双曲随机图的4/3χ色分布式着色
Distributed Colouring with 4/3 chi Colours for Hyperbolic Random Graphs
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中文总结 AI 辅助
研究双曲随机图的分布式顶点着色问题，提出顺序径向着色算法，该算法能以近乎最优调色板给图着色，不同颜色使用情况下运行轮数不同，相比之前算法大幅减少颜色数量，还改进了团着色分析。
中文摘要 AI 辅助
我们研究双曲随机图（HRGs）上的分布式顶点着色，它是一种捕捉现实世界网络关键结构特征的几何随机图模型。这为分析最坏情况通用图之外的分布式算法提供了自然环境。我们引入顺序径向着色，这是一种仅使用高效局部计算的CONGEST算法。该算法实现了近乎最优的调色板，用4/3χ种颜色给HRGs着色，并以大概率在O((log log n)^2)轮内运行。我们还给出了一个变体，以使用O(χ log log n)种颜色为代价，将其加速到以大概率在O(log log n)轮内运行。最后，对于每个常数ε>0，当使用χ^(1+ε)种颜色时，它以大概率在O(1)轮内运行。这比Maus和Ruff（SODA 2026）之前的常数轮算法大大减少了颜色数量，减少因子至少为n^(1/6)。我们的分析包含一个阶段，其中我们考虑图的一个（大）团上的经典随机着色协议。我们还深入研究了这部分分析，并改进了之前关于给团C着色的结果，根据加法松弛s = |Ψ| - χ（其中Ψ是使用的颜色集）来界定所需的轮数。特别地，当且仅当s = |C|^(1+Ω(1))时才可能进行常数轮着色，而s = |C| / log |C|已经给出最优的Θ(log log |C|)轮复杂度。
英文摘要
We study distributed vertex colouring on Hyperbolic Random Graphs (HRGs), a geometric random graph model capturing key structural features of real-world networks. This provides a natural setting for analysing distributed algorithms beyond worst-case general graphs. We introduce Sequential Radial Colouring, a CONGEST algorithm using only efficient local computation. The algorithm achieves a near-optimal palette, colouring HRGs with $\frac{4}{3}χ$ colours and running in $O((\log\log n)^2)$ rounds a.a.s. We also give a variant that speeds this up to $O(\log\log n)$ rounds a.a.s., at the price of using $O(χ\log\log n)$ colours. Finally, for every constant $\varepsilon>0$, it runs in $O(1)$ rounds a.a.s. when $χ^{1+\varepsilon}$ colours are used. This greatly reduces the number of colours over the previous constant-round algorithm of Maus and Ruff (SODA 2026) by a factor of at least $n^{1/6}$. Our analysis contains a phase in which we consider a classical randomised colouring protocol on a (large) clique of the graph. We also delve deeper into this part of the analysis and improve upon previous results for colouring a clique $C$, bounding the number of rounds required as a function of the additive slack $s = |Ψ| - χ$, where $Ψ$ is the set of colours used. In particular, constant-round colouring is possible if and only if $s=|C|^{1+Ω(1)}$, while $s=|C|/\log |C|$ already gives the optimal $Θ(\log\log |C|)$ round complexity.
