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This work is licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0).
© 2025 Alexander S. Kulikov, Ivan Mikhailin, Andrey Mokhov, and Vladimir V. Podolskii.Let A be an n-by-n 0/1-matrix with z zeroes and u ones and let x be an n-dimensional vector of formal variables over a semigroup (S, ◦). How many semigroup operations are required to compute the linear operator Ax? 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? For the case when the semigroup is commutative, we give a constructive proof of an O(z) upper bound. This implies that in the commutative settings, the complements of sparse matrices can be processed as efficiently as sparse matrices, though the corresponding algorithms are more involved. This covers the cases of Boolean and tropical semirings that have numerous applications, e. g., in graph theory. On the other hand, we prove that in general this is not possible: for faithful non-commutative semigroups there exists an n-by-n 0/1-matrix with exactly two zeroes in every row (hence z = 2n) whose complexity is Θ(nα(n)) where α(n) is the inverse Ackermann function. As a simple application of the linear-size construction presented, we show how to multiply two n × n matrices over an arbitrary semiring in O(n2) time if one of these matrices is a 0/1-matrix with O(n) zeroes (i. e., the complement of a sparse matrix).
Author(s): Kulikov AS, Mikhailin I, Mokhov A, Podolskii VV
Publication type: Article
Publication status: Published
Journal: Theory of Computing
Year: 2025
Volume: 21
Issue: 9
Pages: 1-32
Online publication date: 10/11/2025
Acceptance date: 02/04/2018
Date deposited: 24/11/2025
ISSN (electronic): 1557-2862
Publisher: University of Chicago, Department of Computer Science
URL: http://dx.doi.org/10.4086/toc.2025.v021a009
DOI: 10.4086/toc.2025.v021a009
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