New Tradeoffs for Decremental Approximate All-Pairs Shortest Paths

Authors Michal Dory , Sebastian Forster , Yasamin Nazari , Tijn de Vos



PDF
Thumbnail PDF

File

LIPIcs.ICALP.2024.58.pdf
  • Filesize: 0.88 MB
  • 19 pages

Document Identifiers

Author Details

Michal Dory
  • University of Haifa, Israel
Sebastian Forster
  • Department of Computer Science, University of Salzburg, Austria
Yasamin Nazari
  • Vrije Universiteit Amsterdam, The Netherlands
Tijn de Vos
  • Department of Computer Science, University of Salzburg, Austria

Cite AsGet BibTex

Michal Dory, Sebastian Forster, Yasamin Nazari, and Tijn de Vos. New Tradeoffs for Decremental Approximate All-Pairs Shortest Paths. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 58:1-58:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.58

Abstract

We provide new tradeoffs between approximation and running time for the decremental all-pairs shortest paths (APSP) problem. For undirected graphs with m edges and n nodes undergoing edge deletions, we provide four new approximate decremental APSP algorithms, two for weighted and two for unweighted graphs. Our first result is (2+ε)-APSP with total update time Õ(m^{1/2}n^{3/2}) (when m = n^{1+c} for any constant 0 < c < 1). Prior to our work the fastest algorithm for weighted graphs with approximation at most 3 had total Õ(mn) update time for (1+ε)-APSP [Bernstein, SICOMP 2016]. Our second result is (2+ε, W_{u,v})-APSP with total update time Õ(nm^{3/4}), where the second term is an additive stretch with respect to W_{u,v}, the maximum weight on the shortest path from u to v. Our third result is (2+ε)-APSP for unweighted graphs in Õ(m^{7/4}) update time, which for sparse graphs (m = o(n^{8/7})) is the first subquadratic (2+ε)-approximation. Our last result for unweighted graphs is (1+ε, 2(k-1))-APSP, for k ≥ 2, with Õ(n^{2-1/k}m^{1/k}) total update time (when m = n^{1+c} for any constant c > 0). For comparison, in the special case of (1+ε, 2)-approximation, this improves over the state-of-the-art algorithm by [Henzinger, Krinninger, Nanongkai, SICOMP 2016] with total update time of Õ(n^{2.5}). All of our results are randomized, work against an oblivious adversary, and have constant query time.

Subject Classification

ACM Subject Classification
  • Theory of computation → Dynamic graph algorithms
  • Theory of computation → Shortest paths
Keywords
  • Decremental Shortest Path
  • All-Pairs Shortest Paths

Metrics

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

References

  1. Amir Abboud and Greg Bodwin. The 4/3 additive spanner exponent is tight. Journal of the ACM (JACM), 64(4):1-20, 2017. Google Scholar
  2. Amir Abboud, Greg Bodwin, and Seth Pettie. A hierarchy of lower bounds for sublinear additive spanners. SIAM Journal on Computing, 47(6):2203-2236, 2018. Google Scholar
  3. Amir Abboud, Karl Bringmann, and Nick Fischer. Stronger 3-sum lower bounds for approximate distance oracles via additive combinatorics. In Proc. of the 55th Annual ACM Symposium on Theory of Computing (STOC 2023), 2023. URL: https://doi.org/10.48550/arXiv.2211.07058.
  4. Amir Abboud, Karl Bringmann, Seri Khoury, and Or Zamir. Hardness of approximation in P via short cycle removal: cycle detection, distance oracles, and beyond. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing (STOC 2022), pages 1487-1500, 2022. URL: https://doi.org/10.1145/3519935.3520066.
  5. Ittai Abraham and Shiri Chechik. Dynamic decremental approximate distance oracles with (1+ε, 2) stretch. CoRR, abs/1307.1516, 2013. URL: https://arxiv.org/abs/1307.1516.
  6. Ittai Abraham, Shiri Chechik, and Kunal Talwar. Fully dynamic all-pairs shortest paths: Breaking the o(n) barrier. In Proceedings of the 17th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX 2014) and the 18th International Workshop on Randomization and Computation (APPROX/RANDOM 2014), pages 1-16, 2014. URL: https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.1.
  7. Donald Aingworth, Chandra Chekuri, Piotr Indyk, and Rajeev Motwani. Fast estimation of diameter and shortest paths (without matrix multiplication). SIAM Journal on Computing, 28(4):1167-1181, 1999. Announced at SODA 1996. URL: https://doi.org/10.1137/S0097539796303421.
  8. Maor Akav and Liam Roditty. An almost 2-approximation for all-pairs of shortest paths in subquadratic time. In Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, (SODA 2020), pages 1-11, 2020. URL: https://doi.org/10.1137/1.9781611975994.1.
  9. Maor Akav and Liam Roditty. A unified approach for all pairs approximate shortest paths in weighted undirected graphs. In Proceedings of the 29th Annual European Symposium on Algorithms (ESA 2021), volume 204, pages 4:1-4:18, 2021. URL: https://doi.org/10.4230/LIPIcs.ESA.2021.4.
  10. 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 (SICOMP), 50(4):1155-1199, 2021. Google Scholar
  11. Surender Baswana, Vishrut Goyal, and Sandeep Sen. All-pairs nearly 2-approximate shortest paths in o(n²polylogn) time. Theoretical Computer Science, 410(1):84-93, 2009. Announced at STACS 2005. URL: https://doi.org/10.1016/j.tcs.2008.10.018.
  12. Surender Baswana, Ramesh Hariharan, and Sandeep Sen. Maintaining all-pairs approximate shortest paths under deletion of edges. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2003), pages 394-403, 2003. Google Scholar
  13. Surender Baswana, Ramesh Hariharan, and Sandeep Sen. Improved decremental algorithms for maintaining transitive closure and all-pairs shortest paths. Journal of Algorithms, 62(2):74-92, 2007. Announced at STOC 2002. URL: https://doi.org/10.1016/j.jalgor.2004.08.004.
  14. Surender Baswana and Telikepalli Kavitha. Faster algorithms for all-pairs approximate shortest paths in undirected graphs. SIAM Journal on Computing, 39(7):2865-2896, 2010. Announced at FOCS 2006. URL: https://doi.org/10.1137/080737174.
  15. Surender Baswana, Sumeet Khurana, and Soumojit Sarkar. Fully dynamic randomized algorithms for graph spanners. ACM Transactions on Algorithms, 8(4):35:1-35:51, 2012. URL: https://doi.org/10.1145/2344422.2344425.
  16. Piotr Berman and Shiva Prasad Kasiviswanathan. Faster approximation of distances in graphs. In Frank K. H. A. Dehne, Jörg-Rüdiger Sack, and Norbert Zeh, editors, Proceedings of the 10th International Workshop on Algorithms and Data Structures (WADS 2007, pages 541-552, 2007. URL: https://doi.org/10.1007/978-3-540-73951-7_47.
  17. Aaron Bernstein. Fully dynamic (2 + ε) approximate all-pairs shortest paths with fast query and close to linear update time. In Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2009, pages 693-702. IEEE Computer Society, 2009. URL: https://doi.org/10.1109/FOCS.2009.16.
  18. Aaron Bernstein. Maintaining shortest paths under deletions in weighted directed graphs. SIAM Journal on Computing, 45(2):548-574, 2016. Announced at STOC 2013. URL: https://doi.org/10.1137/130938670.
  19. Aaron Bernstein. Deterministic partially dynamic single source shortest paths in weighted graphs. In Proceedings of the 44th International Colloquium on Automata, Languages, and Programming, (ICALP 2017), pages 44:1-44:14, 2017. URL: https://doi.org/10.4230/LIPIcs.ICALP.2017.44.
  20. Aaron Bernstein and Shiri Chechik. Deterministic decremental single source shortest paths: beyond the O(mn) bound. In Daniel Wichs and Yishay Mansour, editors, Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing (STOC 2016), pages 389-397, 2016. URL: https://doi.org/10.1145/2897518.2897521.
  21. Aaron Bernstein, Maximilian Probst Gutenberg, and Thatchaphol Saranurak. Deterministic decremental SSSP and approximate min-cost flow in almost-linear time. In Proceedings of the 62nd IEEE Annual Symposium on Foundations of Computer Science (FOCS 2021), pages 1000-1008, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00100.
  22. Aaron Bernstein and Liam Roditty. Improved dynamic algorithms for maintaining approximate shortest paths under deletions. In Dana Randall, editor, Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, pages 1355-1365. SIAM, 2011. URL: https://doi.org/10.1137/1.9781611973082.104.
  23. Keren Censor-Hillel, Michal Dory, Janne H Korhonen, and Dean Leitersdorf. Fast approximate shortest paths in the congested clique. Distributed Computing, 34(6):463-487, 2021. Google Scholar
  24. Shiri Chechik. Near-optimal approximate decremental all pairs shortest paths. In Mikkel Thorup, editor, Proceedings of the 59th IEEE Annual Symposium on Foundations of Computer Science (FOCS 2018), pages 170-181, 2018. URL: https://doi.org/10.1109/FOCS.2018.00025.
  25. Shiri Chechik and Tianyi Zhang. Nearly 2-approximate distance oracles in subquadratic time. In Proc. of the 2022 ACM-SIAM Symposium on Discrete Algorithms, SODA 2022, pages 551-580. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.26.
  26. Julia Chuzhoy. Decremental all-pairs shortest paths in deterministic near-linear time. In Samir Khuller and Virginia Vassilevska Williams, editors, Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing (STOC 2021), pages 626-639. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451025.
  27. Julia Chuzhoy and Thatchaphol Saranurak. Deterministic algorithms for decremental shortest paths via layered core decomposition. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA 2021), pages 2478-2496, 2021. URL: https://doi.org/10.1137/1.9781611976465.147.
  28. Edith Cohen and Uri Zwick. All-pairs small-stretch paths. Journal of Algorithms, 38(2):335-353, 2001. Announced at SODA 1997. URL: https://doi.org/10.1006/jagm.2000.1117.
  29. Camil Demetrescu and Giuseppe F. Italiano. A new approach to dynamic all pairs shortest paths. Journal of the ACM, 51(6):968-992, 2004. Announced at STOC 2003. URL: https://doi.org/10.1145/1039488.1039492.
  30. Camil Demetrescu and Giuseppe F. Italiano. Fully dynamic all pairs shortest paths with real edge weights. Journal of Computer and System Sciences, 72(5):813-837, 2006. Announced at FOCS 2001. URL: https://doi.org/10.1016/j.jcss.2005.05.005.
  31. Mingyang Deng, Yael Kirkpatrick, Victor Rong, Virginia Vassilevska Williams, and Ziqian Zhong. New additive approximations for shortest paths and cycles. In Mikolaj Bojanczyk, Emanuela Merelli, and David P. Woodruff, editors, 49th International Colloquium on Automata, Languages, and Programming, ICALP 2022, July 4-8, 2022, Paris, France, volume 229 of LIPIcs, pages 50:1-50:10. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.ICALP.2022.50.
  32. Dorit Dor, Shay Halperin, and Uri Zwick. All-pairs almost shortest paths. SIAM Journal on Computing, 29(5):1740-1759, 2000. Announced at FOCS 1996. URL: https://doi.org/10.1137/S0097539797327908.
  33. Michal Dory, Sebastian Forster, Yael Kirkpatrick, Yasamin Nazari, Virginia Vassilevska Williams, and Tijn de Vos. Fast 2-approximate all-pairs shortest paths. In Proc. of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 4728-4757. SIAM, 2024. URL: https://arxiv.org/abs/2307.09258.
  34. Michal Dory, Sebastian Forster, Yasamin Nazari, and Tijn de Vos. New tradeoffs for decremental approximate all-pairs shortest paths. CoRR, abs/2211.01152, 2022. URL: https://doi.org/10.48550/arXiv.2211.01152.
  35. Michal Dory and Merav Parter. Exponentially faster shortest paths in the congested clique. ACM Journal of the ACM (JACM), 69(4):1-42, 2022. Google Scholar
  36. Anita Dürr. Improved bounds for rectangular monotone min-plus product and applications. Information Processing Letters, page 106358, 2023. Google Scholar
  37. Michael Elkin and Ofer Neiman. Near-additive spanners and near-exact hopsets, a unified view. Bulletin of EATCS, 1(130), 2020. Google Scholar
  38. Michael Elkin and David Peleg. (1+ε,β)-spanner constructions for general graphs. SIAM Journal on Computing, 33(3):608-631, 2004. Google Scholar
  39. Jacob Evald, Viktor Fredslund-Hansen, Maximilian Probst Gutenberg, and Christian Wulff-Nilsen. Decremental APSP in unweighted digraphs versus an adaptive adversary. In Proc. of the 48th International Colloquium on Automata, Languages, and Programming, (ICALP 2021), pages 64:1-64:20, 2021. URL: https://doi.org/10.4230/LIPIcs.ICALP.2021.64.
  40. Shimon Even and Yossi Shiloach. An on-line edge-deletion problem. J. ACM, 28(1):1-4, 1981. URL: https://doi.org/10.1145/322234.322235.
  41. Lisa Fleischer. Approximating fractional multicommodity flow independent of the number of commodities. SIAM Journal of Discrete Mathematics, 13(4):505-520, 2000. Announced at FOCS 1999. URL: https://doi.org/10.1137/S0895480199355754.
  42. Sebastian Forster, Gramoz Goranci, and Monika Henzinger. Dynamic maintenance of low-stretch probabilistic tree embeddings with applications. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, (SODA 2021), pages 1226-1245, 2021. URL: https://doi.org/10.1137/1.9781611976465.75.
  43. Sebastian Forster, Gramoz Goranci, Yasamin Nazari, and Antonis Skarlatos. Bootstrapping dynamic distance oracles. In Proceedings of the 31th Annual European Symposium on Algorithms (ESA 2023), 2023. Google Scholar
  44. Naveen Garg and Jochen Könemann. Faster and simpler algorithms for multicommodity flow and other fractional packing problems. SIAM Journal on Computing, 37(2):630-652, 2007. Announced at FOCS 1998. URL: https://doi.org/10.1137/S0097539704446232.
  45. Monika Henzinger, Sebastian Krinninger, and Danupon Nanongkai. A subquadratic-time algorithm for decremental single-source shortest paths. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, pages 1053-1072. SIAM, 2014. URL: https://doi.org/10.1137/1.9781611973402.79.
  46. Monika Henzinger, Sebastian Krinninger, and Danupon Nanongkai. Dynamic approximate all-pairs shortest paths: Breaking the o(mn) barrier and derandomization. SIAM Journal on Computing, 45(3):947-1006, 2016. Announced at FOCS 2013. Google Scholar
  47. Monika Henzinger, Sebastian Krinninger, and Danupon Nanongkai. Decremental single-source shortest paths on undirected graphs in near-linear total update time. Journal of the ACM, 65(6):36:1-36:40, 2018. Announced at FOCS 2014. URL: https://doi.org/10.1145/3218657.
  48. Monika Henzinger, Sebastian Krinninger, Danupon Nanongkai, and Thatchaphol Saranurak. Unifying and strengthening hardness for dynamic problems via the online matrix-vector multiplication conjecture. In Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing (STOC 2015), pages 21-30, 2015. URL: https://doi.org/10.1145/2746539.2746609.
  49. Adam Karczmarz and Jakub Łącki. Reliable hubs for partially-dynamic all-pairs shortest paths in directed graphs. In Proceedings of the 27th Annual European Symposium on Algorithms (ESA 2019), pages 65:1-65:15, 2019. URL: https://doi.org/10.4230/LIPIcs.ESA.2019.65.
  50. Adam Karczmarz and Jakub Łącki. Simple label-correcting algorithms for partially dynamic approximate shortest paths in directed graphs. In Proceedings of the 3rd Symposium on Simplicity in Algorithms (SOSA 2020), pages 106-120, 2020. URL: https://doi.org/10.1137/1.9781611976014.15.
  51. Telikepalli Kavitha. Faster algorithms for all-pairs small stretch distances in weighted graphs. Algorithmica, 63(1-2):224-245, 2012. Announced at FSTTCS 2007. URL: https://doi.org/10.1007/s00453-011-9529-y.
  52. Mathias Bæk Tejs Knudsen. Additive spanners and distance oracles in quadratic time. In Proceedings of the 44th International Colloquium on Automata, Languages, and Programming, (ICALP 2017), pages 64:1-64:12, 2017. URL: https://doi.org/10.4230/LIPIcs.ICALP.2017.64.
  53. Jakub Łącki and Yasamin Nazari. Near-Optimal Decremental Hopsets with Applications. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. Google Scholar
  54. Aleksander Mądry. Faster approximation schemes for fractional multicommodity flow problems via dynamic graph algorithms. In Proceedings of the 42nd ACM Symposium on Theory of Computing (STOC 2010), pages 121-130, 2010. URL: https://doi.org/10.1145/1806689.1806708.
  55. Mihai Patrascu and Liam Roditty. Distance oracles beyond the thorup-zwick bound. SIAM Journal on Computing, 43(1):300-311, 2014. Announced at FOCS 2010. URL: https://doi.org/10.1137/11084128X.
  56. Mihai Patrascu, Liam Roditty, and Mikkel Thorup. A new infinity of distance oracles for sparse graphs. In Proc. of the 53rd Annual IEEE Symposium on Foundations of Computer Science (FOCS 2012), pages 738-747. IEEE Computer Society, 2012. URL: https://doi.org/10.1109/FOCS.2012.44.
  57. Liam Roditty. New algorithms for all pairs approximate shortest paths. In Barna Saha and Rocco A. Servedio, editors, Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, Orlando, FL, USA, June 20-23, 2023, pages 309-320. ACM, 2023. URL: https://doi.org/10.1145/3564246.3585197.
  58. Liam Roditty and Uri Zwick. On dynamic shortest paths problems. Algorithmica, 61(2):389-401, 2011. Announced at ESA 2004. URL: https://doi.org/10.1007/s00453-010-9401-5.
  59. Liam Roditty and Uri Zwick. Dynamic approximate all-pairs shortest paths in undirected graphs. SIAM J. Comput., 41(3):670-683, 2012. Announced at FOCS 2004. URL: https://doi.org/10.1137/090776573.
  60. Barna Saha and Christopher Ye. Faster approximate all pairs shortest paths. In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 4758-4827. SIAM, 2024. URL: https://arxiv.org/abs/2309.13225.
  61. Christian Sommer. All-pairs approximate shortest paths and distance oracle preprocessing. In Proceedings of the 43rd International Colloquium on Automata, Languages, and Programming, (ICALP 2016), volume 55, pages 55:1-55:13, 2016. URL: https://doi.org/10.4230/LIPIcs.ICALP.2016.55.
  62. Mikkel Thorup. Fully-dynamic all-pairs shortest paths: Faster and allowing negative cycles. In Proceedings of the 9th Scandinavian Workshop on Algorithm Theory (SWAT 2004), pages 384-396, 2004. URL: https://doi.org/10.1007/978-3-540-27810-8_33.
  63. Mikkel Thorup and Uri Zwick. Compact routing schemes. In Arnold L. Rosenberg, editor, Proceedings of the Thirteenth Annual ACM Symposium on Parallel Algorithms and Architectures, SPAA 2001, pages 1-10. ACM, 2001. URL: https://doi.org/10.1145/378580.378581.
  64. Mikkel Thorup and Uri Zwick. Approximate distance oracles. Journal of the ACM (JACM), 52(1):1-24, 2005. Google Scholar
  65. Mikkel Thorup and Uri Zwick. Spanners and emulators with sublinear distance errors. In Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm, pages 802-809, 2006. Google Scholar
  66. R. Ryan Williams. Faster all-pairs shortest paths via circuit complexity. SIAM Journal on Computing, 47(5):1965-1985, 2018. Announced at STOC 2014. URL: https://doi.org/10.1137/15M1024524.
  67. Uri Zwick. Exact and approximate distances in graphs - A survey. In Proceedings of the 9th Annual European Symposium on Algorithms (ESA 2001), volume 2161, pages 33-48, 2001. URL: https://doi.org/10.1007/3-540-44676-1_3.