图谱稀疏化属于催化对数空间（Catalytic Logspace）

中文摘要 AI 辅助

我们提出了一种用于图谱稀疏化问题的催化对数空间算法。给定一个含n个顶点的无向图$G$以及$\boldsymbol{\u03b5>0}$，我们的算法输出$G$的一个$\boldsymbol{\u03b5}$-谱稀疏化结果，其边数为$\boldsymbol{O(n\u03b5^{-2}\log n)}$，与Spielman和Srivastava（STOC 2008）的有效电阻采样方法的结果相当。这为催化对数空间领域提供了一个新的自然问题，目前尚不清楚该问题是否属于确定性$\boldsymbol{\mathbf{NC}}$或$\boldsymbol{\mathbf{SC}}$类。我们的主要贡献是提出了催化对数空间“压缩-或-随机”范式下的一种全新技术，我们认为该技术将有更广泛的应用。\n 我们首先利用一种本身可在催化对数空间中计算的悲观估计量，对有效电阻稀疏化方法进行了分析。该估计量的灵感来源于图拟随机性的视角，并且可以直接得到一种简单的确定性贪心图稀疏化算法。\n 随后我们证明，这类悲观估计量可以转化为一种算法，该算法能够对具有不良势的字符串执行原位压缩。我们的算法基于势函数定义字符串上的测度，并利用该测度原位实现算术编码。这种压缩技术与催化计算领域现有的所有工具都有显著区别。

英文摘要

We give a catalytic logspace algorithm for the problem of graph spectral sparsification. Given an undirected graph $G$ on $n$ vertices and $\varepsilon>0$, our algorithm outputs an $\varepsilon$-spectral sparsifier of $G$ with $O(n\varepsilon^{-2}\log n)$ edges, matching the effective resistance sampling of Spielman and Srivastava (STOC 2008). This gives a new, natural problem in catalytic logspace that is not known to be in deterministic $\mathbf{NC}$ or $\mathbf{SC}$. Our main contribution is an entirely new technique in the compress--or--random paradigm for catalytic logspace that we believe will have further applications. We first analyze effective-resistance sparsification using a pessimistic estimator that can itself be computed in catalytic logspace. The estimator is motivated by the viewpoint of graph quasirandomness and immediately gives a simple, deterministic greedy algorithm for graph sparsification. Subsequently, we show that such a pessimistic estimator can be transformed into an algorithm that performs an in-place compression of a string with bad potential. Our algorithm is based on using the potential function to define a measure over strings, and implementing arithmetic coding using this measure in-place. This compression technique is substantially distinct from all prior tools in the field of catalytic computation.

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29 pages