Document Open Access Logo

Streaming Edge Coloring with Subquadratic Palette Size

Authors Shiri Chechik, Doron Mukhtar, Tianyi Zhang

PDF

File

Thumbnail PDF
PDF
LIPIcs.ICALP.2024.40.pdf
  • Filesize: 0.65 MB
  • 12 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords
  • graph algorithms
  • streaming algorithms
  • edge coloring

Metrics

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

Abstract

In this paper, we study the problem of computing an edge-coloring in the (one-pass) W-streaming model. In this setting, the edges of an n-node graph arrive in an arbitrary order to a machine with a relatively small space, and the goal is to design an algorithm that outputs, as a stream, a proper coloring of the edges using the fewest possible number of colors.
Behnezhad et al. [Behnezhad et al., 2019] devised the first non-trivial algorithm for this problem, which computes in Õ(n) space a proper O(Δ²)-coloring w.h.p. (here Δ is the maximum degree of the graph). Subsequent papers improved upon this result, where latest of them [Ansari et al., 2022] showed that it is possible to deterministically compute an O(Δ²/s)-coloring in O(ns) space. However, none of the improvements succeeded in reducing the number of colors to O(Δ^{2-ε}) while keeping the same space bound of Õ(n). In particular, no progress was made on the question of whether computing an O(Δ)-coloring is possible with roughly O(n) space, which was stated in [Behnezhad et al., 2019] to be an interesting open problem.
In this paper we bypass the quadratic bound by presenting a new randomized Õ(n)-space algorithm that uses Õ(Δ^{1.5}) colors.

Cite As Get BibTex

Shiri Chechik, Doron Mukhtar, and Tianyi Zhang. Streaming Edge Coloring with Subquadratic Palette Size. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 40:1-40:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ICALP.2024.40

Author Details

Shiri Chechik
  • Tel Aviv University, Israel
Doron Mukhtar
  • Tel Aviv University, Israel
Tianyi Zhang
  • Tel Aviv University, Israel

Funding

This publication is part of a project that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 803118 UncertainENV).

References

  1. Mohammad Ansari, Mohammad Saneian, and Hamid Zarrabi-Zadeh. Simple streaming algorithms for edge coloring. In Proccedings of the 30th Annual European Symposium on Algorithms (ESA `22), pages 8:1-8:4, 2022. Google Scholar
  2. Soheil Behnezhad, Mahsa Derakhshan, MohammadTaghi Hajiaghayi, Marina Knittel, and Hamed Saleh. Streaming and massively parallel algorithms for edge coloring. In Proccedings of the 27th Annual European Symposium on Algorithms (ESA `19), pages 15:1-15:14, 2019. Google Scholar
  3. Soheil Behnezhad and Mohammad Saneian. Streaming edge coloring with asymptotically optimal colors. arXiv preprint, 2023. URL: https://arxiv.org/abs/2305.01714.
  4. Sayan Bhattacharya, Fabrizio Grandoni, and David Wajc. Online edge coloring algorithms via the nibble method. In Proceedings of the 32nd Symposium on Discrete Algorithms (SODA `21), pages 2830-2842, 2021. Google Scholar
  5. Andrea Ribichini Camil Demetrescu, Irene Finocchi. Trading off space for passes in graph streaming problems. In Proccedings of the 17th Annual Symposium on Discrete Algorithms (SODA `06), pages 714-723, 2006. Google Scholar
  6. Yi-Jun Chang, Qizheng He, Wenzheng Li, Seth Pettie, and Jara Uitto. The complexity of distributed edge coloring with small palettes. In Proceedings of the Twenty-Ninth Annual Symposium on Discrete Algorithms (SODA `18), pages 2633-2652, 2018. Google Scholar
  7. Moses Charikar and Paul Liu. Improved algorithms for edge colouring in the W-streaming model. In Proccedings of the 4th Symposium on Simplicity in Algorithms (SOSA `21), pages 181-183, 2021. Google Scholar
  8. Ilan Reuven Cohen, Binghui Peng, and David Wajc. Tight bounds for online edge coloring. In 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS), pages 1-25. IEEE, 2019. Google Scholar
  9. Prantar Ghosh and Manuel Stoeckl. Low-memory algorithms for online and w-streaming edge coloring. arXiv preprint, 2023. URL: https://arxiv.org/abs/2304.12285.
  10. David B Shmoys Howard J Karloff. Efficient parallel algorithms for edge coloring problems. Journal of Algorithms, 8(1):39-52, 1987. Google Scholar
  11. Janardhan Kulkarni, Yang P Liu, Ashwin Sah, Mehtaab Sawhney, and Jakub Tarnawski. Online edge coloring via tree recurrences and correlation decay. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, pages 104-116, 2022. Google Scholar
  12. J. Misra and David Gries. A constructive proof of Vizing’s theorem. Information Processing Letters, 41(3):131-133, 1992. Google Scholar
  13. Amin Saberi and David Wajc. The greedy algorithm is not optimal for on-line edge coloring. In Proceedings of the 48th International Colloquium on Automata, Languages, and Programming (ICALP `21), pages 109:1-109:18, 2021. Google Scholar
  14. Claude E Shannon. A theorem on coloring the lines of a network. Journal of Mathematics and Physics, 28(1-4):148-152, 1949. Google Scholar
  15. Aravind Srinivasan, David Wajc, et al. Online dependent rounding schemes. arXiv preprint, 2023. URL: https://arxiv.org/abs/2301.08680.
  16. Vadim G Vizing. Critical graphs with given chromatic class (in russian). Metody Discret. Analiz., 5:9-17, 1965. Google Scholar
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