We present a simple deterministic distributed algorithm that computes a $(\Delta+1)$-vertex coloring in $O(\log^2 \Delta \cdot \log n)$ rounds. The algorithm can be implemented with $O(\log n)$-bit messages. The algorithm can also be extended to the more general $(degree+1)$-list coloring problem. Obtaining a polylogarithmic-time deterministic algorithm for $(\Delta+1)$-vertex coloring had remained a central open question in the area of distributed graph algorithms since the 1980s, until a recent network decomposition algorithm of Rozho\v{n} and Ghaffari [STOC'20]. The current state of the art is based on an improved variant of their decomposition, which leads to an $O(\log^5 n)$-round algorithm for $(\Delta+1)$-vertex coloring. Our coloring algorithm is completely different and considerably simpler and faster. It solves the coloring problem in a direct way, without using network decomposition, by gradually rounding a certain fractional color assignment until reaching an integral color assignments. Moreover, via the approach of Chang, Li, and Pettie [STOC'18], this improved deterministic algorithm also leads to an improvement in the complexity of randomized algorithms for $(\Delta+1)$-coloring, now reaching the bound of $O(\log^3\log n)$ rounds. As a further application, we also provide faster deterministic distributed algorithms for the following variants of the vertex coloring problem. In graphs of arboricity $a$, we show that a $(2+\epsilon)a$-vertex coloring can be computed in $O(\log^3 a\cdot\log n)$ rounds. We also show that for $\Delta\geq 3$, a $\Delta$-coloring of a $\Delta$-colorable graph $G$ can be computed in $O(\log^2 \Delta\cdot\log^2 n)$ rounds.
翻译:我们提出了一个简单的确定式分布算法, 该算法计算了 $( delta+1) $( delta+1) 的确定式算法, 在美元( log2) 2\ Delta\ cdot\ log n) 圆环中, 以美元( log2 2\ ddolt\ log n) 来计算一个 美元( del2) 的确定式算法。 该算法可以使用$( 度+1) 美元( delta+1) 美元( verex) 的确定式算法, 在 美元( $( delta+1) 美元( 美元) 的分布式图表算法方面, 直至最近 Rozho2\ vn) 和 Ghaffari [ST\c'20] 的网络解析算法。 艺术的目前状态基于一个更普通的解算法, 也就是 $( delta+5 n) 美元( 美元) 美元( 美元) 美元( 美元) 的计算算法 的多的确定算算法 的计算法可以提供。 我们的颜色算算法 的更简单化的解算法,,,, 也通过网络的解解解算法 来显示一个更分。