# Fast n-fold Boolean Convolution via Additive Combinatorics

We consider the problem of computing the Boolean convolution (with wraparound) of n vectors of dimension m, or, equivalently, the problem of computing the sumset A_1+A_2+…+A_n for A_1,…,A_n ⊆ℤ_m. Boolean convolution formalizes the frequent task of combining two subproblems, where the whole problem has a solution of size k if for some i the first subproblem has a solution of size i and the second subproblem has a solution of size k-i. Our problem formalizes a natural generalization, namely combining solutions of n subproblems subject to a modular constraint. This simultaneously generalises Modular Subset Sum and Boolean Convolution (Sumset Computation). Although nearly optimal algorithms are known for special cases of this problem, not even tiny improvements are known for the general case. We almost resolve the computational complexity of this problem, shaving essentially a factor of n from the running time of previous algorithms. Specifically, we present a deterministic algorithm running in almost linear time with respect to the input plus output size k. We also present a Las Vegas algorithm running in nearly linear expected time with respect to the input plus output size k. Previously, no deterministic or randomized o(nk) algorithm was known. At the heart of our approach lies a careful usage of Kneser's theorem from Additive Combinatorics, and a new deterministic almost linear output-sensitive algorithm for non-negative sparse convolution. In total, our work builds a solid toolbox that could be of independent interest.

• 43 publications
• 18 publications
07/15/2021

### Deterministic and Las Vegas Algorithms for Sparse Nonnegative Convolution

Computing the convolution A⋆ B of two length-n integer vectors A,B is a ...
07/28/2021

### Top-k-Convolution and the Quest for Near-Linear Output-Sensitive Subset Sum

In the classical Subset Sum problem we are given a set X and a target t,...
10/22/2020

### Deterministic Approximation for Submodular Maximization over a Matroid in Nearly Linear Time

We study the problem of maximizing a non-monotone, non-negative submodul...
05/12/2021

### Sparse Nonnegative Convolution Is Equivalent to Dense Nonnegative Convolution

Computing the convolution A⋆ B of two length-n vectors A,B is an ubiquit...
08/24/2020

### Fast and Simple Modular Subset Sum

We revisit the Subset Sum problem over the finite cyclic group ℤ_m for s...
07/12/2018

### Fast Modular Subset Sum using Linear Sketching

Given n positive integers, the Modular Subset Sum problem asks if a subs...
10/29/2021

### Computing Lewis Weights to High Precision

We present an algorithm for computing approximate ℓ_p Lewis weights to h...