Dynamic Graph Coloring: Sequential, Parallel, and Distributed

TL;DR

This paper introduces a simple randomized algorithm for maintaining a proper graph coloring efficiently across sequential, parallel, and distributed models, handling dynamic updates with optimal or near-optimal performance.

Contribution

It provides a unified framework for dynamic graph coloring that achieves $O(1)$ expected update time sequentially, efficient parallel batch processing, and quick convergence in distributed networks.

Findings

Sequential updates processed in $O(1)$ expected time.

Parallel batch updates handled with $O(1)$ work per update.

Distributed setting maintains proper coloring with $O( ext{log} n)$ rounds.

Abstract

We present a simple randomized algorithm that can efficiently maintain a $(\Delta+1)$ coloring as the graph undergoes edge insertion and deletion updates, where $\Delta$ denotes an upper bound on the maximum degree. A key advantage is the algorithm's ability to process many updates simultaneously, which makes it naturally adaptable to the parallel and distributed models. Concretely, it gives a unified framework across the models, leading to the following results: - In the sequential setting, the algorithm processes each update in $O(1)$ expected time, worst-case. This matches and strengthens the results of Henzinger and Peng [TALG 2022] and Bhattacharya et al. [TALG 2022], who achieved an $O(1)$ bound but amortized (in expectation and with high probability, respectively), whose work was an improvement of the $O(\log \Delta)$ expected amortized bound of Bhattacharya et al. [SODA'18]. - In the parallel setting, the algorithm processes each (arbitrary size) batch of updates using $O(1)$ work per update in the batch in expectation, and in $\text{poly}(\log n)$ depth with high probability. This is, in a sense, an ideal parallelization of the above results. - In the distributed setting, the algorithm can maintain a coloring of the network graph as (potentially many) edges are added or deleted. The maintained coloring is always proper; it may become partial upon updates, i.e., some nodes may temporarily lose their colors, but quickly converges to a full, proper coloring. Concretely, each insertion and deletion causes at most $O(1)$ nodes to become uncolored, but this is resolved within $O(\log n)$ rounds with high probability (e.g., in the absence of further updates nearby--the precise guarantee is stronger, but technical). Importantly, the algorithm incurs only $O(1)$ expected message complexity and computation per update.