Complexity of Linear Operators

Authors Alexander S. Kulikov , Ivan Mikhailin, Andrey Mokhov, Vladimir Podolskii

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Alexander S. Kulikov
  • Steklov Mathematical Institute at St. Petersburg, Russian Academy of Sciences, St. Petersburg State University, Russia
Ivan Mikhailin
  • University of California, San Diego, CA, USA
Andrey Mokhov
  • School of Engineering, Newcastle University, UK
Vladimir Podolskii
  • Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia


We thank Paweł Gawrychowski for pointing us out to the paper [Bernard Chazelle and Burton Rosenberg, 1991]. We thank Alexey Talambutsa for fruitful discussions on the theory of semigroups.

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Alexander S. Kulikov, Ivan Mikhailin, Andrey Mokhov, and Vladimir Podolskii. Complexity of Linear Operators. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 17:1-17:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


Let A in {0,1}^{n x n} be a matrix with z zeroes and u ones and x be an n-dimensional vector of formal variables over a semigroup (S, o). 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 x 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 x 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).

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
  • algorithms
  • linear operators
  • commutativity
  • range queries
  • circuit complexity
  • lower bounds
  • upper bounds


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