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# Fully-Dynamic α + 2 Arboricity Decompositions and Implicit Colouring

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## Acknowledgements

We thank Krzysztof Nowicki for helpful discussions and ideas.

## Cite As

Aleksander B. G. Christiansen and Eva Rotenberg. Fully-Dynamic α + 2 Arboricity Decompositions and Implicit Colouring. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 42:1-42:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ICALP.2022.42

## Abstract

The arboricity α of a graph is the smallest number of forests necessary to cover its edges, and an arboricity decomposition of a graph is a decomposition of its edges into forests. The best near-linear time algorithm for arboricity decomposition guarantees at most α +2 forests if the graph has arboricity α (Blumenstock and Fischer [Markus Blumenstock and Frank Fischer, 2020]). In this paper, we study arboricity decomposition for dynamic graphs, that is, graphs that are subject to insertions and deletions of edges. We give an algorithm that, provided the arboricity of the dynamic graph never exceeds α, maintains an α+2 arboricity decomposition of the graph in poly(log n,α) update time, thus matching the number of forests currently obtainable in near-linear time for static (non-changing) graphs. Our construction goes via dynamic bounded out-degree orientations, and we present a fully-dynamic algorithm that explicitly orients the edges of the dynamic graph, such that no vertex has an out-degree exceeding ⌊ (1+ε)α ⌋ + 2. Our algorithm is deterministic and has a worst-case update time of O(ε^{-6}α² log³ n). The state-of-the-art explicit, deterministic, worst-case algorithm for bounded out-degree orientations maintains a β⋅ α + log_β n out-orientation in O(β²α²+βαlog_β n) time [Tsvi Kopelowitz et al., 2014]. As a consequence, we get an algorithm that maintains an implicit vertex colouring with 4⋅ 2^α colours, in amortised poly-log n update time, and with O(α log n) worst-case query time. Thus, at the expense of log n-factors in the update time, we improve on the number of colours from 2^O(α) to O(2^α) compared to the state-of-the-art for implicit dynamic colouring [Monika Henzinger et al., 2020].

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Dynamic graph algorithms
##### Keywords
• Dynamic graphs
• bounded arboricity
• graph colouring
• data structures

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## References

1. Oswin Aichholzer, Franz Aurenhammer, and Günter Rote. Optimal graph orientation with storage applications. SFB-Report , SFB 'Optimierung und Kontrolle'. TU Graz, 1995. Reportnr.: F003-51.
2. Stephen Alstrup, Jacob Holm, Kristian De Lichtenberg, and Mikkel Thorup. Maintaining information in fully dynamic trees with top trees. ACM Trans. Algorithms, 1(2):243-264, October 2005. URL: https://doi.org/10.1145/1103963.1103966.
3. K. Appel and W. Haken. A proof of the four color theorem. Discret. Math., 16(2):179-180, 1976. URL: https://doi.org/10.1016/0012-365X(76)90147-3.
4. Srinivasa Rao Arikati, Anil Maheshwari, and Christos D. Zaroliagis. Efficient computation of implicit representations of sparse graphs. Discret. Appl. Math., 78(1-3):1-16, 1997. URL: https://doi.org/10.1016/S0166-218X(97)00007-3.
5. Niranka Banerjee, Venkatesh Raman, and Saket Saurabh. Fully dynamic arboricity maintenance. In Computing and Combinatorics - 25th International Conference, COCOON 2019, Xi'an, China, July 29-31, 2019, Proceedings, volume 11653 of Lecture Notes in Computer Science, pages 1-12. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-26176-4_1.
6. L. Barba, J. Cardinal, M. Korman, S. Langerman, A. van Renssen, M. Roeloffzen, and S. Verdonschot. Dynamic graph coloring. Algorithmica, 81(4):1319-1341, 2019.
7. Leonid Barenboim and Michael Elkin. Sublogarithmic distributed mis algorithm for sparse graphs using nash-williams decomposition. In Proceedings of the Twenty-Seventh ACM Symposium on Principles of Distributed Computing, PODC '08, pages 25-34, New York, NY, USA, 2008. Association for Computing Machinery. URL: https://doi.org/10.1145/1400751.1400757.
8. Edvin Berglin and Gerth Stolting Brodal. A Simple Greedy Algorithm for Dynamic Graph Orientation. In 28th International Symposium on Algorithms and Computation (ISAAC 2017), volume 92 of Leibniz International Proceedings in Informatics (LIPIcs), pages 12:1-12:12, Dagstuhl, Germany, 2017. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2017.12.
9. Sayan Bhattacharya, Deeparnab Chakrabarty, Monika Henzinger, and Danupon Nanongkai. Dynamic algorithms for graph coloring. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 1-20. SIAM, 2018. URL: https://doi.org/10.1137/1.9781611975031.1.
10. Sayan Bhattacharya, Fabrizio Grandoni, Janardhan Kulkarni, Quanquan C. Liu, and Shay Solomon. Fully dynamic (Δ+1)-coloring in constant update time. CoRR, abs/1910.02063, 2019. URL: http://arxiv.org/abs/1910.02063.
11. Markus Blumenstock. Fast algorithms for pseudoarboricity. In Proceedings of the Eighteenth Workshop on Algorithm Engineering and Experiments, ALENEX 2016, Arlington, Virginia, USA, January 10, 2016, pages 113-126. SIAM, 2016. URL: https://doi.org/10.1137/1.9781611974317.10.
12. Markus Blumenstock and Frank Fischer. A constructive arboricity approximation scheme. In SOFSEM 2020: Theory and Practice of Computer Science - 46th International Conference on Current Trends in Theory and Practice of Informatics, SOFSEM 2020, Limassol, Cyprus, January 20-24, 2020, Proceedings, volume 12011 of Lecture Notes in Computer Science, pages 51-63. Springer, 2020. URL: https://doi.org/10.1007/978-3-030-38919-2_5.
13. Gerth Stolting Brodal and Rolf Fagerberg. Dynamic representations of sparse graphs. In In Proc. 6th International Workshop on Algorithms and Data Structures (WADS), pages 342-351. Springer-Verlag, 1999.
14. Norishige Chiba, Takao Nishizeki, and Nobuji Saito. A linear 5-coloring algorithm of planar graphs. J. Algorithms, 2(4):317-327, 1981. URL: https://doi.org/10.1016/0196-6774(81)90031-6.
15. Aleksander B. G. Christiansen. Dynamic algorithms for implicit vertex-colouring of graphs with bounded arboricity. Master’s thesis, Technical University of Denmark, Kgs. Lyngby, Denmark, October 2021.
16. Aleksander B. G. Christiansen and Eva Rotenberg. Fully-dynamic α+2 arboricity decomposition and implicit colouring. CoRR, abs/2203.06039, 2022. URL: https://doi.org/10.48550/arXiv.2203.06039.
17. O. Coudert. Exact coloring of real-life graphs is easy. Proceedings of 34th Design Automation Conference. ACM, 35(1):121-126, 1997.
18. Jack Edmonds. Minimum partition of a matroid into independent subsets. Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics, page 67, 1965.
19. David Eppstein. Arboricity and bipartite subgraph listing algorithms. Inf. Process. Lett., 51(4):207-211, August 1994. URL: https://doi.org/10.1016/0020-0190(94)90121-X.
20. Greg N. Frederickson. On linear-time algorithms for five-coloring planar graphs. Inf. Process. Lett., 19(5):219-224, 1984. URL: https://doi.org/10.1016/0020-0190(84)90056-5.
21. Harold Gabow and Herbert Westermann. Forests, frames, and games: Algorithms for matroid sums and applications. In Proceedings of the Twentieth Annual ACM Symposium on Theory of Computing, STOC '88, pages 407-421, New York, NY, USA, 1988. Association for Computing Machinery. URL: https://doi.org/10.1145/62212.62252.
22. Harold N. Gabow. Algorithms for graphic polymatroids and parametric s-sets. In Proceedings of the Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '95, pages 88-97, USA, 1995. Society for Industrial and Applied Mathematics.
23. Mohsen Ghaffari and Hsin-Hao Su. Distributed degree splitting, edge coloring, and orientations. 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 2505-2523. SIAM, 2017. URL: https://doi.org/10.1137/1.9781611974782.166.
24. David G. Harris. Distributed local approximation algorithms for maximum matching in graphs and hypergraphs. In 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS), pages 700-724, 2019. URL: https://doi.org/10.1109/FOCS.2019.00048.
25. David G. Harris, Hsin-Hao Su, and Hoa T. Vu. On the locality of nash-williams forest decomposition and star-forest decomposition. In Proceedings of the 2021 ACM Symposium on Principles of Distributed Computing, PODC'21, pages 295-305, New York, NY, USA, 2021. Association for Computing Machinery. URL: https://doi.org/10.1145/3465084.3467908.
26. Meng He, Ganggui Tang, and N. Zeh. Orienting dynamic graphs, with applications to maximal matchings and adjacency queries. In ISAAC, 2014.
27. Monika Henzinger, Stefan Neumann, and Andreas Wiese. Explicit and implicit dynamic coloring of graphs with bounded arboricity. CoRR, abs/2002.10142, 2020. URL: http://arxiv.org/abs/2002.10142.
28. Monika Henzinger and Pan Peng. Constant-time dynamic (Δ+1)-coloring. In 37th International Symposium on Theoretical Aspects of Computer Science, STACS 2020, March 10-13, 2020, Montpellier, France, volume 154 of LIPIcs, pages 53:1-53:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.STACS.2020.53.
29. Subhash Khot and Ashok Kumar Ponnuswami. Better inapproximability results for maxclique, chromatic number and min-3lin-deletion. In Automata, Languages and Programming, 33rd International Colloquium, ICALP 2006, Venice, Italy, July 10-14, 2006, Proceedings, Part I, volume 4051 of Lecture Notes in Computer Science, pages 226-237. Springer, 2006. URL: https://doi.org/10.1007/11786986_21.
30. Tsvi Kopelowitz, Robert Krauthgamer, Ely Porat, and Shay Solomon. Orienting fully dynamic graphs with worst-case time bounds. In Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part II, volume 8573 of Lecture Notes in Computer Science, pages 532-543. Springer, 2014. URL: https://doi.org/10.1007/978-3-662-43951-7_45.
31. Łukasz Kowalik. Approximation scheme for lowest outdegree orientation and graph density measures. In Proceedings of the 17th International Conference on Algorithms and Computation, ISAAC'06, pages 557-566, Berlin, Heidelberg, 2006. Springer-Verlag. URL: https://doi.org/10.1007/11940128_56.
32. Łukasz Kowalik. Adjacency queries in dynamic sparse graphs. Inf. Process. Lett., 102(5):191-195, May 2007. URL: https://doi.org/10.1016/j.ipl.2006.12.006.
33. Yin Tat Lee and Aaron Sidford. Path finding methods for linear programming: Solving linear programs in Õ(vrank) iterations and faster algorithms for maximum flow. In 2014 IEEE 55th Annual Symposium on Foundations of Computer Science, pages 424-433, 2014. URL: https://doi.org/10.1109/FOCS.2014.52.
34. Aleksander Madry. Navigating central path with electrical flows: From flows to matchings, and back. In 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, pages 253-262, 2013. URL: https://doi.org/10.1109/FOCS.2013.35.
35. D. Matula, Y. Shiloach, and R. Tarjan. Two linear-time algorithms for five-coloring a planar graph, 1980. Reportnr.: STAN-CS-80-830.
36. C. St.J. A. Nash-Williams. Decomposition of finite graphs into forests. Journal of the London Mathematical Society, s1-39(1):12-12, 1964. URL: https://doi.org/10.1112/jlms/s1-39.1.12.
37. Jean-Claude Picard and Maurice Queyranne. A network flow solution to some nonlinear 0-1 programming problems, with applications to graph theory. Networks, 12(2):141-159, 1982. URL: https://doi.org/10.1002/net.3230120206.
38. Neil Robertson, Daniel P. Sanders, Paul D. Seymour, and Robin Thomas. Efficiently four-coloring planar graphs. In Gary L. Miller, editor, Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing, Philadelphia, Pennsylvania, USA, May 22-24, 1996, pages 571-575. ACM, 1996. URL: https://doi.org/10.1145/237814.238005.
39. Saurabh Sawlani and Junxing Wang. Near-optimal fully dynamic densest subgraph. In Proccedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, Chicago, IL, USA, June 22-26, 2020, pages 181-193. ACM, 2020. URL: https://doi.org/10.1145/3357713.3384327.
40. Daniel D. Sleator and Robert Endre Tarjan. A data structure for dynamic trees. In Proceedings of the Thirteenth Annual ACM Symposium on Theory of Computing, STOC '81, pages 114-122, New York, NY, USA, 1981. Association for Computing Machinery. URL: https://doi.org/10.1145/800076.802464.
41. Shay Solomon and Nicole Wein. Improved dynamic graph coloring. ACM Trans. Algorithms, 16(3), June 2020. URL: https://doi.org/10.1145/3392724.
42. Hsin-Hao Su and Hoa T. Vu. Distributed Dense Subgraph Detection and Low Outdegree Orientation. In Hagit Attiya, editor, 34th International Symposium on Distributed Computing (DISC 2020), volume 179 of Leibniz International Proceedings in Informatics (LIPIcs), pages 15:1-15:18, Dagstuhl, Germany, 2020. Schloss Dagstuhl-Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.DISC.2020.15.
43. David Zuckerman. Linear degree extractors and the inapproximability of max clique and chromatic number. In Proceedings of the Thirty-Eighth Annual ACM Symposium on Theory of Computing, STOC '06, pages 681-690, New York, NY, USA, 2006. Association for Computing Machinery. URL: https://doi.org/10.1145/1132516.1132612.