# Complexity of Linear Operators

## TL;DR

This paper investigates the complexity of computing linear operators over semigroups, revealing fundamental limits and providing efficient algorithms for sparse matrices, especially in commutative semigroup settings with applications in graph algorithms and matrix multiplication.

## Contribution

It establishes lower bounds for non-commutative cases and provides constructive upper bounds for commutative semigroups, enabling efficient processing of sparse matrix complements.

## Key findings

- Non-commutative case requires (n (n \u03b1(n))) operations.
- In commutative semigroups, complements of sparse matrices can be processed in O(z) operations.
- Matrix multiplication with a sparse matrix can be done in O(n^2) time when one matrix is a 0/1-matrix with O(n) zeroes.

## Abstract

Let $A \in \{0,1\}^{n \times n}$ be a matrix with $z$ zeroes and $u$ ones and $x$ be an $n$-dimensional vector of formal variables over a semigroup $(S, \circ)$. How many semigroup operations are required to compute the linear operator $Ax$? As we observe in this paper, this problem contains as a special case the well-known range queries problem and has a rich variety of applications in such areas as graph algorithms, functional programming, circuit complexity, and others. It is easy to compute $Ax$ using $O(u)$ semigroup operations. The main question studied in this paper is: can $Ax$ be computed using $O(z)$ semigroup operations? We prove that in general this is not possible: there exists a matrix $A \in \{0,1\}^{n \times n}$ with exactly two zeroes in every row (hence $z=2n$) whose complexity is $\Theta(n\alpha(n))$ where $\alpha(n)$ is the inverse Ackermann function. However, for the case when the semigroup is commutative, we give a constructive proof of an $O(z)$ upper bound. This implies that in commutative settings, complements of sparse matrices can be processed as efficiently as sparse matrices (though the corresponding algorithms are more involved). Note that this covers the cases of Boolean and tropical semirings that have numerous applications, e.g., in graph theory. As a simple application of the presented linear-size construction, we show how to multiply two $n\times n$ matrices over an arbitrary semiring in $O(n^2)$ time if one of these matrices is a 0/1-matrix with $O(n)$ zeroes (i.e., a complement of a sparse matrix).