Solving Square-Submatrix Equation Systems

2026-08-13

AI Summary

This work addresses the problem of solving systems of equations defined by equality constraints among square submatrices of an input matrix. Inspired by Gawrychowski et al.'s recursive algorithm for one-dimensional string reconstruction, we present the first extension to the two-dimensional setting, introducing a linear-time algorithm based on recursive decomposition and submatrix matching. The proposed method efficiently resolves any system of square submatrix equalities and directly yields an optimal-time decompression algorithm for copy operations in two-dimensional macro compression schemes. This advancement significantly enhances the efficiency of compression and reconstruction for structured two-dimensional data.

Abstract

We consider systems of submatrix equations, that is, sets of equality constraints over square submatrices of the input. By generalising the recursive algorithm of Gawrychowski et al. [Universal reconstruction of a string, Theoretical Computer Science 2020] to two dimensions, we obtain a linear-time procedure that finds a solution for any such input system. As an immediate by-product, this yields an optimal-time algorithm for decompressing any two-dimensional macro scheme based on copy operations of sub-squares.

Problem

• submatrix equations
• square submatrices
• equality constraints
• two-dimensional macro scheme
• string reconstruction

Innovation

• submatrix equations
• two-dimensional macro scheme
• linear-time algorithm
• recursive reconstruction
• square submatrices