发表机构
Georgetown University（乔治城大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对$Δ+1$边着色问题，提出跨度$\tilde{O}(Δ^3)$、工作量$\tilde{O}(mΔ^3)$的确定性并行算法，以及高概率下跨度$\tilde{O}(Δ^2)$、工作量$\tilde{O}(mΔ^2)$的随机算法，大幅优于现有算法且未增大对数因子。
AI 中文摘要
本文提出两种用于$Δ+1$边着色的并行算法，其中$Δ$表示任意顶点的最大度。第一种是确定性并行算法，跨度为$\tilde{O}(Δ^3)$，工作量为$\tilde{O}(m Δ^3)$。我们的第二种算法也是核心成果，是一种更高效的随机算法，在高概率下实现$\tilde{O}(Δ^2)$的跨度和$\tilde{O}(m Δ^2)$的工作量。这些界相比Elkin和Khuzman近期提出的、跨度为$\tilde{O}(Δ^4)$、工作量为$\tilde{O}(m Δ^5)$的确定性并行算法有显著提升。我们的确定性算法在跨度和工作量上较其算法分别有$\tilde{O}(Δ)$和$\tilde{O}(Δ^2)$的改进，随机算法则分别将跨度和工作量提升了$\tilde{O}(Δ^2)$和$\tilde{O}(Δ^3)$倍。此外，我们的改进并未以增大对数因子为代价。
英文摘要
This paper gives two parallel algorithms for $Δ+1$ edge coloring, where $Δ$ denotes the maximum degree of any vertex. The first is a deterministic parallel algorithm with $\tilde{O}(Δ^3)$ span and $\tilde{O}(m Δ^3)$ work. Our second algorithm and our main result is a more efficient randomized algorithm, achieving $\tilde{O}(Δ^2)$ span and $\tilde{O}(m Δ^2 )$ work both with high probability. These bounds substantially improve over the recent deterministic parallel algorithm of Elkin and Khuzman, which has $\tilde{O}(Δ^4)$ span and $\tilde{O}(m Δ^5)$ work. Our deterministic algorithm thus represents a $\tilde{O}(Δ)$ improvement on span and $\tilde{O}(Δ^2)$ on work compared to their algorithm, and our randomized algorithm improves the span and work by $\tilde{O}(Δ^2)$ and $\tilde{O}(Δ^3)$ factors, respectively. Moreover, our improvements do not come at the expense of larger logarithmic factors.
