Document Open Access Logo

Fault-Tolerant ST-Diameter Oracles

Authors Davide Bilò , Keerti Choudhary , Sarel Cohen , Tobias Friedrich , Simon Krogmann , Martin Schirneck

PDF
Thumbnail PDF

File

LIPIcs.ICALP.2023.24.pdf
  • Filesize: 0.86 MB
  • 20 pages

Document Identifiers

Author Details

Davide Bilò
  • Department of Information Engineering, Computer Science and Mathematics, University of L'Aquila, Italy
Keerti Choudhary
  • Department of Computer Science and Engineering, Indian Institute of Technology Delhi, India
Sarel Cohen
  • School of Computer Science, Tel-Aviv-Yaffo Academic College, Israel
Tobias Friedrich
  • Hasso Plattner Institute, Universität Potsdam, Germany
Simon Krogmann
  • Hasso Plattner Institute, Universität Potsdam, Germany
Martin Schirneck
  • Faculty of Computer Science, Universität Wien, Austria

Cite AsGet BibTex

Davide Bilò, Keerti Choudhary, Sarel Cohen, Tobias Friedrich, Simon Krogmann, and Martin Schirneck. Fault-Tolerant ST-Diameter Oracles. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 24:1-24:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ICALP.2023.24

Abstract

We study the problem of estimating the ST-diameter of a graph that is subject to a bounded number of edge failures. An f-edge fault-tolerant ST-diameter oracle (f-FDO-ST) is a data structure that preprocesses a given graph G, two sets of vertices S,T, and positive integer f. When queried with a set F of at most f edges, the oracle returns an estimate D̂ of the ST-diameter diam(G-F,S,T), the maximum distance between vertices in S and T in G-F. The oracle has stretch σ ⩾ 1 if diam(G-F,S,T) ⩽ D̂ ⩽ σ diam(G-F,S,T). If S and T both contain all vertices, the data structure is called an f-edge fault-tolerant diameter oracle (f-FDO). An f-edge fault-tolerant distance sensitivity oracles (f-DSO) estimates the pairwise graph distances under up to f failures. We design new f-FDOs and f-FDO-STs by reducing their construction to that of all-pairs and single-source f-DSOs. We obtain several new tradeoffs between the size of the data structure, stretch guarantee, query and preprocessing times for diameter oracles by combining our black-box reductions with known results from the literature. We also provide an information-theoretic lower bound on the space requirement of approximate f-FDOs. We show that there exists a family of graphs for which any f-FDO with sensitivity f ⩾ 2 and stretch less than 5/3 requires Ω(n^{3/2}) bits of space, regardless of the query time.

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
Keywords
  • diameter oracles
  • distance sensitivity oracles
  • space lower bounds
  • fault-tolerant data structures

Metrics

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

Related Versions

References

  1. Donald Aingworth, Chandra Chekuri, Piotr Indyk, and Rajeev Motwani. Fast Estimation of Diameter and Shortest Paths (Without Matrix Multiplication). SIAM Journal on Computing, 28:1167-1181, 1999. URL: https://doi.org/10.1137/S0097539796303421.
  2. 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.
  3. 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.
  4. Arturs Backurs, Liam Roditty, Gilad Segal, Virginia Vassilevska Williams, and Nicole Wein. Toward Tight Approximation Bounds for Graph Diameter and Eccentricities. SIAM Journal on Computing, 50:1155-1199, 2021. URL: https://doi.org/10.1137/18M1226737.
  5. Surender Baswana, Keerti Choudhary, Moazzam Hussain, and Liam Roditty. Approximate Single-Source Fault Tolerant Shortest Path. ACM Transactions on Algorithms, 16:44:1-44:22, 2020. URL: https://doi.org/10.1145/3397532.
  6. Surender Baswana, Keerti Choudhary, and Liam Roditty. Fault-Tolerant Subgraph for Single-Source Reachability: General and Optimal. SIAM Journal on Computing, 47:80-95, 2018. URL: https://doi.org/10.1137/16M1087643.
  7. Surender Baswana and Telikepalli Kavitha. Faster Algorithms for Approximate Distance Oracles and All-Pairs Small Stretch Paths. In Proceedings of the 47th Symposium on Foundations of Computer Science (FOCS), pages 591-602, 2006. URL: https://doi.org/10.1109/FOCS.2006.29.
  8. Surender Baswana and Neelesh Khanna. Approximate Shortest Paths Avoiding a Failed Vertex: Near Optimal Data Structures for Undirected Unweighted Graphs. Algorithmica, 66:18-50, 2013. URL: https://doi.org/10.1007/s00453-012-9621-y.
  9. Michael A. Bender and Martin Farach-Colton. The LCA problem revisited. In Gaston H. Gonnet, Daniel Panario, and Alfredo Viola, editors, LATIN 2000: Theoretical Informatics, 4th Latin American Symposium, Punta del Este, Uruguay, April 10-14, 2000, Proceedings, volume 1776 of Lecture Notes in Computer Science, pages 88-94. Springer, 2000. URL: https://doi.org/10.1007/10719839_9.
  10. Aaron Bernstein and David R. Karger. Improved Distance Sensitivity Oracles via Random Sampling. In Proceedings of the 19th Symposium on Discrete Algorithms (SODA), pages 34-43, 2008. URL: https://dl.acm.org/citation.cfm?id=1347082.1347087.
  11. 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.
  12. Davide Bilò, Keerti Choudhary, Luciano Gualà, Stefano Leucci, Merav Parter, and Guido Proietti. Efficient Oracles and Routing Schemes for Replacement Paths. In Proceedings of the 35th Symposium on Theoretical Aspects of Computer Science (STACS), pages 13:1-13:15, 2018. URL: https://doi.org/10.4230/LIPIcs.STACS.2018.13.
  13. Davide Bilò, Luciano Gualà, Stefano Leucci, and Guido Proietti. Compact and Fast Sensitivity Oracles for Single-Source Distances. In Piotr Sankowski and Christos D. Zaroliagis, editors, Proceedings of the 24th European Symposium on Algorithms (ESA), pages 13:1-13:14, 2016. URL: https://doi.org/10.4230/LIPIcs.ESA.2016.13.
  14. Davide Bilò, Luciano Gualà, Stefano Leucci, and Guido Proietti. Multiple-Edge-Fault-Tolerant Approximate Shortest-Path Trees. Algorithmica, 84:37-59, 2022. URL: https://doi.org/10.1007/s00453-021-00879-8.
  15. Davide Bilò, Keerti Choudhary, Sarel Cohen, Tobias Friedrich, and Martin Schirneck. Deterministic Sensitivity Oracles for Diameter, Eccentricities and All Pairs Distances. In Proceedings of the 49th International Colloquium on Automata, Languages, and Programming (ICALP), pages 22:1-22:19, 2022. URL: https://doi.org/10.4230/LIPIcs.ICALP.2022.22.
  16. Davide Bilò, Sarel Cohen, Tobias Friedrich, and Martin Schirneck. Near-Optimal Deterministic Single-Source Distance Sensitivity Oracles. In Proceedings of the 29th European Symposium on Algorithms (ESA), pages 18:1-18:17, 2021. URL: https://doi.org/10.4230/LIPIcs.ESA.2021.18.
  17. Davide Bilò, Sarel Cohen, Tobias Friedrich, and Martin Schirneck. Space-Efficient Fault-Tolerant Diameter Oracles. In Proceedings of the 46th International Symposium on Mathematical Foundations of Computer Science (MFCS), pages 18:1-18:16, 2021. URL: https://doi.org/10.4230/LIPIcs.MFCS.2021.18.
  18. Jan van den Brand and Thatchaphol Saranurak. Sensitive Distance and Reachability Oracles for Large Batch Updates. In Proceedings of the 60th Symposium on Foundations of Computer Science (FOCS), pages 424-435, 2019. URL: https://doi.org/10.1109/FOCS.2019.00034.
  19. Sergio Cabello, Erin W. Chambers, and Jeff Erickson. Multiple-Source Shortest Paths in Embedded Graphs. SIAM J. Comput., 42:1542-1571, 2013. URL: https://doi.org/10.1137/120864271.
  20. Diptarka Chakraborty and Keerti Choudhary. New Extremal Bounds for Reachability and Strong-Connectivity Preservers Under Failures. In Proceedings of the 47th International Colloquium on Automata, Languages, and Programming (ICALP), pages 25:1-25:20, 2020. URL: https://doi.org/10.4230/LIPIcs.ICALP.2020.25.
  21. Abraham Charnes. Optimality and Degeneracy in Linear Programming. Econometrica, 20:160-170, 1952. Google Scholar
  22. 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.
  23. 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.
  24. Shiri Chechik, Michael Langberg, David Peleg, and Liam Roditty. f-Sensitivity Distance Oracles and Routing Schemes. Algorithmica, 63:861-882, 2012. URL: https://doi.org/10.1007/s00453-011-9543-0.
  25. 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.
  26. 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.
  27. Pierluigi Crescenzi, Roberto Grossi, Leonardo Lanzi, and Andrea Marino. On Computing the Diameter of Real-World Directed (Weighted) Graphs. In Ralf Klasing, editor, Proceedings of the 11th Symposium on Experimental Algorithms (SEA), pages 99-110, 2012. URL: https://doi.org/10.1007/978-3-642-30850-5_10.
  28. Mina Dalirrooyfard, Virginia Vassilevska Williams, Nikhil Vyas, and Nicole Wein. Tight Approximation Algorithms for Bichromatic Graph Diameter and Related Problems. In Proceedings of the 46th International Colloquium on Automata, Languages, and Programming (ICALP), pages 47:1-47:15, 2019. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.47.
  29. Camil Demetrescu and Mikkel Thorup. Oracles for Distances Avoiding a Link-Failure. In Proceedings of the 13th Symposium on Discrete Algorithms (SODA), pages 838-843, 2002. URL: https://dl.acm.org/citation.cfm?id=545381.545490.
  30. Camil Demetrescu, Mikkel Thorup, Rezaul A. 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.
  31. Ran Duan, Yong Gu, and Hanlin Ren. Approximate Distance Oracles Subject to Multiple Vertex Failures. In Proceedings of the 32nd Symposium on Discrete Algorithms (SODA), pages 2497-2516, 2021. URL: https://doi.org/10.1137/1.9781611976465.148.
  32. 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: http://dl.acm.org/citation.cfm?id=1496770.1496826.
  33. Ran Duan and Seth Pettie. Connectivity Oracles for Failure Prone Graphs. In Leonard J. Schulman, editor, Proceedings of the 42nd Symposium on Theory of Computing (STOC), pages 465-474, 2010. URL: https://doi.org/10.1145/1806689.1806754.
  34. Ran Duan and Hanlin Ren. Maintaining Exact Distances under Multiple Edge Failures. In Proceedings of the 54th Symposium on Theory of Computing (STOC), pages 1093-1101, 2022. URL: https://doi.org/10.1145/3519935.3520002.
  35. 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.
  36. 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.
  37. 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), pages 76:1-76:20, 2021. URL: https://doi.org/10.4230/LIPIcs.ICALP.2021.76.
  38. Manoj Gupta and Aditi Singh. Generic Single Edge Fault Tolerant Exact Distance Oracle. In Proceedings of the 45th International Colloquium on Automata, Languages, and Programming, (ICALP), pages 72:1-72:15, 2018. URL: https://doi.org/10.4230/LIPIcs.ICALP.2018.72.
  39. David Hartvigsen and Russell Mardon. The All-Pairs Min Cut Problem and the Minimum Cycle Basis Problem on Planar Graphs. SIAM J. Discret. Math., 7:403-418, 1994. URL: https://doi.org/10.1137/S0895480190177042.
  40. 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.
  41. Ketan Mulmuley, Umesh V. Vazirani, and Vijay V. Vazirani. Matching Is as Easy as Matrix Inversion. Comb., 7:105-113, 1987. URL: https://doi.org/10.1007/BF02579206.
  42. Hanlin Ren. Improved Distance Sensitivity Oracles with Subcubic Preprocessing Time. Journal of Computer and System Sciences, 123:159-170, 2022. URL: https://doi.org/10.1016/j.jcss.2021.08.005.
  43. 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.
  44. Liam Roditty and Virginia Vassilevska Williams. Fast Approximation Algorithms for the Diameter and Radius of Sparse Graphs. In Proceedings of the 45th Symposium on Theory of Computing (STOC), pages 515-524, 2013. URL: https://doi.org/10.1145/2488608.2488673.
  45. Frank W. Takes and Walter A. Kosters. Determining the Diameter of Small World Networks. In Craig Macdonald, Iadh Ounis, and Ian Ruthven, editors, Proceedings of the 20th Conference on Information and Knowledge Management (CIKM), pages 1191-1196, 2011. URL: https://doi.org/10.1145/2063576.2063748.
  46. Mikkel Thorup. Undirected Single-Source Shortest Paths with Positive Integer Weights in Linear Time. Journal of the ACM, 46:362-394, 1999. URL: https://doi.org/10.1145/316542.316548.
  47. 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.
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
© 2023 Schloss Dagstuhl - LZI GmbH Imprint Privacy Contact