Explicit unbalanced 1-expanders with small degree and right size

2026-09-13

AI Summary

研究通过构建一个显式的不平衡1-扩展图，提出了一种新算法，该算法能在多项式时间内生成一个程序列表，其中至少有一个程序能打印输入x且长度接近x的Kolmogorov复杂度。

Abstract

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.

Problem

• 1-expander
• Kolmogorov complexity
• explicit graph
• program length
• algorithm efficiency

Innovation

• explicit 1-expander
• small degree
• right size
• Kolmogorov complexity
• shortlist algorithm