Equivalence of Systematic Linear Data Structures and Matrix Rigidity

TL;DR

This paper establishes a fundamental equivalence between matrix rigidity and systematic linear data structures, leading to new lower bounds and insights for problems like vector-matrix-vector and inner product queries.

Contribution

It proves an equivalence between matrix rigidity and systematic linear data structures, enabling transfer of lower bounds and explicit constructions between the two areas.

Findings

Lower bounds on query time imply rigidity bounds for query sets.

Explicit rigidity bounds lead to improved lower bounds for vector-matrix-vector problems.

Cell probe lower bounds are established for high error regimes in systematic models.

Abstract

Recently, Dvir, Golovnev, and Weinstein have shown that sufficiently strong lower bounds for linear data structures would imply new bounds for rigid matrices. However, their result utilizes an algorithm that requires an $NP$ oracle, and hence, the rigid matrices are not explicit. In this work, we derive an equivalence between rigidity and the systematic linear model of data structures. For the $n$-dimensional inner product problem with $m$ queries, we prove that lower bounds on the query time imply rigidity lower bounds for the query set itself. In particular, an explicit lower bound of $\omega\left(\frac{n}{r}\log m\right)$ for $r$ redundant storage bits would yield better rigidity parameters than the best bounds due to Alon, Panigrahy, and Yekhanin. We also prove a converse result, showing that rigid matrices directly correspond to hard query sets for the systematic linear model. As an application, we prove that the set of vectors obtained from rank one binary matrices is rigid with parameters matching the known results for explicit sets. This implies that the vector-matrix-vector problem requires query time $\Omega(n^{3/2}/r)$ for redundancy $r \geq \sqrt{n}$ in the systematic linear model, improving a result of Chakraborty, Kamma, and Larsen. Finally, we prove a cell probe lower bound for the vector-matrix-vector problem in the high error regime, improving a result of Chattopadhyay, Kouck\'{y}, Loff, and Mukhopadhyay.