Computing Generalized Convolutions Faster Than Brute Force

Abstract

AbstractIn this paper, we consider a general notion of convolution. Let $$D$$ D be a finite domain and let $$D^n$$ D n be the set of n-length vectors (tuples) of $$D$$ D . Let $$f :D\times D\rightarrow D$$ f : D × D → D be a function and let $$\oplus _f$$ ⊕ f be a coordinate-wise application of f. The $$f$$ f -Convolution of two functions $$g,h :D^n \rightarrow \{-M,\ldots ,M\}$$ g , h : D n → { - M , … , M } is $$\begin{aligned} (g \mathbin {\circledast _{f}}h)(\textbf{v}) {:}{=}\sum _{\begin{array}{c} \textbf{v}_g,\textbf{v}_h \in D^n\\ \text {s.t. } \textbf{v}= \textbf{v}_g \oplus _f \textbf{v}_h \end{array}} g(\textbf{v}_g) \cdot h(\textbf{v}_h) \end{aligned}$$ ( g ⊛ f h ) ( v ) : = ∑ v g , v h ∈ D n s.t. v = v g ⊕ f v h g ( v g ) · h ( v h ) for every $$\textbf{v}\in D^n$$ v ∈ 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 $$D$$ D we can compute $$f$$ f -Convolution via brute-force enumeration in $$\widetilde{{\mathcal {O}}}(|D|^{2n} \cdot \textrm{polylog}(M))$$ O ~ ( | D | 2 n · polylog ( M ) ) time. Our main result is an improvement over this naive algorithm. We show that $$f$$ f -Convolution can be computed exactly in $$\widetilde{{\mathcal {O}}}( (c \cdot |D|^2)^{n} \cdot \textrm{polylog}(M))$$ O ~ ( ( c · | D | 2 ) n · polylog ( M ) ) for constant $$c {:}{=}3/4$$ c : = 3 / 4 when $$D$$ D has even cardinality. Our main observation is that a cyclic partition of a function $$f :D\times D\rightarrow D$$ f : D × D → D can be used to speed up the computation of $$f$$ f -Convolution, and we show that an appropriate cyclic partition exists for every f. Furthermore, we demonstrate that a single entry of the $$f$$ f -Convolution can be computed more efficiently. In this variant, we are given two functions $$g,h :D^n \rightarrow \{-M,\ldots ,M\}$$ g , h : D n → { - M , … , M } alongside with a vector $$\textbf{v}\in D^n$$ v ∈ D n and the task of the $$f$$ f -Query problem is to compute integer $$(g \mathbin {\circledast _{f}}h)(\textbf{v})$$ ( g ⊛ f h ) ( v ) . This is a generalization of the well-known Orthogonal Vectors problem. We show that $$f$$ f -Query can be computed in $$\widetilde{{\mathcal {O}}}(|D|^{\frac{\omega }{2} n} \cdot \textrm{polylog}(M))$$ O ~ ( | D | ω 2 n · polylog ( M ) ) time, where $$\omega \in [2,2.372)$$ ω ∈ [ 2 , 2.372 ) is the exponent of currently fastest matrix multiplication algorithm.