Document

# Streaming Edge Coloring with Asymptotically Optimal Colors

## File

LIPIcs.ICALP.2024.121.pdf
• Filesize: 0.74 MB
• 20 pages

## Cite As

Mohammad Saneian and Soheil Behnezhad. Streaming Edge Coloring with Asymptotically Optimal Colors. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 121:1-121:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.121

## Abstract

Given a graph G, an edge-coloring is an assignment of colors to edges of G such that any two edges sharing an endpoint receive different colors. By Vizing’s celebrated theorem, any graph of maximum degree Δ needs at least Δ and at most (Δ + 1) colors to be properly edge colored. In this paper, we study edge colorings in the streaming setting. The edges arrive one by one in an arbitrary order. The algorithm takes a single pass over the input and must output a solution using a much smaller space than the input size. Since the output of edge coloring is as large as its input, the assigned colors should also be reported in a streaming fashion. The streaming edge coloring problem has been studied in a series of works over the past few years. The main challenge is that the algorithm cannot "remember" all the color assignments that it returns. To ensure the validity of the solution, existing algorithms use many more colors than Vizing’s bound. Namely, in n-vertex graphs, the state-of-the-art algorithm with Õ(n s) space requires O(Δ²/s + Δ) colors. Note, in particular, that for an asymptotically optimal O(Δ) coloring, this algorithm requires Ω(nΔ) space which is as large as the input. Whether such a coloring can be achieved with sublinear space has been left open. In this paper, we answer this question in the affirmative. We present a randomized algorithm that returns an asymptotically optimal O(Δ) edge coloring using Õ(n √{Δ}) space. More generally, our algorithm returns a proper O(Δ^{1.5}/s + Δ) edge coloring with Õ(n s) space, improving prior algorithms for the whole range of s.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Design and analysis of algorithms
• Mathematics of computing → Graph coloring
##### Keywords
• Streaming
• Edge coloring

## Metrics

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

## References

1. Kook Jin Ahn, Graham Cormode, Sudipto Guha, Andrew McGregor, and Anthony Wirth. Correlation clustering in data streams. In Proceedings of the 32nd International Conference on Machine Learning, ICML 2015, Lille, France, 6-11 July 2015, pages 2237-2246, 2015. URL: http://proceedings.mlr.press/v37/ahn15.html.
2. Kook Jin Ahn, Sudipto Guha, and Andrew McGregor. Analyzing graph structure via linear measurements. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17-19, 2012, pages 459-467, 2012. URL: https://doi.org/10.1137/1.9781611973099.40.
3. Kook Jin Ahn, Sudipto Guha, and Andrew McGregor. Graph sketches: sparsification, spanners, and subgraphs. In Proceedings of the 31st ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, PODS 2012, Scottsdale, AZ, USA, May 20-24, 2012, pages 5-14, 2012. URL: https://doi.org/10.1145/2213556.2213560.
4. Mohammad Ansari, Mohammad Saneian, and Hamid Zarrabi-Zadeh. Simple streaming algorithms for edge coloring. In 30th Annual European Symposium on Algorithms, ESA 2022, September 5-9, 2022, Berlin/Potsdam, Germany, pages 8:1-8:4, 2022. URL: https://doi.org/10.4230/LIPICS.ESA.2022.8.
5. Sepehr Assadi. A two-pass (conditional) lower bound for semi-streaming maximum matching. In Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms, SODA 2022, Virtual Conference / Alexandria, VA, USA, January 9 - 12, 2022, pages 708-742, 2022. URL: https://doi.org/10.1137/1.9781611977073.32.
6. Sepehr Assadi and Soheil Behnezhad. Beating two-thirds for random-order streaming matching. In 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021, July 12-16, 2021, Glasgow, Scotland (Virtual Conference), pages 19:1-19:13, 2021. URL: https://doi.org/10.4230/LIPICS.ICALP.2021.19.
7. Sepehr Assadi, Soheil Behnezhad, Sanjeev Khanna, and Huan Li. On regularity lemma and barriers in streaming and dynamic matching. In Proccedings of the 55th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2023, to appear, 2023.
8. Sepehr Assadi, Amit Chakrabarti, Prantar Ghosh, and Manuel Stoeckl. Coloring in graph streams via deterministic and adversarially robust algorithms. CoRR, abs/2212.10641, 2022. URL: https://doi.org/10.48550/arXiv.2212.10641.
9. Sepehr Assadi, Andrew Chen, and Glenn Sun. Deterministic graph coloring in the streaming model. In STOC '22: 54th Annual ACM SIGACT Symposium on Theory of Computing, Rome, Italy, June 20 - 24, 2022, pages 261-274, 2022. URL: https://doi.org/10.1145/3519935.3520016.
10. Sepehr Assadi, Yu Chen, and Sanjeev Khanna. Sublinear algorithms for (δ + 1) vertex coloring. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pages 767-786, 2019. URL: https://doi.org/10.1137/1.9781611975482.48.
11. Sepehr Assadi and Aditi Dudeja. A simple semi-streaming algorithm for global minimum cuts. In 4th Symposium on Simplicity in Algorithms, SOSA 2021, Virtual Conference, January 11-12, 2021, pages 172-180. SIAM, 2021.
12. Sepehr Assadi, Pankaj Kumar, and Parth Mittal. Brooks' theorem in graph streams: a single-pass semi-streaming algorithm for δ-coloring. In STOC '22: 54th Annual ACM SIGACT Symposium on Theory of Computing, Rome, Italy, June 20 - 24, 2022, pages 234-247, 2022. URL: https://doi.org/10.1145/3519935.3520005.
13. Sepehr Assadi and Janani Sundaresan. (Noisy) gap cycle counting strikes back: Random order streaming lower bounds for connected components and beyond. In Proccedings of the 55th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2023, to appear, 2023.
14. Amotz Bar-Noy, Rajeev Motwani, and Joseph Naor. The greedy algorithm is optimal for on-line edge coloring. Inf. Process. Lett., 44(5):251-253, 1992. URL: https://doi.org/10.1016/0020-0190(92)90209-E.
15. Soheil Behnezhad, Moses Charikar, Weiyun Ma, and Li-Yang Tan. Almost 3-approximate correlation clustering in constant rounds. In 63rd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2022, Denver, CO, USA, October 31 - November 3, 2022, pages 720-731, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00074.
16. Soheil Behnezhad, Moses Charikar, Weiyun Ma, and Li-Yang Tan. Single-pass streaming algorithms for correlation clustering. In Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms, SODA 2023, to appear, 2023.
17. Soheil Behnezhad, Mahsa Derakhshan, MohammadTaghi Hajiaghayi, Marina Knittel, and Hamed Saleh. Streaming and massively parallel algorithms for edge coloring. In 27th Annual European Symposium on Algorithms, ESA 2019, September 9-11, 2019, Munich/Garching, Germany, pages 15:1-15:14, 2019. URL: https://doi.org/10.4230/LIPICS.ESA.2019.15.
18. Soheil Behnezhad and Mohammad Saneian. Streaming edge coloring with asymptotically optimal colors. CoRR, abs/2305.01714, 2023. URL: https://doi.org/10.48550/arXiv.2305.01714.
19. Suman K. Bera, Amit Chakrabarti, and Prantar Ghosh. Graph coloring via degeneracy in streaming and other space-conscious models. In 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020, July 8-11, 2020, Saarbrücken, Germany (Virtual Conference), pages 11:1-11:21, 2020. URL: https://doi.org/10.4230/LIPICS.ICALP.2020.11.
20. Sayan Bhattacharya, Fabrizio Grandoni, and David Wajc. Online edge coloring algorithms via the nibble method. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10 - 13, 2021, pages 2830-2842. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.168.
21. Amit Chakrabarti, Prantar Ghosh, and Manuel Stoeckl. Adversarially robust coloring for graph streams. In 13th Innovations in Theoretical Computer Science Conference, ITCS 2022, January 31 - February 3, 2022, Berkeley, CA, USA, pages 37:1-37:23, 2022. URL: https://doi.org/10.4230/LIPICS.ITCS.2022.37.
22. Moses Charikar and Paul Liu. Improved algorithms for edge colouring in the w-streaming model. In 4th Symposium on Simplicity in Algorithms, SOSA 2021, Virtual Conference, January 11-12, 2021, pages 181-183, 2021. URL: https://doi.org/10.1137/1.9781611976496.20.
23. Shiri Chechik, Doron Mukhtar, and Tianyi Zhang. Streaming Edge Coloring with Subquadratic Palette Size. CoRR, abs/2305.07090, 2023. https://arxiv.org/abs/2305.07090, URL: https://doi.org/10.48550/arXiv.2305.07090.
24. Yu Chen, Sanjeev Khanna, and Huan Li. On weighted graph sparsification by linear sketching. In 63rd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2022, Denver, CO, USA, October 31 - November 3, 2022, pages 474-485, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00052.
25. Ilan Reuven Cohen, Binghui Peng, and David Wajc. Tight bounds for online edge coloring. In 60th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2019, Baltimore, Maryland, USA, November 9-12, 2019, pages 1-25. IEEE Computer Society, 2019. URL: https://doi.org/10.1109/FOCS.2019.00010.
26. Vincent Cohen-Addad, Silvio Lattanzi, Slobodan Mitrovic, Ashkan Norouzi-Fard, Nikos Parotsidis, and Jakub Tarnawski. Correlation clustering in constant many parallel rounds. In Proceedings of the 38th International Conference on Machine Learning, ICML 2021, 18-24 July 2021, Virtual Event, pages 2069-2078, 2021. URL: http://proceedings.mlr.press/v139/cohen-addad21b.html.
27. Camil Demetrescu, Irene Finocchi, and Andrea Ribichini. Trading off space for passes in graph streaming problems. ACM Trans. Algorithms, 6(1), December 2010. URL: https://doi.org/10.1145/1644015.1644021.
28. Joan Feigenbaum, Sampath Kannan, Andrew McGregor, Siddharth Suri, and Jian Zhang. On graph problems in a semi-streaming model. In Automata, Languages and Programming: 31st International Colloquium, ICALP 2004, Turku, Finland, July 12-16, 2004. Proceedings, pages 531-543, 2004. URL: https://doi.org/10.1007/978-3-540-27836-8_46.
29. Alan M. Frieze. Maximum matchings in a class of random graphs. J. Comb. Theory, Ser. B, 40(2):196-212, 1986. URL: https://doi.org/10.1016/0095-8956(86)90077-8.
30. Harold N Gabow. Algorithms for edge-coloring graphs. Technical Report, 1985.
31. Mohsen Ghaffari, Krzysztof Nowicki, and Mikkel Thorup. Faster algorithms for edge connectivity via random 2-out contractions. In Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020, Salt Lake City, UT, USA, January 5-8, 2020, pages 1260-1279. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611975994.77.
32. Prantar Ghosh and Manuel Stoeckl. Low-memory algorithms for online and w-streaming edge coloring. CoRR, abs/2304.12285, 2023.
33. Christian Glazik, Jan Schiemann, and Anand Srivastav. Finding euler tours in one pass in the w-streaming model with o(n log(n)) RAM. CoRR, abs/1710.04091, 2017. URL: https://arxiv.org/abs/1710.04091.
34. Jacob Holm, Valerie King, Mikkel Thorup, Or Zamir, and Uri Zwick. Random k-out subgraph leaves only o(n/k) inter-component edges. In 60th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2019, Baltimore, Maryland, USA, November 9-12, 2019, pages 896-909. IEEE Computer Society, 2019. URL: https://doi.org/10.1109/FOCS.2019.00058.
35. John E. Hopcroft and Richard M. Karp. A n5/2 algorithm for maximum matchings in bipartite. In 12th Annual Symposium on Switching and Automata Theory (SWAT 1971), pages 122-125, 1971.
36. Michael Kapralov. Space lower bounds for approximating maximum matching in the edge arrival model. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10 - 13, 2021, pages 1874-1893, 2021. URL: https://doi.org/10.1137/1.9781611976465.112.
37. Michael Kapralov, Yin Tat Lee, Cameron Musco, Christopher Musco, and Aaron Sidford. Single pass spectral sparsification in dynamic streams. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 561-570, 2014. URL: https://doi.org/10.1109/FOCS.2014.66.
38. Michael Kapralov, Aida Mousavifar, Cameron Musco, Christopher Musco, Navid Nouri, Aaron Sidford, and Jakab Tardos. Fast and space efficient spectral sparsification in dynamic streams. In Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020, Salt Lake City, UT, USA, January 5-8, 2020, pages 1814-1833, 2020. URL: https://doi.org/10.1137/1.9781611975994.111.
39. Howard J. Karloff and David B. Shmoys. Efficient parallel algorithms for edge coloring problems. J. Algorithms, 8(1):39-52, 1987. URL: https://doi.org/10.1016/0196-6774(87)90026-5.
40. Janardhan Kulkarni, Yang P. Liu, Ashwin Sah, Mehtaab Sawhney, and Jakub Tarnawski. Online edge coloring via tree recurrences and correlation decay. In STOC '22: 54th Annual ACM SIGACT Symposium on Theory of Computing, Rome, Italy, June 20 - 24, 2022, pages 104-116. ACM, 2022. URL: https://doi.org/10.1145/3519935.3519986.
41. Amin Saberi and David Wajc. The greedy algorithm is not optimal for on-line edge coloring. In 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021, July 12-16, 2021, Glasgow, Scotland (Virtual Conference), volume 198 of LIPIcs, pages 109:1-109:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPICS.ICALP.2021.109.