近均衡着色的分布式算法
Distributed Algorithms for Near-Equitable Coloring
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中文总结 AI 辅助
针对分布式网络中的近均衡图着色问题，本文提出一组对应不同权衡点的快速随机分布式算法，并分析其在串行、CONGEST及CC模型中的时间复杂度。
中文摘要 AI 辅助
对于顶点数为n、最大度为Δ、直径为D的图，均衡(Δ+1)-着色是一种顶点着色，其中每种颜色的出现频率（即该颜色着色的顶点数）均等于σ=n/(Δ+1)（允许取整）。Hajnal-Szemerédi定理保证了所有图都存在此类着色，且已知存在时间复杂度为O(n²Δ)的串行算法可计算该着色。本文研究分布式网络中的近均衡图着色，核心问题是在少量分布式轮次内计算出频率接近σ的着色时，能在多大程度上接近所需的Δ+1调色板大小，且这两个相互冲突的参数存在权衡关系，本文尝试对此进行探索。本文提出一组快速随机分布式算法，对应该权衡曲线上的不同点，分析其性质，并在串行、CONGEST和Congested Clique（CC）模型中研究其时间复杂度。
英文摘要
For an $n$-vertex graph of maximum degree $Δ$ and diameter $D$, an equitable $(Δ+1)$-coloring is a vertex coloring where the frequency of each color (namely, the number of vertices it colors) are all equal to $σ=n/(Δ+1)$ (up to rounding). The Hajnal-Szemerédi Theorem guarantees the existence of such a coloring for every graph, and an $O(n^2Δ)$ time sequential algorithm is known for computing such a coloring. Here, we study near-equitable graph coloring in distributed networks. The main question of interest is how close one can remain to the desired palette size of $Δ+1$ while computing, in few distributed rounds, a coloring whose frequencies are close to $σ$. It appears that these two conflicting parameters exhibit a tradeoff, which we attempt to explore. We present a suite of fast randomized distributed algorithms representing varying points on this tradeoff, analyze their properties, and study their time complexity in the sequential, CONGEST and Congested Clique (CC) models.
