显式非平衡1-扩张图：小度数与合适大小

中文摘要 AI 辅助

本文给出一个显式图，其左侧大小为$N$，左侧度数为$\widetilde O(\log^2 N)$，右侧大小为$(1+o(1))K$，并且具有直到$K$的$1$-扩张性质，即任意大小$K' \le K$的左侧子集至少有$K'$个邻居。设$\C(x)$为打印$x$的程序的最短长度（即Kolmogorov复杂度的核心概念）。该$1$-扩张图被用于获得一个算法，该算法在输入$x$时，在$\poly(|x|)$时间内计算一个包含$\widetilde O(|x|^3)$个程序的列表，使得至少有一个程序打印$x$且其长度为$\C(x) + O(1)$。这改进了文献[zim:c:shortlistshortproof]中$O(|x|^{6+\eps})$的上界，并接近文献[bmvz:j:shortlist]定理4中$\Omega(|x|^2)$的下界。在配套论文“Online matching games in bipartite expanders: applications to data structures”中，该$1$-扩张图被用于获得动态字典，其中查询操作具有非自适应内存访问。

英文摘要

An explicit graph is given with left size $N$, left degree $\widetilde O(\log^2 N)$, right size $(1+o(1))K$ and $1$-expansion up to~$K$, meaning that every left subset of size $K' \le K$ has at least $K'$ neighbors. Let $\C(x)$ be the minimal length of a program that prints~$x$ (i.e., the central concept in Kolmogorov complexity). The $1$-expander is used to obtain an algorithm that on input $x$ computes in time $\poly(|x|)$ a list with $\widetilde O(|x|^3)$ programs such that at least 1 program prints $x$ and has length $\C(x) + O(1)$. This improves on the $O(|x|^{6+\eps})$ upper bound in~\cite{zim:c:shortlistshortproof} and is close to the $Ω(|x|^2)$ lower bound from~\cite[theorem 4]{bmvz:j:shortlist}. In the companion paper ``Online matching games in bipartite expanders: applications to data structures," the $1$-expander is used to obtain dynamic dictionaries in which the query operation has non-adaptive memory access.

发表机构

• National Research University Higher School of Economics, Faculty of Computer Science（国立研究大学高等经济大学计算机学院）
• Department of Computer and Information Sciences, Towson University（托森大学计算机与信息科学系）

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