Document Open Access Logo

Constructing Long Paths in Graph Streams

Authors Christian Konrad , Chhaya Trehan

PDF

File

Thumbnail PDF
PDF
LIPIcs.ESA.2025.22.pdf
  • Filesize: 0.84 MB
  • 19 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming models
  • Mathematics of computing → Paths and connectivity problems
Keywords
  • Longest Path Problem
  • Streaming Algorithms
  • One-way Two-party Communication Complexity

Metrics

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

Abstract

In the graph stream model of computation, an algorithm processes the edges of an n-vertex input graph in one or more sequential passes while using a memory that is sublinear in the input size. The streaming model poses significant challenges for algorithmically constructing long paths. Many known algorithms that are tasked with extending an existing path as a subroutine require an entire pass over the input to add a single additional edge. This raises a fundamental question: Are multiple passes inherently necessary to construct paths of non-trivial lengths, or can a single pass suffice? To address this question, we systematically study the Longest Path problem in the one-pass streaming model. In this problem, given a desired approximation factor α, the objective is to compute a path of length at least lp(G)/α, where lp(G) is the length of a longest path in the input graph G.
We study the problem in the insertion-only and the insertion-deletion streaming models, and we give algorithms as well as space lower bounds for both undirected and directed graphs. Our results are:  
1) We show that for undirected graphs, in both the insertion-only and the insertion-deletion models, there are semi-streaming algorithms, i.e., algorithms that use space O(n poly log n), that compute a path of length at least d/3 with high probability, where d is the average degree of the input graph. These algorithms can also yield an α-approximation to Longest Path using space Õ(n²/α).
2) Next, we show that such a result cannot be achieved for directed graphs, even in the insertion-only model. We show that computing a (n^{1-o(1)})-approximation to Longest Path in directed graphs in the insertion-only model requires space Ω(n²). This result is in line with recent results that demonstrate that processing directed graphs is often significantly harder than undirected graphs in the streaming model.
3) We further complement our results with two additional lower bounds. First, we show that semi-streaming space is insufficient for small constant factor approximations to Longest Path for undirected graphs in the insertion-only model. Last, in undirected graphs in the insertion-deletion model, we show that computing an α-approximation requires space Ω(n²/α³).

Cite As Get BibTex

Christian Konrad and Chhaya Trehan. Constructing Long Paths in Graph Streams. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 22:1-22:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ESA.2025.22

Author Details

Christian Konrad
  • School of Computer Science, University of Bristol, UK
Chhaya Trehan
  • Unaffiliated Researcher, Durham, UK

Funding

  • Konrad, Christian: Supported by EPSRC New Investigator Award EP/V010611/1.
  • Trehan, Chhaya: Most of the work was done while C.T. was at the University of Bristol where she was supported by EPSRC New Investigator Award EP/V010611/1.

References

  1. Kook Jin Ahn, Graham Cormode, Sudipto Guha, Andrew McGregor, and Anthony Wirth. Correlation clustering in data streams. Algorithmica, 83(7):1980-2017, 2021. URL: https://doi.org/10.1007/S00453-021-00816-9.
  2. Noga Alon, Ankur Moitra, and Benny Sudakov. Nearly complete graphs decomposable into large induced matchings and their applications. In Howard J. Karloff and Toniann Pitassi, editors, Proceedings of the 44th Symposium on Theory of Computing Conference, STOC 2012, New York, NY, USA, May 19-22, 2012, pages 1079-1090. ACM, 2012. URL: https://doi.org/10.1145/2213977.2214074.
  3. Sepehr Assadi. A two-pass (conditional) lower bound for semi-streaming maximum matching. 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 708-742. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.32.
  4. Sepehr Assadi, Soheil Behnezhad, Christian Konrad, Kheeran K. Naidu, and Janani Sundaresan. Settling the pass complexity of approximate matchings in dynamic graph streams. In Yossi Azar and Debmalya Panigrahi, editors, Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2025, New Orleans, LA, USA, January 12-15, 2025, pages 864-904. SIAM, 2025. URL: https://doi.org/10.1137/1.9781611978322.25.
  5. Sepehr Assadi, Aaron Bernstein, Zachary Langley, Lap Chi Lau, and Robert Wang. Streaming and communication complexity of load-balancing via matching contractors. In Yossi Azar and Debmalya Panigrahi, editors, Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2025, New Orleans, LA, USA, January 12-15, 2025, pages 3423-3449. SIAM, 2025. URL: https://doi.org/10.1137/1.9781611978322.113.
  6. Sepehr Assadi, Yu Chen, and Sanjeev Khanna. Sublinear algorithms for (Δ + 1) vertex coloring. In Timothy M. Chan, editor, Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pages 767-786. SIAM, 2019. URL: https://doi.org/10.1137/1.9781611975482.48.
  7. Sepehr Assadi, Sanjeev Khanna, and Yang Li. On estimating maximum matching size in graph streams. In Philip N. Klein, editor, Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 1723-1742. SIAM, 2017. URL: https://doi.org/10.1137/1.9781611974782.113.
  8. Sepehr Assadi, Sanjeev Khanna, Yang Li, and Grigory Yaroslavtsev. Maximum matchings in dynamic graph streams and the simultaneous communication model. 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 1345-1364. SIAM, 2016. URL: https://doi.org/10.1137/1.9781611974331.CH93.
  9. Sepehr Assadi, Christian Konrad, Kheeran K. Naidu, and Janani Sundaresan. O(log log n) passes is optimal for semi-streaming maximal independent set. In Bojan Mohar, Igor Shinkar, and Ryan O'Donnell, editors, Proceedings of the 56th Annual ACM Symposium on Theory of Computing, STOC 2024, Vancouver, BC, Canada, June 24-28, 2024, pages 847-858. ACM, 2024. URL: https://doi.org/10.1145/3618260.3649763.
  10. Andreas Björklund, Thore Husfeldt, Petteri Kaski, and Mikko Koivisto. Narrow sieves for parameterized paths and packings. Journal of Computer and System Sciences, 87:119-139, 2017. URL: https://doi.org/10.1016/j.jcss.2017.03.003.
  11. Andreas Björklund, Thore Husfeldt, and Sanjeev Khanna. Approximating longest directed paths and cycles. In Josep Díaz, Juhani Karhumäki, Arto Lepistö, and Donald Sannella, editors, Automata, Languages and Programming, pages 222-233, Berlin, Heidelberg, 2004. Springer Berlin Heidelberg. URL: https://doi.org/10.1007/978-3-540-27836-8_21.
  12. Amit Chakrabarti, Prantar Ghosh, Andrew McGregor, and Sofya Vorotnikova. Vertex ordering problems in directed graph streams. 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 1786-1802. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611975994.109.
  13. Yi-Jun Chang, Martin Farach-Colton, Tsan-sheng Hsu, and Meng-Tsung Tsai. Streaming complexity of spanning tree computation. In Christophe Paul and Markus Bläser, editors, 37th International Symposium on Theoretical Aspects of Computer Science, STACS 2020, March 10-13, 2020, Montpellier, France, volume 154 of LIPIcs, pages 34:1-34:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPICS.STACS.2020.34.
  14. Graham Cormode, Jacques Dark, and Christian Konrad. Independent sets in vertex-arrival streams. In Christel Baier, Ioannis Chatzigiannakis, Paola Flocchini, and Stefano Leonardi, editors, 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019, July 9-12, 2019, Patras, Greece, volume 132 of LIPIcs, pages 45:1-45:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPICS.ICALP.2019.45.
  15. Jacques Dark and Christian Konrad. Optimal lower bounds for matching and vertex cover in dynamic graph streams. In Shubhangi Saraf, editor, 35th Computational Complexity Conference, CCC 2020, July 28-31, 2020, Saarbrücken, Germany (Virtual Conference), volume 169 of LIPIcs, pages 30:1-30:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPICS.CCC.2020.30.
  16. Joan Feigenbaum, Sampath Kannan, Andrew McGregor, Siddharth Suri, and Jian Zhang. On graph problems in a semi-streaming model. Theor. Comput. Sci., 348(2-3):207-216, 2005. URL: https://doi.org/10.1016/J.TCS.2005.09.013.
  17. Eldar Fischer, Eric Lehman, Ilan Newman, Sofya Raskhodnikova, Ronitt Rubinfeld, and Alex Samorodnitsky. Monotonicity testing over general poset domains. In John H. Reif, editor, Proceedings on 34th Annual ACM Symposium on Theory of Computing, May 19-21, 2002, Montréal, Québec, Canada, pages 474-483. ACM, 2002. URL: https://doi.org/10.1145/509907.509977.
  18. Ashish Goel, Michael Kapralov, and Sanjeev Khanna. On the communication and streaming complexity of maximum bipartite matching. In Yuval Rabani, editor, Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17-19, 2012, pages 468-485. SIAM, 2012. URL: https://doi.org/10.1137/1.9781611973099.41.
  19. S. W. Golomb. Random permutations. Bull. Amer. Math. Soc. 70, 1964. Google Scholar
  20. S. W. Golomb, L. R. Welch, and R. M. Goldstein. Cycles from nonlinear shift registers. Technical report, Progress Rep. No. 20-389, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, 1959. Google Scholar
  21. Prahladh Harsha, Rahul Jain, David A. McAllester, and Jaikumar Radhakrishnan. The communication complexity of correlation. In 22nd Annual IEEE Conference on Computational Complexity (CCC 2007), 13-16 June 2007, San Diego, California, USA, pages 10-23. IEEE Computer Society, 2007. URL: https://doi.org/10.1109/CCC.2007.32.
  22. Hossein Jowhari, Mert Saglam, and Gábor Tardos. Tight bounds for lp samplers, finding duplicates in streams, and related problems. In Maurizio Lenzerini and Thomas Schwentick, editors, Proceedings of the 30th ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, PODS 2011, June 12-16, 2011, Athens, Greece, pages 49-58. ACM, 2011. URL: https://doi.org/10.1145/1989284.1989289.
  23. Ramkumar G. D. S. Karger D, Motwani R. On approximating the longest path in a graph. Algorithmica, pages 82-98, 1997. URL: https://doi.org/10.1007/BF02523689.
  24. Shahbaz Khan and Shashank K. Mehta. Depth First Search in the Semi-streaming Model. In Rolf Niedermeier and Christophe Paul, editors, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019), volume 126 of Leibniz International Proceedings in Informatics (LIPIcs), pages 42:1-42:16, Dagstuhl, Germany, 2019. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.STACS.2019.42.
  25. Lasse Kliemann, Christian Schielke, and Anand Srivastav. A streaming algorithm for the undirected longest path problem. In Piotr Sankowski and Christos D. Zaroliagis, editors, 24th Annual European Symposium on Algorithms, ESA 2016, August 22-24, 2016, Aarhus, Denmark, volume 57 of LIPIcs, pages 56:1-56:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPICS.ESA.2016.56.
  26. Christian Konrad and Kheeran K. Naidu. On two-pass streaming algorithms for maximum bipartite matching. In Mary Wootters and Laura Sanità, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2021, August 16-18, 2021, University of Washington, Seattle, Washington, USA (Virtual Conference), volume 207 of LIPIcs, pages 19:1-19:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPICS.APPROX/RANDOM.2021.19.
  27. Christian Konrad and Kheeran K. Naidu. An unconditional lower bound for two-pass streaming algorithms for maximum matching approximation. In David P. Woodruff, editor, Proceedings of the 2024 ACM-SIAM Symposium on Discrete Algorithms, SODA 2024, Alexandria, VA, USA, January 7-10, 2024, pages 2881-2899. SIAM, 2024. URL: https://doi.org/10.1137/1.9781611977912.102.
  28. Christian Konrad, Kheeran K. Naidu, and Arun Steward. Maximum matching via maximal matching queries. In Petra Berenbrink, Patricia Bouyer, Anuj Dawar, and Mamadou Moustapha Kanté, editors, 40th International Symposium on Theoretical Aspects of Computer Science, STACS 2023, March 7-9, 2023, Hamburg, Germany, volume 254 of LIPIcs, pages 41:1-41:22. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPICS.STACS.2023.41.
  29. Andrew McGregor. Finding graph matchings in data streams. In Chandra Chekuri, Klaus Jansen, José D. P. Rolim, and Luca Trevisan, editors, Approximation, Randomization and Combinatorial Optimization, Algorithms and Techniques, 8th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2005 and 9th InternationalWorkshop on Randomization and Computation, RANDOM 2005, Berkeley, CA, USA, August 22-24, 2005, Proceedings, volume 3624 of Lecture Notes in Computer Science, pages 170-181. Springer, 2005. URL: https://doi.org/10.1007/11538462_15.
  30. Slobodan Mitrovi?, Anish Mukherjee, Piotr Sankowski, and Wen-Horng Sheu. Faster semi-streaming matchings via alternating trees, 2025. URL: https://doi.org/10.48550/arXiv.2412.19057.
  31. Anup Rao and Amir Yehudayoff. Communication Complexity: and Applications. Cambridge University Press, 2020. Google Scholar
  32. Jeffrey Scott Vitter. Random sampling with a reservoir. ACM Trans. Math. Softw., 11(1):37-57, 1985. URL: https://doi.org/10.1145/3147.3165.
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
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact