Efficient Parallel $(\Delta+1)$-Edge-Coloring

We study the $(\Delta+1)$-edge-coloring problem in the parallel $\left(\mathrm{PRAM}\right)$ model of computation. The celebrated Vizing's theorem [Viz64] states that every simple graph $G = (V,E)$ can be properly $(\Delta+1)$-edge-colored. In a seminal paper, Karloff and Shmoys [KS87] devised a parallel algorithm with time $O\left(\Delta^5\cdot\log n\cdot\left(\log^3 n+\Delta^2\right)\right)$ and $O(m\cdot\Delta)$ processors. This result was improved by Liang et al. [LSH96] to time $O\left(\Delta^{4.5}\cdot \log^3\Delta\cdot \log n + \Delta^4 \cdot\log^4 n\right)$ and $O\left(n\cdot\Delta^{3} +n^2\right)$ processors. [LSH96] claimed $O\left(\Delta^{3.5} \cdot\log^3\Delta\cdot \log n + \Delta^3\cdot \log^4 n\right)$ time, but we point out a flaw in their analysis, which once corrected, results in the above bound. We devise a faster parallel algorithm for this fundamental problem. Specifically, our algorithm uses $O\left(\Delta^4\cdot \log^4 n\right)$ time and $O(m\cdot \Delta)$ processors. Another variant of our algorithm requires $O\left(\Delta^{4+o(1)}\cdot\log^2 n\right)$ time, and $O\left(m\cdot\Delta\cdot\log n\cdot\log^{\delta}\Delta\right)$ processors, for an arbitrarily small $\delta>0$. We also devise a few other tradeoffs between the time and the number of processors, and devise an improved algorithm for graphs with small arboricity. On the way to these results, we also provide a very fast parallel algorithm for updating $(\Delta+1)$-edge-coloring. Our algorithm for this problem is dramatically faster and simpler than the previous state-of-the-art algorithm (due to [LSH96]) for this problem.