# Computing Generalized Convolutions Faster Than Brute Force.

Algorithmica（2023）

Abstract

In this paper, we consider a general notion of convolution. Let D be a finite domain and let Dn be the set of n-length vectors (tuples) of D. Let f:D×D→D be a function and let ⊕f be a coordinate-wise application of f. The f-Convolution of two functions g,h:Dn→{-M,…,M} is (g⊛fh)(v):=∑vg,vh∈Dns.t.v=vg⊕fvhg(vg)·h(vh)for every v∈Dn. 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 O~(|D|2n·polylog(M)) time. Our main result is an improvement over this naive algorithm. We show that f-Convolution can be computed exactly in O~((c·|D|2)n·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×D→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:Dn→{-M,…,M} alongside with a vector v∈Dn and the task of the f-Query problem is to compute integer (g⊛fh)(v). This is a generalization of the well-known Orthogonal Vectors problem. We show that f-Query can be computed in O~(|D|ω2n·polylog(M)) time, where ω∈[2,2.372) is the exponent of currently fastest matrix multiplication algorithm.

MoreTranslated text

Key words

Generalized Convolution,Fast Fourier Transform,Fast Subset Convolution,Orthogonal Vectors,Theory of computation,Parameterized complexity and exact algorithms,Theory of computation,Algorithm design techniques

AI Read Science

Must-Reading Tree

Example

Generate MRT to find the research sequence of this paper

Chat Paper

Summary is being generated by the instructions you defined