一种新的随机扩散算法的离散性分析
Discrepancy Analysis of a New Randomized Diffusion Algorithm
摘要：
对于分布图顶点上的离散载荷的任意初始配置，我们考虑最小化所有顶点之间的最大和最小载荷之间的{ \em差异}的问题.针对这一问题，本文研究了基于自然扩散的迭代算法的能力：在算法的每个离散和同步时间步，允许每个顶点将其负载分配给每个邻居（包括自身），而不发生负负载或使用先前时间步的信息。在这种情况下，本文提出了一种新的{ \em随机}扩散算法，如多重随机游动。我们的算法档案$O(\sqrt{d \log N})$差异的任何$d$ -正则图的$N$顶点具有高概率，而{ \em确定性}扩散算法有$Ω(d)$下界。此外，我们成功地将我们的算法推广到任何对称轮矩阵。这产生了任意图的$O(\sqrt{ d_{\max} \log N})$差异，而不使用最大度$d_{\max}$的信息。
英文摘要：
For an arbitrary initial configuration of discrete loads over vertices of a distributed graph, we consider the problem of minimizing the {\em discrepancy} between the maximum and minimum loads among all vertices. For this problem, this paper is concerned with the ability of natural diffusion-based iterative algorithms: at each discrete and synchronous time step on an algorithm, each vertex is allowed to distribute its loads to each neighbor (including itself) without occurring negative loads or using the information of previous time steps. In this setting, this paper presents a new {\em randomized} diffusion algorithm like multiple random walks. Our algorithm archives $O(\sqrt{d \log N})$ discrepancy for any $d$-regular graph with $N$ vertices with high probability, while {\em deterministic} diffusion algorithms have $Ω(d)$ lower bound. Furthermore, we succeed in generalizing our algorithm to any symmetric round matrix. This yields that $O(\sqrt{ d_{\max} \log N})$ discrepancy for arbitrary graphs without using the information of maximum degree $d_{\max}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
