Computing Generalized Convolutions Faster Than Brute Force

Barış Can Esmer; Ariel Kulik; Dániel Marx; Philipp Schepper; Karol Węgrzycki

doi:10.1007/s00453-023-01176-2

Computing Generalized Convolutions Faster Than Brute Force

Algorithmica. 2024;86(1):334-366. doi: 10.1007/s00453-023-01176-2. Epub 2023 Oct 6.

Authors

Barış Can Esmer¹, Ariel Kulik¹, Dániel Marx¹, Philipp Schepper¹, Karol Węgrzycki²

Affiliations

¹ CISPA Helmholtz Center for Information Security, Saarbrücken, Germany.
² Max Planck Institute for Informatics, Saarland University, Saarbrücken, Germany.

Abstract

In this paper, we consider a general notion of convolution. Let $D$ be a finite domain and let $D^{n}$ be the set of n-length vectors (tuples) of $D$ . Let $f : D \times D \to D$ be a function and let $\oplus_{f}$ be a coordinate-wise application of f. The $f$ -Convolution of two functions $g, h : D^{n} \to {- M, \dots, M}$ is $\begin{matrix} (g ⊛_{f} h) (v) : = \sum_{\begin{matrix} v_{g}, v_{h} \in D^{n} \\ s.t. v = v_{g} \oplus_{f} v_{h} \end{matrix}} g (v_{g}) \cdot h (v_{h}) \end{matrix}$ for every $v \in D^{n}$ . This problem generalizes many fundamental convolutions such as Subset Convolution, XOR Product, Covering Product or Packing Product, etc. For arbitrary function f and domain $D$ we can compute $f$ -Convolution via brute-force enumeration in $\tilde{O} {(| D |}^{2 n} \cdot polylog (M))$ time. Our main result is an improvement over this naive algorithm. We show that $f$ -Convolution can be computed exactly in $\tilde{O} ((c \cdot {| D |}^{2})^{n} \cdot polylog (M))$ for constant $c : = 3 / 4$ when $D$ has even cardinality. Our main observation is that a cyclic partition of a function $f : D \times D \to D$ can be used to speed up the computation of $f$ -Convolution, and we show that an appropriate cyclic partition exists for every f. Furthermore, we demonstrate that a single entry of the $f$ -Convolution can be computed more efficiently. In this variant, we are given two functions $g, h : D^{n} \to {- M, \dots, M}$ alongside with a vector $v \in D^{n}$ and the task of the $f$ -Query problem is to compute integer $(g ⊛_{f} h) (v)$ . This is a generalization of the well-known Orthogonal Vectors problem. We show that $f$ -Query can be computed in $\tilde{O} {(| D |}^{\frac{ω}{2} n} \cdot polylog (M))$ time, where $ω \in [2, 2.372)$ is the exponent of currently fastest matrix multiplication algorithm.

Keywords: Fast Fourier Transform; Fast Subset Convolution; Generalized Convolution; Orthogonal Vectors.