CONGEST模型中用于图着色的无直径分布式频率控制
Diameter-Free Distributed Frequency Control for Graph Coloring in the CONGEST Model
浏览论文内容
中文总结 AI 辅助
本文在CONGEST模型中提出两种不依赖网络直径的分布式图着色算法，分别实现双向颜色频率相对平衡与单向频率上限，均在O((λ+1)lg n)轮内运行。
中文摘要 AI 辅助
本文提出了两种在同步CONGEST模型中控制颜色频率的随机正确着色算法，无需支付依赖于直径的协调成本。设λ≥1为期望的失败指数，对于任意固定的δ>0，第一种算法使用χ=⌈(2+δ)Δ⌉种颜色，且以至少1−n^−λ的概率输出正确着色，该着色将每种颜色频率与n/χ的偏差限制为O_δ(√((λ+1)(n/χ)lg n)+(λ+1)lg n)。在显式负载条件下，该加性保证产生双向相对平衡。第二种算法适用于所有χ>Δ的情况，提供由调色板余量χ−Δ控制的单向频率上限，特别地，它使用Δ+⌈(Δ+1)/⌈ln n⌉⌉种颜色，并将每个使用的颜色类上限设为O((λ+1)(σ lg²n+lg n))，其中σ=n/(Δ+1)。两种算法均在O((λ+1)lg n)轮内运行，不依赖网络直径；对于第一种算法，时间复杂度中的乘法常数取决于δ。
英文摘要
This paper presents two randomized proper-coloring algorithms that control color frequencies in the synchronous CONGEST model without paying a diameter-dependent coordination cost. Let $λ\geq 1$ denote the desired failure exponent. For every fixed $δ> 0$, the first algorithm uses $χ= \lceil (2+δ)Δ\rceil$ colors and, with probability at least $1 - n^{-λ}$, outputs a proper coloring that bounds the deviation of every color frequency from $n/χ$ by $O_δ(\sqrt{(λ+1)(n/χ)\lg n} + (λ+1)\lg n)$. Under an explicit load condition, this additive guarantee yields two-sided relative balance. The second algorithm works with every $χ> Δ$ and gives a one-sided frequency cap controlled by the palette slack $χ- Δ$. In particular, it uses $Δ+ \lceil (Δ+1)/\lceil \ln n \rceil \rceil$ colors and caps every used color class by $O((λ+1)(σ\lg^2 n + \lg n))$, where $σ= n/(Δ+1)$. Both algorithms run in $O((λ+1)\lg n)$ rounds, with no dependence on the network diameter; for the first algorithm, the multiplicative constant in the time bound depends on $δ$.
