# Sparse nonnegative convolution is equivalent to dense nonnegative convolution

@article{Bringmann2021SparseNC, title={Sparse nonnegative convolution is equivalent to dense nonnegative convolution}, author={Karl Bringmann and Nick Fischer and Vasileios Nakos}, journal={Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing}, year={2021} }

Computing the convolution A ⋆ B of two length-n vectors A,B is an ubiquitous computational primitive, with applications in a variety of disciplines. Within theoretical computer science, applications range from string problems to Knapsack-type problems, and from 3SUM to All-Pairs Shortest Paths. These applications often come in the form of nonnegative convolution, where the entries of A,B are nonnegative integers. The classical algorithm to compute A⋆ B uses the Fast Fourier Transform (FFT) and…

## 4 Citations

### Fast n-fold Boolean Convolution via Additive Combinatorics

- Computer Science, MathematicsICALP
- 2021

A deterministic or randomized o(nk) algorithm running in almost linear time with respect to the input plus output size k is presented, and a new deterministic almost linear output-sensitive algorithm for non-negative sparse convolution is presented.

### Removing Additive Structure in 3SUM-Based Reductions

- Mathematics, Computer ScienceArXiv
- 2022

It is shown that solving 3SUM on a size-n integer set that avoids solutions to a + b = c + d for {a, b} 6 = {c, d} still requires n time, under the3SUM hypothesis, which implies that the All-Edges Sparse Triangle problem on nvertex graphs with maximum degree √ n and at mostn k-cycles for every k ≥ 3 requiresn time.

### Stronger 3-SUM Lower Bounds for Approximate Distance Oracles via Additive Combinatorics

- Computer Science, Mathematics
- 2022

The “short cycle removal” technique was recently introduced by Abboud, Bringmann, Khoury and Zamir to prove fine-grained hardness of approximation and is settled by showing that the Õ(min(m, n)+ t) upper bound is tight up to n factors.

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

- Computer ScienceSODA
- 2022

This paper presents the first deterministic near-linear-time algorithm for computing sparse nonnegative convolutions, which immediately gives improved deterministic algorithms for the state-of-the-art of output-sensitive Subset Sum, block-mass pattern matching, N -fold Boolean convolution, and others, matching up to log-factors the fastest known randomized algorithms for these problems.

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