Efficient Oracles and Routing Schemes for Replacement Paths

Authors Davide Bilò, Keerti Choudhary, Luciano Gualà, Stefano Leucci, Merav Parter, Guido Proietti

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Davide Bilò
Keerti Choudhary
Luciano Gualà
Stefano Leucci
Merav Parter
Guido Proietti

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Davide Bilò, Keerti Choudhary, Luciano Gualà, Stefano Leucci, Merav Parter, and Guido Proietti. Efficient Oracles and Routing Schemes for Replacement Paths. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 13:1-13:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


Real life graphs and networks are prone to failure of nodes (vertices) and links (edges). In particular, for a pair of nodes s and t and a failing edge e in an n-vertex unweighted graph G=(V(G),E(G)), the replacement path pi_{G-e}(s,t) is a shortest s-t path that avoids e. In this paper we present several efficient constructions that, for every (s,t) \in S x T, where S, T \subseteq V(G), and every e \in E(G), maintain the collection of all pi_{G-e}(s,t), either implicitly (i.e., through compact data structures a.k.a. distance sensitivity oracles (DSO)), or explicitly (i.e., through sparse subgraphs a.k.a. fault-tolerant preservers (FTP)). More precisely, we provide the following results: (1) DSO: For every S,T \subseteq V(G), we construct a DSO for maintaining S x T distances under single edge (or vertex) faults. This DSO has size tilde{O}(n\sqrt{|S||T|}) and query time of O(\sqrt{|S||T|}). At the expense of having quasi-polynomial query time, the size of the oracle can be improved to tilde{O}(n|S|+|T|\sqrt{|S|n}), which is optimal for |T| = Omega(sqrt{n|S|}). When |T| = Omega(n^frac{3}{4} |S|^frac{1}{4}), the construction can be further refined in order to get a polynomial query time. We also consider the approximate additive setting, and show a family of DSOs that exhibits a tradeoff between the additive stretch and the size of the oracle. Finally, for the meaningful single-source case, the above result is complemented by a lower bound conditioned on the Set-Intersection conjecture. This lower bound establishes a separation between the oracle and the subgraph settings. (2) FTP: We show the construction of a path-reporting DSO of size tilde{O}(n^{4/3}(|S||T|)^{1/3}) reporting pi_{G-e}(s,t) in O(|pi_{G-e}(s,t)|+(n|S||T|)^{1/3}) time. Such a DSO can be transformed into a FTP having the same size, and moreover it can be elaborated in order to make it optimal (up to a poly-logarithmic factor) both in space and query time for the special case in which T=V(G). Our FTP improves over previous constructions when |T|=O(sqrt{|S|n}) (up to inverse poly-logarithmic factors). (3) Routing and Labeling Schemes: For the well-studied single-source setting, we present a novel routing scheme, that allows to route messages on pi_{G-e}(s,t) by using edge labels and routing tables of size tilde{O}(\sqrt{n}), and a header message of poly-logarithmic size. We also present a labeling scheme for the setting which is optimal in space up to constant factors.
  • Fault tolerant
  • Shortest path
  • Oracle
  • Routing


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  1. A. Abboud and G. Bodwin. The 4/3 additive spanner exponent is tight. In STOC, pages 351-361, 2016. Google Scholar
  2. Ittai Abraham, Shiri Chechik, Cyril Gavoille, and David Peleg. Forbidden-set distance labels for graphs of bounded doubling dimension. ACM Transactions on Algorithms (TALG), 12(2):22, 2016. Google Scholar
  3. 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
  4. Michael A. Bender and Martin Farach-Colton. The level ancestor problem simplified. Theor. Comput. Sci., 321(1):5-12, 2004. URL: http://dx.doi.org/10.1016/j.tcs.2003.05.002.
  5. Aaron Bernstein and David R. Karger. A nearly optimal oracle for avoiding failed vertices and edges. In Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, Bethesda, MD, USA, May 31 - June 2, 2009, pages 101-110, 2009. Google Scholar
  6. 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, Patras, Greece, September 14-16, 2015, Proceedings, pages 167-178, 2015. Google Scholar
  7. Davide Bilò, Luciano Gualà, Stefano Leucci, and Guido Proietti. Fault-tolerant approximate shortest-path trees. In Algorithms - ESA 2014 - 22th Annual European Symposium, Wroclaw, Poland, September 8-10, 2014. Proceedings, pages 137-148, 2014. Google Scholar
  8. Davide Bilò, Luciano Gualà, Stefano Leucci, and Guido Proietti. Compact and fast sensitivity oracles for single-source distances. In 24th Annual European Symposium on Algorithms, ESA 2016, August 22-24, 2016, Aarhus, Denmark, pages 13:1-13:14, 2016. 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, February 17-20, 2016, Orléans, France, pages 18:1-18:14, 2016. Google Scholar
  10. Greg Bodwin, Fabrizio Grandoni, Merav Parter, and Virginia Vassilevska Williams. Preserving Distances in Very Faulty Graphs. In Ioannis Chatzigiannakis, Piotr Indyk, Fabian Kuhn, and Anca Muscholl, editors, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017), volume 80 of Leibniz International Proceedings in Informatics (LIPIcs), pages 73:1-73:14, Dagstuhl, Germany, 2017. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. Google Scholar
  11. Jean Bourgain. On Lipschitz embedding of finite metric spaces in Hilbert space. Israel Journal of Mathematics, 52(1-2):46-52, 1985. Google Scholar
  12. Shiri Chechik. Fault-tolerant compact routing schemes for general graphs. Inf. Comput., 222:36-44, 2013. Google Scholar
  13. Shiri Chechik, Sarel Cohen, Amos Fiat, and Haim Kaplan. (1 + epsilon)-approximate f-sensitive distance oracles. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 1479-1496, 2017. Google Scholar
  14. Shiri Chechik, Michael Langberg, David Peleg, and Liam Roditty. f-sensitivity distance oracles and routing schemes. Algorithmica, 63(4):861-882, 2012. Google Scholar
  15. Don Coppersmith and Michael Elkin. Sparse source-wise and pair-wise distance preservers. In Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2005, Vancouver, British Columbia, Canada, January 23-25, 2005, pages 660-669. SIAM, 2005. URL: http://dl.acm.org/citation.cfm?id=1070432.1070524.
  16. Bruno Courcelle and Andrew Twigg. Compact forbidden-set routing. In Annual Symposium on Theoretical Aspects of Computer Science, pages 37-48. Springer, 2007. Google Scholar
  17. 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
  18. Ran Duan and Seth Pettie. Dual-failure distance and connectivity oracles. In SODA'09: Proceedings of 19th Annual ACM -SIAM Symposium on Discrete Algorithms, pages 506-515, Philadelphia, PA, USA, 2009. Society for Industrial and Applied Mathematics. Google Scholar
  19. Fabrizio Grandoni and Virginia Vassilevska Williams. Improved distance sensitivity oracles via fast single-source replacement paths. In 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, New Brunswick, NJ, USA, October 20-23, 2012, pages 748-757, 2012. Google Scholar
  20. Telikepalli Kavitha. New pairwise spanners. In Ernst W. Mayr and Nicolas Ollinger, editors, 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015, March 4-7, 2015, Garching, Germany, volume 30 of LIPIcs, pages 513-526. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2015.513.
  21. Kavindra Malik, A.K. Mittal, and Sumit K. Gupta. The k most vital arcs in the shortest path problem. Oper. Res. Lett., 8:223-227, 1989. Google Scholar
  22. Jiří Matoušek. On the distortion required for embedding finite metric spaces into normed spaces. Israel Journal of Mathematics, 93(1):333-344, 1996. Google Scholar
  23. Merav Parter. Vertex fault tolerant additive spanners. In International Symposium on Distributed Computing, pages 167-181. Springer, 2014. Google Scholar
  24. Merav Parter and David Peleg. Sparse fault-tolerant BFS trees. In Algorithms - ESA 2013 - 21st Annual European Symposium, Sophia Antipolis, France, September 2-4, 2013. Proceedings, pages 779-790, 2013. Google Scholar
  25. 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, Portland, Oregon, USA, January 5-7, 2014, pages 1073-1092, 2014. Google Scholar
  26. Mihai Patrascu and Liam Roditty. Distance oracles beyond the thorup-zwick bound. In Foundations of Computer Science (FOCS), 2010 51st Annual IEEE Symposium on, pages 815-823. IEEE, 2010. Google Scholar
  27. Mihai Patrascu, Liam Roditty, and Mikkel Thorup. A new infinity of distance oracles for sparse graphs. In Foundations of Computer Science (focs), 2012 Ieee 53rd Annual Symposium on, pages 738-747. IEEE, 2012. Google Scholar
  28. Christian Sommer, Elad Verbin, and Wei Yu. Distance oracles for sparse graphs. In Foundations of Computer Science, 2009. FOCS'09. 50th Annual IEEE Symposium on, pages 703-712. IEEE, 2009. Google Scholar
  29. Mikkel Thorup and Uri Zwick. Compact routing schemes. In SPAA, pages 1-10, 2001. Google Scholar
  30. Oren Weimann and Raphael Yuster. Replacement paths via fast matrix multiplication. In 51th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2010, October 23-26, 2010, Las Vegas, Nevada, USA, pages 655-662, 2010. Google Scholar
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