorithms (ESA 2021), volume 204 of LIPIcs, pages 70:1-70:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ESA.2021.70.
  26. 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, 2007. URL: https://doi.org/10.1145/1290672.1290680.
  27. Omer Reingold. Undirected connectivity in log-space. J. ACM, 55(4), September 2008. URL: https://doi.org/10.1145/1391289.1391291.
  28. Jens M. Schmidt. A simple test on 2-vertex- and 2-edge-connectivity. Information Processing Letters, 113(7):241-244, 2013. URL: https://doi.org/10.1016/j.ipl.2013.01.016.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact