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New Extremal Bounds for Reachability and Strong-Connectivity Preservers Under Failures

Authors Diptarka Chakraborty, Keerti Choudhary



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Diptarka Chakraborty
  • National University of Singapore, Singapore
Keerti Choudhary
  • Tel Aviv University, Israel

Acknowledgements

Authors would like to thank Robert Krauthgamer and Greg Bodwin for some useful discussion.

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Diptarka Chakraborty and Keerti Choudhary. New Extremal Bounds for Reachability and Strong-Connectivity Preservers Under Failures. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 25:1-25:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ICALP.2020.25

Abstract

In this paper, we consider the question of computing sparse subgraphs for any input directed graph G = (V,E) on n vertices and m edges, that preserves reachability and/or strong connectivity structures. - We show O(n+min{|P|√n, n√|P|}) bound on a subgraph that is an 1-fault-tolerant reachability preserver for a given vertex-pair set P ⊆ V× V, i.e., it preserves reachability between any pair of vertices in P under single edge (or vertex) failure. Our result is a significant improvement over the previous best O(n |P|) bound obtained as a corollary of single-source reachability preserver construction. We prove our upper bound by exploiting the special structure of single fault-tolerant reachability preserver for any pair, and then considering the interaction among such structures for different pairs. - In the lower bound side, we show that a 2-fault-tolerant reachability preserver for a vertex-pair set P ⊆ V×V of size Ω(n^ε), for even any arbitrarily small ε, requires at least Ω(n^(1+ε/8)) edges. This refutes the existence of linear-sized dual fault-tolerant preservers for reachability for any polynomial sized vertex-pair set. - We also present the first sub-quadratic bound of at most Õ(k 2^k n^(2-1/k)) size, for strong-connectivity preservers of directed graphs under k failures. To the best of our knowledge no non-trivial bound for this problem was known before, for a general k. We get our result by adopting the color-coding technique of Alon, Yuster, and Zwick [JACM'95].

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
Keywords
  • Preservers
  • Strong-connectivity
  • Reachability
  • Fault-tolerant
  • Graph sparsification

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References

  1. Amir Abboud and Greg Bodwin. Reachability preservers: New extremal bounds and approximation algorithms. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 1865-1883, 2018. Google Scholar
  2. Noga Alon, Raphael Yuster, and Uri Zwick. Color-coding. J. ACM, 42(4):844-856, 1995. Google Scholar
  3. Surender Baswana, Shreejit Ray Chaudhury, Keerti Choudhary, and Shahbaz Khan. Dynamic DFS in undirected graphs: Breaking the o(m) barrier. SIAM J. Comput., 48(4):1335-1363, 2019. Google Scholar
  4. Surender Baswana, Keerti Choudhary, Moazzam Hussain, and Liam Roditty. Approximate single source fault tolerant shortest path. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, pages 1901-1915, 2018. Google Scholar
  5. Surender Baswana, Keerti Choudhary, and Liam Roditty. Fault tolerant subgraph for single source reachability: generic and optimal. In Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, pages 509-518, 2016. Google Scholar
  6. Surender Baswana, Keerti Choudhary, and Liam Roditty. An efficient strongly connected components algorithm in the fault tolerant model. Algorithmica, 81(3):967-985, 2019. Google Scholar
  7. Surender Baswana and Neelesh Khanna. Approximate shortest paths avoiding a failed vertex: Near optimal data structures for undirected unweighted graphs. Algorithmica, 66(1):18-50, 2013. Google Scholar
  8. Davide Bilò, Fabrizio Grandoni, Luciano Gualà, Stefano Leucci, and Guido Proietti. Improved purely additive fault-tolerant spanners. In Algorithms - ESA 2015 - 23rd Annual European Symposium, Proceedings, pages 167-178, 2015. Google Scholar
  9. Davide Bilò, Luciano Gualà, Stefano Leucci, and Guido Proietti. Multiple-edge-fault-tolerant approximate shortest-path trees. In 33rd Symposium on Theoretical Aspects of Computer Science, STACS 2016, pages 18:1-18:14, 2016. Google Scholar
  10. Greg Bodwin. Linear size distance preservers. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 600-615, 2017. Google Scholar
  11. Greg Bodwin, Fabrizio Grandoni, Merav Parter, and Virginia Vassilevska Williams. Preserving distances in very faulty graphs. In 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, pages 73:1-73:14, 2017. Google Scholar
  12. Shiri Chechik. Fault-tolerant compact routing schemes for general graphs. Inf. Comput., 222:36-44, 2013. Google Scholar
  13. Shiri Chechik, Michael Langberg, David Peleg, and Liam Roditty. Fault-tolerant spanners for general graphs. In Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, pages 435-444, 2009. Google Scholar
  14. Shiri Chechik, Michael Langberg, David Peleg, and Liam Roditty. f-sensitivity distance oracles and routing schemes. In 18th Annual European Symposium on Algorithms - ESA (1), pages 84-96, 2010. Google Scholar
  15. Keerti Choudhary. An optimal dual fault tolerant reachability oracle. In 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, pages 130:1-130:13, 2016. Google Scholar
  16. Don Coppersmith and Michael Elkin. Sparse sourcewise and pairwise distance preservers. SIAM J. Discrete Math., 20(2):463-501, 2006. Google Scholar
  17. Artur Czumaj and Hairong Zhao. Fault-tolerant geometric spanners. Discrete & Computational Geometry, 32(2):207-230, 2004. Google Scholar
  18. Camil Demetrescu, Mikkel Thorup, Rezaul Alam Chowdhury, and Vijaya Ramachandran. Oracles for distances avoiding a failed node or link. SIAM J. Comput., 37(5):1299-1318, 2008. Google Scholar
  19. Michael Dinitz and Robert Krauthgamer. Fault-tolerant spanners: better and simpler. In Proceedings of the 30th Annual ACM Symposium on Principles of Distributed Computing, PODC 2011, pages 169-178, 2011. Google Scholar
  20. Ran Duan and Seth Pettie. Dual-failure distance and connectivity oracles. In Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2009, pages 506-515, 2009. Google Scholar
  21. Loukas Georgiadis, Giuseppe F. Italiano, Aikaterini Karanasiou, Charis Papadopoulos, and Nikos Parotsidis. Sparse certificates for 2-connectivity in directed graphs. Theor. Comput. Sci., 698:40-66, 2017. Google Scholar
  22. Loukas Georgiadis, Giuseppe F. Italiano, Luigi Laura, and Nikos Parotsidis. 2-edge connectivity in directed graphs. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 1988-2005, 2015. Google Scholar
  23. Loukas Georgiadis, Giuseppe F. Italiano, Luigi Laura, and Nikos Parotsidis. 2-vertex connectivity in directed graphs. In Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6-10, 2015, Proceedings, Part I, pages 605-616, 2015. Google Scholar
  24. Loukas Georgiadis, Giuseppe F. Italiano, and Nikos Parotsidis. Strong connectivity in directed graphs under failures, with applications. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 1880-1899, 2017. Google Scholar
  25. Manoj Gupta and Shahbaz Khan. Multiple source dual fault tolerant BFS trees. In 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, pages 127:1-127:15, 2017. Google Scholar
  26. Tamás Lukovszki. New results of fault tolerant geometric spanners. In Algorithms and Data Structures, 6th International Workshop, WADS '99, Proceedings, pages 193-204, 1999. Google Scholar
  27. Hiroshi Nagamochi and Toshihide Ibaraki. A linear-time algorithm for finding a sparse k-connected spanning subgraph of a k-connected graph. Algorithmica, 7(5&6):583-596, 1992. Google Scholar
  28. Merav Parter. Dual failure resilient BFS structure. In Proceedings of the 2015 ACM Symposium on Principles of Distributed Computing, PODC 2015, pages 481-490, 2015. Google Scholar
  29. Merav Parter and David Peleg. Sparse fault-tolerant BFS trees. In Algorithms - ESA 2013 - 21st Annual European Symposium, Proceedings, pages 779-790, 2013. Google Scholar
  30. Merav Parter and David Peleg. Fault tolerant approximate BFS structures. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, pages 1073-1092, 2014. Google Scholar
  31. Jan van den Brand and Thatchaphol Saranurak. Sensitive distance and reachability oracles for large batch updates. In 60th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2019, Baltimore, Maryland, USA, November 9-12, 2019, pages 424-435, 2019. Google Scholar
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