Document Open Access Logo

Space-Efficient Fault-Tolerant Diameter Oracles

Authors Davide Bilò , Sarel Cohen, Tobias Friedrich , Martin Schirneck



PDF
Thumbnail PDF

File

LIPIcs.MFCS.2021.18.pdf
  • Filesize: 0.8 MB
  • 16 pages

Document Identifiers

Author Details

Davide Bilò
  • Department of Humanities and Social Sciences, University of Sassari, Italy
Sarel Cohen
  • Hasso Plattner Institute, University of Potsdam, Germany
Tobias Friedrich
  • Hasso Plattner Institute, University of Potsdam, Germany
Martin Schirneck
  • Hasso Plattner Institute, University of Potsdam, Germany

Cite AsGet BibTex

Davide Bilò, Sarel Cohen, Tobias Friedrich, and Martin Schirneck. Space-Efficient Fault-Tolerant Diameter Oracles. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 18:1-18:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.MFCS.2021.18

Abstract

We design f-edge fault-tolerant diameter oracles (f-FDO, or simply FDO if f = 1). For a given directed or undirected and possibly edge-weighted graph G with n vertices and m edges and a positive integer f, we preprocess the graph and construct a data structure that, when queried with a set F of edges, where |F| ⩽ f, returns the diameter of G-F. An f-FDO has stretch σ ⩾ 1 if the returned value D^ satisfies diam(G-F) ⩽ D^ ⩽ σ diam(G-F). For the case of a single edge failure (f = 1) in an unweighted directed graph, there exists an approximate FDO by Henzinger et al. [ITCS 2017] with stretch (1+ε), constant query time, space O(m), and a combinatorial preprocessing time of Õ(mn + n^{1.5} √{Dm/ε}), where D is the diameter. We present an FDO for directed graphs with the same stretch, query time, and space. It has a preprocessing time of Õ(mn + n²/ε), which is better for constant ε > 0. The preprocessing time nearly matches a conditional lower bound for combinatorial algorithms, also by Henzinger et al. With fast matrix multiplication, we achieve a preprocessing time of Õ(n^{2.5794} + n²/ε). We further prove an information-theoretic lower bound showing that any FDO with stretch better than 3/2 requires Ω(m) bits of space. Thus, for constant 0 < ε < 3/2, our combinatorial (1+ε)-approximate FDO is near-optimal in all parameters. In the case of multiple edge failures (f > 1) in undirected graphs with non-negative edge weights, we give an f-FDO with stretch (f+2), query time O(f²log²{n}), Õ(fn) space, and preprocessing time Õ(fm). We complement this with a lower bound excluding any finite stretch in o(fn) space. Many real-world networks have polylogarithmic diameter. We show that for those graphs and up to f = o(log n/ log log n) failures one can swap approximation for query time and space. We present an exact combinatorial f-FDO with preprocessing time mn^{1+o(1)}, query time n^o(1), and space n^{2+o(1)}. When using fast matrix multiplication instead, the preprocessing time can be improved to n^{ω+o(1)}, where ω < 2.373 is the matrix multiplication exponent.

Subject Classification

ACM Subject Classification
  • Theory of computation → Shortest paths
  • Theory of computation → Data structures design and analysis
  • Theory of computation → Cell probe models and lower bounds
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • derandomization
  • diameter
  • distance sensitivity oracle
  • fault-tolerant data structure
  • space lower bound

Metrics

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

References

  1. Yehuda Afek, Anat Bremler-Barr, Haim Kaplan, Edith Cohen, and Michael Merritt. Restoration by Path Concatenation: Fast Recovery of MPLS Paths. Distributed Computing, 15:273-283, 2002. URL: https://doi.org/10.1007/s00446-002-0080-6.
  2. Josh Alman and Virginia Vassilevska Williams. A Refined Laser Method and Faster Matrix Multiplication. In Proceedings of the 32nd Symposium on Discrete Algorithms (SODA), pages 522-539, 2021. URL: https://doi.org/10.1137/1.9781611976465.32.
  3. Noga Alon, Shiri Chechik, and Sarel Cohen. Deterministic Combinatorial Replacement Paths and Distance Sensitivity Oracles. In Proceedings of the 46th International Colloquium on Automata, Languages, and Programming, (ICALP), pages 12:1-12:14, 2019. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.12.
  4. Noga Alon and Moni Naor. Derandomization, Witnesses for Boolean Matrix Multiplication and Construction of Perfect Hash Functions. Algorithmica, 16:434-449, 1996. URL: https://doi.org/10.1007/BF01940874.
  5. Ingo Althöfer, Gautam Das, David P. Dobkin, Deborah Joseph, and José Soares. On Sparse Spanners of Weighted Graphs. Discrete and Computational Geometry, 9:81-100, 1993. URL: https://doi.org/10.1007/BF02189308.
  6. Bertie Ancona, Monika Henzinger, Liam Roditty, Virginia Vassilevska Williams, and Nicole Wein. Algorithms and Hardness for Diameter in Dynamic Graphs. In Proceedings of the 46th International Colloquium on Automata, Languages, and Programming (ICALP), pages 13:1-13:14, 2019. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.13.
  7. Surender Baswana and Telikepalli Kavitha. Faster Algorithms for All-pairs Approximate Shortest Paths in Undirected Graphs. SIAM Journal on Computing, 39:2865-2896, 2010. URL: https://doi.org/10.1137/080737174.
  8. Aaron Bernstein and David R. Karger. A Nearly Optimal Oracle for Avoiding Failed Vertices and Edges. In Proceedings of the 41st Symposium on Theory of Computing (STOC), pages 101-110, 2009. URL: https://doi.org/10.1145/1536414.1536431.
  9. Davide Bilò, Luciano Gualà, Stefano Leucci, and Guido Proietti. Multiple-Edge-Fault-Tolerant Approximate Shortest-Path Trees. In Proceedings of the 33rd Symposium on Theoretical Aspects of Computer Science (STACS), pages 18:1-18:14, 2016. URL: https://doi.org/10.4230/LIPIcs.STACS.2016.18.
  10. Édouard Bonnet. 4 vs 7 Sparse Undirected Unweighted Diameter is SETH-hard at Time n^4/3. In Proceedings of 48th International Colloquium on Automata, Languages, and Programming, (ICALP), 2021. To appear. Google Scholar
  11. Shiri Chechik and Sarel Cohen. Near Optimal Algorithms for the Single Source Replacement Paths Problem. In Proceedings of the 30th Symposium on Discrete Algorithms (SODA), pages 2090-2109, 2019. URL: https://doi.org/10.1137/1.9781611975482.126.
  12. Shiri Chechik and Sarel Cohen. Distance Sensitivity Oracles with Subcubic Preprocessing Time and Fast Query Time. In Proccedings of the 52nd Symposium on Theory of Computing (STOC), pages 1375-1388, 2020. URL: https://doi.org/10.1145/3357713.3384253.
  13. Shiri Chechik, Sarel Cohen, Amos Fiat, and Haim Kaplan. (1 + ε)-Approximate f-Sensitive Distance Oracles. In Proceedings of the 28th Symposium on Discrete Algorithms (SODA), pages 1479-1496, 2017. URL: https://doi.org/10.1137/1.9781611974782.96.
  14. Shiri Chechik, Daniel H. Larkin, Liam Roditty, Grant Schoenebeck, Robert E. Tarjan, and Virginia Vassilevska Williams. Better Approximation Algorithms for the Graph Diameter. In Proceedings of the 25th Symposium on Discrete Algorithms (SODA), pages 1041-1052, 2014. URL: https://doi.org/10.1137/1.9781611973402.78.
  15. Shiri Chechik and Ofer Magen. Near Optimal Algorithm for the Directed Single Source Replacement Paths Problem. In Proceedings of the 47th International Colloquium on Automata, Languages, and Programming (ICALP), pages 81:1-81:17, 2020. URL: https://doi.org/10.4230/LIPIcs.ICALP.2020.81.
  16. Keerti Choudhary and Omer Gold. Extremal Distances in Directed Graphs: Tight Spanners and Near-Optimal Approximation Algorithms. In Proceedings of the 31st Symposium on Discrete Algorithms (SODA), pages 495-514, 2020. URL: https://doi.org/10.1137/1.9781611975994.30.
  17. Fan Chung and Linyuan Lu. The Average Distances in Random Graphs with Given Expected Degrees. Proceedings of the National Academy of Sciences, 99:15879-15882, 2002. URL: https://doi.org/10.1073/pnas.252631999.
  18. Edith Cohen and Uri Zwick. All-Pairs Small-Stretch Paths. Journal of Algorithms, 38:335-353, 2001. URL: https://doi.org/10.1006/jagm.2000.1117.
  19. Camil Demetrescu, Mikkel Thorup, Rezaul Alam Chowdhury, and Vijaya Ramachandran. Oracles for Distances Avoiding a Failed Node or Link. SIAM Journal on Computing, 37:1299-1318, 2008. URL: https://doi.org/10.1137/S0097539705429847.
  20. Martin Dietzfelbinger, Anna R. Karlin, Kurt Mehlhorn, Friedhelm Meyer auf der Heide, Hans Rohnert, and Robert E. Tarjan. Dynamic Perfect Hashing: Upper and Lower Bounds. SIAM Journal on Computing, 23:738-761, 1994. URL: https://doi.org/10.1137/S0097539791194094.
  21. Ran Duan, Yong Gu, and Hanlin Ren. Approximate Distance Oracles Subject to Multiple Vertex Failures. In PProceedings of the 32nd Symposium on Discrete Algorithms (SODA), pages 2497-2516, 2021. URL: https://doi.org/10.1137/1.9781611976465.148.
  22. Ran Duan and Seth Pettie. Dual-Failure Distance and Connectivity Oracles. In Proceedings of the 20th Symposium on Discrete Algorithms (SODA), pages 506-515, 2009. URL: https://dl.acm.org/citation.cfm?id=1496770.1496826.
  23. Ran Duan and Tianyi Zhang. Improved Distance Sensitivity Oracles via Tree Partitioning. In Proceedings of the 15th Algorithms and Data Structures Symposium (WADS), pages 349-360, 2017. URL: https://doi.org/10.1007/978-3-319-62127-2_30.
  24. Tobias Friedrich and Anton Krohmer. On the Diameter of Hyperbolic Random Graphs. SIAM Journal on Discrete Mathematics, 32:1314-1334, 2018. Google Scholar
  25. Pawel Gawrychowski, Haim Kaplan, Shay Mozes, Micha Sharir, and Oren Weimann. Voronoi Diagrams on Planar Graphs, and Computing the Diameter in Deterministic Õ(n^5/3) Time. In Proceedings of the 29th Symposium on Discrete Algorithms (SODA), pages 495-514, 2018. URL: https://doi.org/10.1137/1.9781611975031.33.
  26. Fabrizio Grandoni and Virginia Vassilevska Williams. Improved Distance Sensitivity Oracles via Fast Single-Source Replacement Paths. In Proceedings of the 53rd Symposium on Foundations of Computer Science (FOCS), pages 748-757, 2012. URL: https://doi.org/10.1109/FOCS.2012.17.
  27. Fabrizio Grandoni and Virginia Vassilevska Williams. Faster Replacement Paths and Distance Sensitivity Oracles. ACM Transaction on Algorithms, 16:15:1-15:25, 2020. URL: https://doi.org/10.1145/3365835.
  28. Yong Gu and Hanlin Ren. Constructing a Distance Sensitivity Oracle in O(n^2.5794M) Time. In Proceedings of the 48th International Colloquium on Automata, Languages, and Programming (ICALP), 2021. To appear. Google Scholar
  29. Torben Hagerup, Peter Bro Miltersen, and Rasmus Pagh. Deterministic Dictionaries. Journal of Algorithms, 41:69-85, 2001. URL: https://doi.org/10.1006/jagm.2001.1171.
  30. Monika Henzinger, Andrea Lincoln, Stefan Neumann, and Virginia Vassilevska Williams. Conditional Hardness for Sensitivity Problems. In Proceedings of the 8th Conference on Innovations in Theoretical Computer Science (ITCS), pages 26:1-26:31, 2017. URL: https://doi.org/10.4230/LIPIcs.ITCS.2017.26.
  31. Remco van der Hofstad. Random Graphs and Complex Networks, volume 1 of Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, UK, 2016. URL: https://doi.org/10.1017/9781316779422.
  32. Giuseppe F. Italiano, Luigi Laura, and Federico Santaroni. Finding Strong Bridges and Strong Articulation Points in Linear Time. Theoretical Computer Science, 447:74-84, 2012. URL: https://doi.org/10.1016/j.tcs.2011.11.011.
  33. Telikepalli Kavitha. Faster Algorithms for All-Pairs Small Stretch Distances in Weighted Graphs. Algorithmica, 63:224-245, 2012. URL: https://doi.org/10.1007/s00453-011-9529-y.
  34. Valerie King. Fully Dynamic Algorithms for Maintaining All-Pairs Shortest Paths and Transitive Closure in Digraphs. In Proceedings of the 40th Symposium on Foundations of Computer Science (FOCS), pages 81-91, 1999. URL: https://doi.org/10.1109/SFFCS.1999.814580.
  35. Jon M. Kleinberg. Navigation in a Small World. Nature, 406:845-845, 2000. URL: https://doi.org/10.1038/35022643.
  36. Rasmus Pagh and Flemming Friche Rodler. Cuckoo Hashing. Journal of Algorithms, 51:122-144, 2004. URL: https://doi.org/10.1016/j.jalgor.2003.12.002.
  37. Seth Pettie. Sensitivity Analysis of Minimum Spanning Trees in Sub-Inverse-Ackermann Time. Journal of Graph Algorithms and Applications, 19:375-391, 2015. URL: https://doi.org/10.7155/jgaa.00365.
  38. Hanlin Ren. Improved Distance Sensitivity Oracles with Subcubic Preprocessing Time. In Proceedings of the 28th European Symposium on Algorithms (ESA), pages 79:1-79:13, 2020. URL: https://doi.org/10.4230/LIPIcs.ESA.2020.79.
  39. Hanlin Ren. Improved Distance Sensitivity Oracles with Subcubic Preprocessing Time. CoRR, abs/2007.11495, 2020. ArXiv preprint. Full version of [Hanlin Ren, 2020]. URL: http://arxiv.org/abs/2007.11495.
  40. Liam Roditty. Approximating the Diameter. In Ming-Yang Kao, editor, Encyclopedia of Algorithms, pages 116-117. Springer, New York City, NY, USA, 2016. URL: https://doi.org/10.1007/978-1-4939-2864-4_566.
  41. Liam Roditty and Uri Zwick. Replacement Paths and k Simple Shortest Paths in Unweighted Directed Graphs. ACM Transaction on Algorithms, 8:33:1-33:11, 2012. URL: https://doi.org/10.1145/2344422.2344423.
  42. Mikkel Thorup and Uri Zwick. Approximate Distance Oracles. In Proceedings on 33rd Symposium on Theory of Computing (STOC), pages 183-192, 2001. URL: https://doi.org/10.1145/380752.380798.
  43. Oren Weimann and Raphael Yuster. Replacement Paths and Distance Sensitivity Oracles via Fast Matrix Multiplication. ACM Transactions on Algorithms, 9:14:1-14:13, 2013. URL: https://doi.org/10.1145/2438645.2438646.
  44. Uri Zwick. All Pairs Shortest Paths Using Bridging Sets and Rectangular Matrix Multiplication. Journal of the ACM, 49:289–317, 2002. URL: https://doi.org/10.1145/567112.567114.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail