比暴力枚举更快地计算广义卷积
数据结构与算法
2023-01-31 v2
摘要
本文考虑卷积的一般概念。设 D D D D n D^n D n D D D n n n f : D × D → D f : D \times D \to D f : D × D → D ⊕ f \oplus_f ⊕ f f f f g , h : D n → { − M , … , M } g,h : D^n \to \{-M,\ldots,M\} g , h : D n → { − M , … , M } f f f ( g ⊗ f h ) ( v ) : = ∑ v g , v h ∈ D n s.t. v g ⊕ f v h g ( v g ) ⋅ h ( v h ) (g \otimes_f h)(\textbf{v}) := \sum_{\substack{\textbf{v}_g,\textbf{v}_h \in D^n\\ \text{s.t. } \textbf{v}_g \oplus_f \textbf{v}_h}} g(\textbf{v}_g) \cdot h(\textbf{v}_h) ( g ⊗ f h ) ( v ) := v g , v h ∈ D n s.t. v g ⊕ f v h ∑ g ( v g ) ⋅ h ( v h ) v ∈ D n \textbf{v} \in D^n v ∈ D n f f f D D D O ~ ( ∣ D ∣ 2 n p o l y l o g ( M ) ) \widetilde{O}(|D|^{2n}\mathrm{polylog}(M)) O ( ∣ D ∣ 2 n polylog ( M )) f f f D D D f f f O ~ ( ( c ⋅ ∣ D ∣ 2 ) n p o l y l o g ( M ) ) \widetilde{O}((c \cdot |D|^2)^{n}\mathrm{polylog}(M)) O (( c ⋅ ∣ D ∣ 2 ) n polylog ( M )) c : = 3 / 4 c := 3/4 c := 3/4 f : D × D → D f : D \times D \to D f : D × D → D f f f f f f f f f g , h : D n → { − M , … , M } g,h : D^n \to \{-M,\ldots,M\} g , h : D n → { − M , … , M } v ∈ D n \textbf{v} \in D^n v ∈ D n f f f ( g ⊗ f h ) ( v ) (g \otimes_f h)(\textbf{v}) ( g ⊗ f h ) ( v ) f f f O ~ ( ∣ D ∣ ω 2 n p o l y l o g ( M ) ) \widetilde{O}(|D|^{\frac{\omega}{2} n}\mathrm{polylog}(M)) O ( ∣ D ∣ 2 ω n polylog ( M )) ω ∈ [ 2 , 2.372 ) \omega \in [2,2.372) ω ∈ [ 2 , 2.372 )
引用
@article{arxiv.2209.01623,
title = {Computing Generalized Convolutions Faster Than Brute Force},
author = {Barış Can Esmer and Ariel Kulik and Dániel Marx and Philipp Schepper and Karol Węgrzycki},
journal= {arXiv preprint arXiv:2209.01623},
year = {2023}
}
备注
We improved constant c, 29 pages, 5 colored figures