Solving systems of linear equations through zero forcing set and application to lights out

TL;DR

This paper introduces a method using zero forcing sets to efficiently solve multiple linear systems and applies it to optimize the Lights Out game solution, achieving significant time improvements.

Contribution

It presents a novel approach leveraging zero forcing sets for fast repeated solutions of linear systems and applies this to improve Lights Out game algorithms.

Findings

Data structure construction in O(mk) time for solving Ax=b

Solution time per instance reduced to O(k^2 + m)

Improved Lights Out game solution from O(n^3) to O(n^ω log n)

Abstract

Let $\mathbb{F}$ be any field, we consider solving $Ax=b$ repeatedly for a matrix $A\in\mathbb{F}^{n\times n}$ of $m$ non-zero elements, and multiple different $b\in\mathbb{F}^{n}$. If we are given a zero forcing set of $A$ of size $k$, we can then build a data structure in $O(mk)$ time, such that each instance of $Ax=b$ can be solved in $O(k^2+m)$ time. As an application, we show how the lights out game in an $n\times n$ grid is solved in $O(n^3)$ time, and then improve the running time to $O(n^\omega\log n)$ by exploiting the repeated structure in grids.