Computing Generalized Convolutions Faster Than Brute Force

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 (Equation Presented) 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^e(|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^e((c · |D|²)ⁿ · polylog(M)) for constant c := 5/6 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⊛_f h)(v). This is a generalization of the well-known Orthogonal Vectors problem. We show that f-Query can be computed in O^e(|D|^{ω2 n} · polylog(M)) time, where ω ∈ [2, 2.373) is the exponent of currently fastest matrix multiplication algorithm.