Document Open Access Logo

Õptimal Dual Vertex Failure Connectivity Labels

Authors Merav Parter, Asaf Petruschka

PDF

File

Thumbnail PDF
PDF
LIPIcs.DISC.2022.32.pdf
  • Filesize: 1.35 MB
  • 19 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Data structures design and analysis
Keywords
  • Fault-Tolerance
  • Heavy-Light Decomposition
  • Labeling Schemes

Metrics

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

Abstract

In this paper we present succinct labeling schemes for supporting connectivity queries under vertex faults. For a given n-vertex graph G, an f-VFT (resp., EFT) connectivity labeling scheme is a distributed data structure that assigns each of the graph edges and vertices a short label, such that given the labels of a vertex pair u and v, and the labels of at most f failing vertices (resp., edges) F, one can determine if u and v are connected in G ⧵ F. The primary complexity measure is the length of the individual labels. Since their introduction by [Courcelle, Twigg, STACS '07], FT labeling schemes have been devised only for a limited collection of graph families. A recent work [Dory and Parter, PODC 2021] provided EFT labeling schemes for general graphs under edge failures, leaving the vertex failure case fairly open. 
We provide the first sublinear f-VFT labeling schemes for f ≥ 2 for any n-vertex graph. Our key result is 2-VFT connectivity labels with O(log³ n) bits. Our constructions are based on analyzing the structure of dual failure replacement paths on top of the well-known heavy-light tree decomposition technique of [Sleator and Tarjan, STOC 1981]. We also provide f-VFT labels with sub-linear length (in |V|) for any f = o(log log n), that are based on a reduction to the existing EFT labels.

Cite As Get BibTex

Merav Parter and Asaf Petruschka. Õptimal Dual Vertex Failure Connectivity Labels. In 36th International Symposium on Distributed Computing (DISC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 246, pp. 32:1-32:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.DISC.2022.32

Author Details

Merav Parter
  • Weizmann Institute, Rehovot, Israel
Asaf Petruschka
  • Weizmann Institute, Rehovot, Israel

Funding

  • Parter, Merav: Supported by the European Research Council (ERC) No. 949083), and by the Israeli Science Foundation (ISF) No. 2084/18.

Acknowledgements

We would like to thank Michal Dory for useful discussions.

References

  1. Ittai Abraham, Shiri Chechik, and Cyril Gavoille. Fully dynamic approximate distance oracles for planar graphs via forbidden-set distance labels. In Proceedings of the forty-fourth annual ACM symposium on Theory of computing, pages 1199-1218, 2012. Google Scholar
  2. Ittai Abraham, Shiri Chechik, Cyril Gavoille, and David Peleg. Forbidden-set distance labels for graphs of bounded doubling dimension. ACM Trans. Algorithms, 12(2):22:1-22:17, 2016. Google Scholar
  3. Surender Baswana, Keerti Choudhary, Moazzam Hussain, and Liam Roditty. Approximate single source fault tolerant shortest path. In Artur Czumaj, editor, Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 1901-1915. SIAM, 2018. Google Scholar
  4. Giuseppe Di Battista and Roberto Tamassia. Incremental planarity testing (extended abstract). In 30th Annual Symposium on Foundations of Computer Science, Research Triangle Park, North Carolina, USA, 30 October - 1 November 1989, pages 436-441. IEEE Computer Society, 1989. Google Scholar
  5. Giuseppe Di Battista and Roberto Tamassia. On-line maintenance of triconnected components with spqr-trees. Algorithmica, 15(4):302-318, 1996. Google Scholar
  6. Li Chen, Rasmus Kyng, Yang P. Liu, Richard Peng, Maximilian Probst Gutenberg, and Sushant Sachdeva. Maximum flow and minimum-cost flow in almost-linear time. CoRR, abs/2203.00671, 2022. URL: https://doi.org/10.48550/arXiv.2203.00671.
  7. Joseph Cheriyan, Ming-Yang Kao, and Ramakrishna Thurimella. Scan-first search and sparse certificates: an improved parallel algorithm for k-vertex connectivity. SIAM Journal on Computing, 22(1):157-174, 1993. Google Scholar
  8. Keerti Choudhary. An optimal dual fault tolerant reachability oracle. In Ioannis Chatzigiannakis, Michael Mitzenmacher, Yuval Rabani, and Davide Sangiorgi, editors, 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, July 11-15, 2016, Rome, Italy, volume 55 of LIPIcs, pages 130:1-130:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. Google Scholar
  9. Bruno Courcelle and Andrew Twigg. Compact forbidden-set routing. In STACS 2007, 24th Annual Symposium on Theoretical Aspects of Computer Science, Aachen, Germany, February 22-24, 2007, Proceedings, pages 37-48, 2007. Google Scholar
  10. Michal Dory, Yuval Efron, Sagnik Mukhopadhyay, and Danupon Nanongkai. Distributed weighted min-cut in nearly-optimal time. In Samir Khuller and Virginia Vassilevska Williams, editors, STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 1144-1153. ACM, 2021. Google Scholar
  11. Michal Dory and Merav Parter. Fault-tolerant labeling and compact routing schemes. In Avery Miller, Keren Censor-Hillel, and Janne H. Korhonen, editors, PODC '21: ACM Symposium on Principles of Distributed Computing, Virtual Event, Italy, July 26-30, 2021, pages 445-455. ACM, 2021. Google Scholar
  12. Ran Duan and Seth Pettie. Dual-failure distance and connectivity oracles. In Proceedings of the twentieth annual ACM-SIAM symposium on Discrete algorithms, pages 506-515. SIAM, 2009. Google Scholar
  13. Ran Duan and Seth Pettie. Connectivity oracles for graphs subject to vertex failures. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 490-509, 2017. Google Scholar
  14. Ran Duan and Seth Pettie. Connectivity oracles for graphs subject to vertex failures. SIAM J. Comput., 49(6):1363-1396, 2020. Google Scholar
  15. Maciej Duleba, Pawel Gawrychowski, and Wojciech Janczewski. Efficient labeling for reachability in directed acyclic graphs. 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 27:1-27:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. Google Scholar
  16. Pawel Gawrychowski, Shay Mozes, and Oren Weimann. Minimum cut in o(m log² n) time. In Artur Czumaj, Anuj Dawar, and Emanuela Merelli, editors, 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020, July 8-11, 2020, Saarbrücken, Germany (Virtual Conference), volume 168 of LIPIcs, pages 57:1-57:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. Google Scholar
  17. Loukas Georgiadis, Giuseppe F. Italiano, Luigi Laura, and Nikos Parotsidis. 2-vertex connectivity in directed graphs. In Magnús M. Halldórsson, Kazuo Iwama, Naoki Kobayashi, and Bettina Speckmann, editors, Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6-10, 2015, Proceedings, Part I, volume 9134 of Lecture Notes in Computer Science, pages 605-616. Springer, 2015. Google Scholar
  18. Loukas Georgiadis and Robert Endre Tarjan. Dominators, directed bipolar orders, and independent spanning trees. In Artur Czumaj, Kurt Mehlhorn, Andrew M. Pitts, and Roger Wattenhofer, editors, Automata, Languages, and Programming - 39th International Colloquium, ICALP 2012, Warwick, UK, July 9-13, 2012, Proceedings, Part I, volume 7391 of Lecture Notes in Computer Science, pages 375-386. Springer, 2012. Google Scholar
  19. Mohsen Ghaffari, Krzysztof Nowicki, and Mikkel Thorup. Faster algorithms for edge connectivity via random 2-out contractions. In Shuchi Chawla, editor, Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020, Salt Lake City, UT, USA, January 5-8, 2020, pages 1260-1279. SIAM, 2020. Google Scholar
  20. Mohsen Ghaffari and Goran Zuzic. Universally-optimal distributed exact min-cut. In Alessia Milani and Philipp Woelfel, editors, PODC '22: ACM Symposium on Principles of Distributed Computing, Salerno, Italy, July 25 - 29, 2022, pages 281-291. ACM, 2022. Google Scholar
  21. Manoj Gupta and Shahbaz Khan. Multiple source dual fault tolerant BFS trees. In Ioannis Chatzigiannakis, Piotr Indyk, Fabian Kuhn, and Anca Muscholl, editors, 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, July 10-14, 2017, Warsaw, Poland, volume 80 of LIPIcs, pages 127:1-127:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. Google Scholar
  22. Zhiyang He, Jason Li, and Magnus Wahlström. Near-linear-time, optimal vertex cut sparsifiers in directed acyclic graphs. In Petra Mutzel, Rasmus Pagh, and Grzegorz Herman, editors, 29th Annual European Symposium on Algorithms, ESA 2021, September 6-8, 2021, Lisbon, Portugal (Virtual Conference), volume 204 of LIPIcs, pages 52:1-52:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. Google Scholar
  23. John E. Hopcroft and Robert Endre Tarjan. Dividing a graph into triconnected components. SIAM J. Comput., 2(3):135-158, 1973. Google Scholar
  24. Arkady Kanevsky, Roberto Tamassia, Giuseppe Di Battista, and Jianer Chen. On-line maintenance of the four-connected components of a graph (extended abstract). In 32nd Annual Symposium on Foundations of Computer Science, San Juan, Puerto Rico, 1-4 October 1991, pages 793-801. IEEE Computer Society, 1991. URL: https://doi.org/10.1109/SFCS.1991.185451.
  25. Sampath Kannan, Moni Naor, and Steven Rudich. Implicit representation of graphs. SIAM Journal on Discrete Mathematics, 5(4):596-603, 1992. Google Scholar
  26. David R. Karger. Global min-cuts in rnc, and other ramifications of a simple min-cut algorithm. In Vijaya Ramachandran, editor, Proceedings of the Fourth Annual ACM/SIGACT-SIAM Symposium on Discrete Algorithms, 25-27 January 1993, Austin, Texas, USA, pages 21-30. ACM/SIAM, 1993. Google Scholar
  27. Neelesh Khanna and Surender Baswana. Approximate shortest paths avoiding a failed vertex: Optimal size data structures for unweighted graph. In 27th International Symposium on Theoretical Aspects of Computer Science-STACS 2010, pages 513-524, 2010. Google Scholar
  28. Jason Li, Danupon Nanongkai, Debmalya Panigrahi, Thatchaphol Saranurak, and Sorrachai Yingchareonthawornchai. Vertex connectivity in poly-logarithmic max-flows. In Samir Khuller and Virginia Vassilevska Williams, editors, STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 317-329. ACM, 2021. Google Scholar
  29. Danupon Nanongkai, Thatchaphol Saranurak, and Sorrachai Yingchareonthawornchai. Breaking quadratic time for small vertex connectivity and an approximation scheme. In Moses Charikar and Edith Cohen, editors, Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, Phoenix, AZ, USA, June 23-26, 2019, pages 241-252. ACM, 2019. Google Scholar
  30. Merav Parter. Dual failure resilient bfs structure. In Proceedings of the 2015 ACM Symposium on Principles of Distributed Computing, pages 481-490, 2015. Google Scholar
  31. Merav Parter. Small cuts and connectivity certificates: A fault tolerant approach. In 33rd International Symposium on Distributed Computing, 2019. Google Scholar
  32. Merav Parter. Distributed constructions of dual-failure fault-tolerant distance preservers. In Hagit Attiya, editor, 34th International Symposium on Distributed Computing, DISC 2020, October 12-16, 2020, Virtual Conference, volume 179 of LIPIcs, pages 21:1-21:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. Google Scholar
  33. Merav Parter and Asaf Petruschka. Near-optimal distributed computation of small vertex cuts. In 36th International Symposium on Distributed Computing DISC, 2022. Google Scholar
  34. Seth Pettie and Longhui Yin. The structure of minimum vertex cuts. In Nikhil Bansal, Emanuela Merelli, and James Worrell, editors, 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021, July 12-16, 2021, Glasgow, Scotland (Virtual Conference), volume 198 of LIPIcs, pages 105:1-105:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. Google Scholar
  35. David Pritchard and Ramakrishna Thurimella. Fast computation of small cuts via cycle space sampling. ACM Transactions on Algorithms (TALG), 7(4):46, 2011. Google Scholar
  36. Daniel Dominic Sleator and Robert Endre Tarjan. A data structure for dynamic trees. J. Comput. Syst. Sci., 26(3):362-391, 1983. Google Scholar
  37. Douglas B. West. Introduction to Graph Theory. Prentice Hall, 2 edition, September 2000. Google Scholar
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