通过有界独立性采样实现小空间谱稀疏化

IF 0.8 Q3 COMPUTER SCIENCE, THEORY & METHODS
Dean Doron, Jack Murtagh, Salil P. Vadhan, David Zuckerman
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引用次数: 0

摘要

我们给出了一种用于无向图温和谱稀疏化的确定性近对数空间算法。给定一个由长度为 N 的二进制字符串描述的 n 个顶点上的加权无向图 G、一个整数 k ≤ log n 和一个误差参数 ϵ > 0,我们的算法在空间 \(\tilde{O}(k\log (N\cdot w_\mathrm{max}}/w_{\mathrm{min}})) 中运行。\),其中 w max 和 w min 是 G 中最大和最小的边权重,生成的加权图 H 带有 \(\tilde{O}(n^{1+2/k}/\epsilon ^2) \)条边,在 Spielmen 和 Teng [54] 的意义上,频谱上近似于 G,误差不超过 ϵ。 我们的算法基于对 Spielman 和 Srivastava 基于有效阻力的边缘采样算法 [53] 的新的有界独立性分析,并使用了最近关于空间有界拉普拉斯求解器 [43] 的研究成果。特别是,我们证明了边缘采样算法中使用的(有界)独立性量(以上用 k 表示)与所能达到的稀疏性之间的内在权衡(通过上下限)。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Small-Space Spectral Sparsification via Bounded-Independence Sampling
We give a deterministic, nearly logarithmic-space algorithm for mild spectral sparsification of undirected graphs. Given a weighted, undirected graph G on n vertices described by a binary string of length N , an integer k ≤ log  n , and an error parameter ϵ > 0, our algorithm runs in space \(\tilde{O}(k\log (N\cdot w_{\mathrm{max}}/w_{\mathrm{min}})) \) where w max and w min are the maximum and minimum edge weights in G , and produces a weighted graph H with \(\tilde{O}(n^{1+2/k}/\epsilon ^2) \) edges that spectrally approximates G , in the sense of Spielmen and Teng [54], up to an error of ϵ. Our algorithm is based on a new bounded-independence analysis of Spielman and Srivastava’s effective resistance based edge sampling algorithm [53] and uses results from recent work on space-bounded Laplacian solvers [43]. In particular, we demonstrate an inherent tradeoff (via upper and lower bounds) between the amount of (bounded) independence used in the edge sampling algorithm, denoted by k above, and the resulting sparsity that can be achieved.
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来源期刊
ACM Transactions on Computation Theory
ACM Transactions on Computation Theory COMPUTER SCIENCE, THEORY & METHODS-
CiteScore
2.30
自引率
0.00%
发文量
10
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