Systematic Data Structure Lower Bounds via the Query-with-Sketch Model

2026-09-15

AI Summary

研究了近似矩阵幂问题的数据结构下界，通过引入查询-草图模型和基于最小熵的方法，证明了系统数据结构中探针-冗余权衡。

Abstract

We study data structure lower bounds for the Approximate Matrix Powering (AMP) problem. Given a substochastic, symmetric matrix $\mathbf{M}\in\mathbb{R}^{n\times n}$ and parameters $k$ and $α$, the goal is to preprocess $\mathbf{M}$ so as to answer entry queries $(u,v)\mapsto \mathbf{M}^{k}[u,v]$ up to additive error $1/n^α$. We focus on AMP in the succinct and systematic regime, in which the data structure stores $\mathbf{M}$ verbatim, uses an additional $r$ bits of redundancy, and must answer queries by probing only a small number of entries of $\mathbf{M}$. Our main conceptual contribution is a general framework for proving probe--redundancy trade-offs for systematic data structures. We introduce the query-with-sketch model and develop a min-entropy-based approach that lifts conditional min-entropy bounds in the absence of redundancy to probe lower bounds in the presence of redundancy. We then establish these min-entropy bounds using problem-specific analytic and algebraic tools, for the downstream applications to AMP and its variants. As a consequence, our results provide new unconditional evidence toward a conjecture of Patrascu and Roditty (2010) on the space required for constant-time set-disjointness queries.

Problem

• Approximate Matrix Powering
• Data Structure Lower Bounds
• Query-with-Sketch Model

Innovation

• query-with-sketch model
• probe-redundancy trade-offs
• min-entropy based approach