Parallel Batch Dynamic Vertex Coloring in $O(\log Δ)$ Amortized Update Time
Chase Hutton, Adam Melrod
TL;DR
The paper advances parallel batch-dynamic vertex coloring by presenting the first randomized algorithm that maintains a proper $(\Delta+1)$-coloring in $\Delta$-bounded graphs with $O(\log \Delta)$ expected amortized update time. It builds on the BCHN sequential framework, introduces a relaxed sequential algorithm to enable parallelization, and then designs a three-phase parallel scheme (initialization, coloring, moving) with token-based amortized analysis. A key innovation is allowing dirty vertices to be recolored without immediate movement and using symmetry-breaking raising/lowering to ensure sufficient token release, yielding provable $O(\log \Delta)$ cost and polylog-span for batches. The approach combines advanced data-structure design (hash tables, dynamic partitioned arrays) with a hierarchical partition and partial list-coloring subroutines to achieve near-optimal parallel performance for dynamic coloring, with potential impact on parallel graph processing under update streams.
Abstract
We present the first parallel batch-dynamic algorithm for maintaining a proper $(Δ+ 1)$-vertex coloring. Our approach builds on a new sequential dynamic algorithm inspired by the work of Bhattacharya et al. (SODA'18). The resulting randomized algorithm achieves $O(\log Δ)$ expected amortized update time and, for any batch of $b$ updates, has parallel span $O(\operatorname{polylog} b + \operatorname{polylog} n)$ with high probability.