Document Open Access Logo

On the Space Usage of Approximate Distance Oracles with Sub-2 Stretch

Authors Tsvi Kopelowitz , Ariel Korin , Liam Roditty

PDF
Thumbnail PDF

File

LIPIcs.ICALP.2024.101.pdf
  • Filesize: 0.77 MB
  • 18 pages

Document Identifiers

Author Details

Tsvi Kopelowitz
  • Bar-Ilan University, Ramat-Gan, Israel
Ariel Korin
  • Bar-Ilan University, Ramat-Gan, Israel
Liam Roditty
  • Bar-Ilan University, Ramat-Gan, Israel

Funding

This work was supported in part by the Israel Science Foundation (ISF) grant 1926/19 and by the United States - Israel Binational Science Foundation (BSF) grants 2018364, 2020356.

Cite As Get BibTex

Tsvi Kopelowitz, Ariel Korin, and Liam Roditty. On the Space Usage of Approximate Distance Oracles with Sub-2 Stretch. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 101:1-101:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ICALP.2024.101

Abstract

For an undirected unweighted graph G = (V,E) with n vertices and m edges, let d(u,v) denote the distance from u ∈ V to v ∈ V in G. An (α,β)-stretch approximate distance oracle (ADO) for G is a data structure that given u,v ∈ V returns in constant (or near constant) time a value dˆ(u,v) such that d(u,v) ≤ dˆ(u,v) ≤ α⋅ d(u,v) + β, for some reals α > 1, β.
Thorup and Zwick [Mikkel Thorup and Uri Zwick, 2005] showed that one cannot beat stretch 3 with subquadratic space (in terms of n) for general graphs. Pǎtraşcu and Roditty [Mihai Pǎtraşcu and Liam Roditty, 2010] showed that one can obtain stretch 2 using O(m^{1/3}n^{4/3}) space, and so if m is subquadratic in n then the space usage is also subquadratic. Moreover, Pǎtraşcu and Roditty [Mihai Pǎtraşcu and Liam Roditty, 2010] showed that one cannot beat stretch 2 with subquadratic space even for graphs where m = Õ(n), based on the set-intersection hypothesis. 
In this paper we explore the conditions for which an ADO can beat stretch 2 while using subquadratic space. In particular, we show that if the maximum degree in G is Δ_G ≤ O(n^{1/k-ε}) for some 0 < ε ≤ 1/k, then there exists an ADO for G that uses Õ(n^{2-(kε)/3) space and has a (2,1-k)-stretch. For k = 2 this result implies a subquadratic sub-2 stretch ADO for graphs with Δ_G ≤ O(n^{1/2-ε}).
Moreover, we prove a conditional lower bound, based on the set intersection hypothesis, which states that for any positive integer k ≤ log n, obtaining a sub-(k+2)/k stretch for graphs with Δ_G = Θ(n^{1/k}) requires Ω̃(n²) space. Thus, for graphs with maximum degree Θ(n^{1/2}), obtaining a sub-2 stretch requires Ω̃(n²) space.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
Keywords
  • Graph algorithms
  • Approximate distance oracle
  • data structures
  • shortest path

Metrics

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

Related Versions

References

  1. 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 Stefano Leonardi and Anupam Gupta, editors, STOC '22: 54th Annual ACM SIGACT Symposium on Theory of Computing, Rome, Italy, June 20 - 24, 2022, pages 1487-1500. ACM, 2022. URL: https://doi.org/10.1145/3519935.3520066.
  2. Ittai Abraham and Cyril Gavoille. On approximate distance labels and routing schemes with affine stretch. In David Peleg, editor, Distributed Computing - 25th International Symposium, DISC 2011, Rome, Italy, September 20-22, 2011. Proceedings, volume 6950 of Lecture Notes in Computer Science, pages 404-415. Springer, 2011. URL: https://doi.org/10.1007/978-3-642-24100-0_39.
  3. Rachit Agarwal. The space-stretch-time tradeoff in distance oracles. In Andreas S. Schulz and Dorothea Wagner, editors, Algorithms - ESA 2014 - 22th Annual European Symposium, Wroclaw, Poland, September 8-10, 2014. Proceedings, volume 8737 of Lecture Notes in Computer Science, pages 49-60. Springer, 2014. URL: https://doi.org/10.1007/978-3-662-44777-2_5.
  4. Rachit Agarwal and Philip Brighten Godfrey. Brief announcement: a simple stretch 2 distance oracle. In Panagiota Fatourou and Gadi Taubenfeld, editors, ACM Symposium on Principles of Distributed Computing, PODC '13, Montreal, QC, Canada, July 22-24, 2013, pages 110-112. ACM, 2013. URL: https://doi.org/10.1145/2484239.2484277.
  5. Rachit Agarwal and Philip Brighten Godfrey. Distance oracles for stretch less than 2. In Sanjeev Khanna, editor, Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013, New Orleans, Louisiana, USA, January 6-8, 2013, pages 526-538. SIAM, 2013. URL: https://doi.org/10.1137/1.9781611973105.38.
  6. Rachit Agarwal, Philip Brighten Godfrey, and Sariel Har-Peled. Approximate distance queries and compact routing in sparse graphs. In INFOCOM 2011. 30th IEEE International Conference on Computer Communications, Joint Conference of the IEEE Computer and Communications Societies, 10-15 April 2011, Shanghai, China, pages 1754-1762. IEEE, 2011. URL: https://doi.org/10.1109/INFCOM.2011.5934973.
  7. Maor Akav and Liam Roditty. An almost 2-approximation for all-pairs of shortest paths in subquadratic time. In Shuchi Chawla, editor, Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020, Salt Lake City, UT, USA, January 5-8, 2020, pages 1-11. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611975994.1.
  8. Surender Baswana, Vishrut Goyal, and Sandeep Sen. All-pairs nearly 2-approximate shortest paths in I time. Theor. Comput. Sci., 410(1):84-93, 2009. URL: https://doi.org/10.1016/J.TCS.2008.10.018.
  9. Surender Baswana and Telikepalli Kavitha. Faster algorithms for approximate distance oracles and all-pairs small stretch paths. In 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06), pages 591-602. IEEE, 2006. Google Scholar
  10. Richard Bellman. On a routing problem. Quarterly of applied mathematics, 16(1):87-90, 1958. Google Scholar
  11. Davide Bilò, Shiri Chechik, Keerti Choudhary, Sarel Cohen, Tobias Friedrich, and Martin Schirneck. Improved approximate distance oracles: Bypassing the thorup-zwick bound in dense graphs. arXiv preprint, 2023. URL: https://arxiv.org/abs/2307.11677.
  12. Timothy M. Chan. More algorithms for all-pairs shortest paths in weighted graphs. SIAM J. Comput., 39(5):2075-2089, 2010. URL: https://doi.org/10.1137/08071990X.
  13. Shiri Chechik. Approximate distance oracles with constant query time. In David B. Shmoys, editor, Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 654-663. ACM, 2014. URL: https://doi.org/10.1145/2591796.2591801.
  14. Shiri Chechik. Approximate distance oracles with improved bounds. In Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, June 14-17, 2015, pages 1-10, 2015. Google Scholar
  15. Shiri Chechik and Tianyi Zhang. Nearly 2-approximate distance oracles in subquadratic time. In Joseph (Seffi) Naor and Niv Buchbinder, editors, Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms, SODA 2022, Virtual Conference / Alexandria, VA, USA, January 9 - 12, 2022, pages 551-580. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.26.
  16. Boris V. Cherkassky, Andrew V. Goldberg, and Tomasz Radzik. Shortest paths algorithms: Theory and experimental evaluation. Math. Program., 73:129-174, 1996. URL: https://doi.org/10.1007/BF02592101.
  17. Hagai Cohen and Ely Porat. On the hardness of distance oracle for sparse graph. CoRR, abs/1006.1117, 2010. URL: https://arxiv.org/abs/1006.1117.
  18. Edsger W. Dijkstra. A note on two problems in connexion with graphs. Numerische Mathematik, 1:269-271, 1959. URL: https://doi.org/10.1007/BF01386390.
  19. Dorit Dor, Shay Halperin, and Uri Zwick. All-pairs almost shortest paths. SIAM J. Comput., 29(5):1740-1759, 2000. URL: https://doi.org/10.1137/S0097539797327908.
  20. Michael L. Fredman and Robert Endre Tarjan. Fibonacci heaps and their uses in improved network optimization algorithms. J. ACM, 34(3):596-615, 1987. URL: https://doi.org/10.1145/28869.28874.
  21. Tsvi Kopelowitz, Seth Pettie, and Ely Porat. Higher lower bounds from the 3sum conjecture. In Robert Krauthgamer, editor, Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 1272-1287. SIAM, 2016. URL: https://doi.org/10.1137/1.9781611974331.CH89.
  22. Amgad Madkour, Walid G. Aref, Faizan Ur Rehman, Mohamed Abdur Rahman, and Saleh M. Basalamah. A survey of shortest-path algorithms. CoRR, abs/1705.02044, 2017. URL: https://arxiv.org/abs/1705.02044.
  23. Manor Mendel and Assaf Naor. Ramsey partitions and proximity data structures. In 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2006), 21-24 October 2006, Berkeley, California, USA, Proceedings, pages 109-118. IEEE Computer Society, 2006. URL: https://doi.org/10.1109/FOCS.2006.65.
  24. Ely Porat and Liam Roditty. Preprocess, set, query! Algorithmica, 67(4):516-528, 2013. Google Scholar
  25. Mihai Puatracscu. Towards polynomial lower bounds for dynamic problems. In Leonard J. Schulman, editor, Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, Cambridge, Massachusetts, USA, 5-8 June 2010, pages 603-610. ACM, 2010. URL: https://doi.org/10.1145/1806689.1806772.
  26. Mihai Puatracscu. Unifying the landscape of cell-probe lower bounds. SIAM J. Comput., 40(3):827-847, 2011. URL: https://doi.org/10.1137/09075336X.
  27. Mihai Puatracscu and Liam Roditty. Distance oracles beyond the thorup-zwick bound. In 51th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2010, October 23-26, 2010, Las Vegas, Nevada, USA, pages 815-823. IEEE Computer Society, 2010. URL: https://doi.org/10.1109/FOCS.2010.83.
  28. Mihai Puatracscu, Liam Roditty, and Mikkel Thorup. A new infinity of distance oracles for sparse graphs. In 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, New Brunswick, NJ, USA, October 20-23, 2012, pages 738-747. IEEE Computer Society, 2012. URL: https://doi.org/10.1109/FOCS.2012.44.
  29. Liam Roditty, Mikkel Thorup, and Uri Zwick. Deterministic constructions of approximate distance oracles and spanners. In Luís Caires, Giuseppe F. Italiano, Luís Monteiro, Catuscia Palamidessi, and Moti Yung, editors, Automata, Languages and Programming, 32nd International Colloquium, ICALP 2005, Lisbon, Portugal, July 11-15, 2005, Proceedings, volume 3580 of Lecture Notes in Computer Science, pages 261-272. Springer, 2005. URL: https://doi.org/10.1007/11523468_22.
  30. Liam Roditty and Roei Tov. Approximate distance oracles with improved stretch for sparse graphs. Theor. Comput. Sci., 943:89-101, 2023. URL: https://doi.org/10.1016/J.TCS.2022.11.016.
  31. Christian Sommer. Shortest-path queries in static networks. ACM Comput. Surv., 46(4):45:1-45:31, 2014. URL: https://doi.org/10.1145/2530531.
  32. Christian Sommer. All-pairs approximate shortest paths and distance oracle preprocessing. In Ioannis Chatzigiannakis, Michael Mitzenmacher, Yuval Rabani, and Davide Sangiorgi, editors, 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, July 11-15, 2016, Rome, Italy, volume 55 of LIPIcs, pages 55:1-55:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPICS.ICALP.2016.55.
  33. 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, Heraklion, Crete Island, Greece, July 4-6, 2001, pages 1-10. ACM, 2001. URL: https://doi.org/10.1145/378580.378581.
  34. Mikkel Thorup and Uri Zwick. Approximate distance oracles. J. ACM, 52(1):1-24, 2005. URL: https://doi.org/10.1145/1044731.1044732.
  35. Virginia Vassilevska Williams and Yinzhan Xu. Monochromatic triangles, triangle listing and APSP. In Sandy Irani, editor, 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020, pages 786-797. IEEE, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00078.
  36. Christian Wulff-Nilsen. Approximate distance oracles with improved query time. In Encyclopedia of Algorithms, pages 94-97, 2016. URL: https://doi.org/10.1007/978-1-4939-2864-4_568.
  37. Uri Zwick. Exact and approximate distances in graphs - A survey. In Friedhelm Meyer auf der Heide, editor, Algorithms - ESA 2001, 9th Annual European Symposium, Aarhus, Denmark, August 28-31, 2001, Proceedings, volume 2161 of Lecture Notes in Computer Science, pages 33-48. Springer, 2001. URL: https://doi.org/10.1007/3-540-44676-1_3.
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
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact