### Complexity of Linear Operators

Alexander S. Kulikov, Ivan Mikhailin, Andrey Mokhov, Vladimir Podolskii, Michael Wagner
2019 International Symposium on Algorithms and Computation
Let A ∈ {0, 1} n×n be a matrix with z zeroes and u ones and 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? 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
more » ... ions. 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 ∈ {0, 1} n×n with exactly two zeroes in every row (hence z = 2n) whose complexity is Θ(nα(n)) where α(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 × 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). ACM Subject Classification Theory of computation → Streaming, sublinear and near linear time algorithms Complexity of Linear Operators 1≤j≤n Aij =1 x j where the summation is over the semigroup operation •. 1 More specifically, we are interested in lower and upper bounds involving z and u. Matrices with u = O(n) are usually called sparse, whereas matrices with z = O(n) are called complements of sparse matrices. Computing all n outputs of Ax directly (i.e. using the above definition) takes O(u) semigroup operations. The main question studied in this paper is: can Ax be computed using O(z) semigroup operations? Note that it is easy to achieve O(z) complexity if • has an inverse. Indeed, in this case Ax can be computed via subtraction: where U is the all-ones matrix whose linear operator can be computed trivially using O(n) semigroup operations, and A is the complement of A and therefore has only z = O(n) ones.