Sparse Nonnegative Convolution Is Equivalent to Dense Nonnegative Convolution
Abstract
Computing the convolution $A\star B$ of two length$n$ vectors $A,B$ is an ubiquitous computational primitive. Applications range from string problems to Knapsacktype problems, and from 3SUM to AllPairs 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 $A\star B$ uses the Fast Fourier Transform and runs in time $O(n\log n)$. However, often $A$ and $B$ satisfy sparsity conditions, and hence one could hope for significant improvements. The ideal goal is an $O(k\log k)$time algorithm, where $k$ is the number of nonzero elements in the output, i.e., the size of the support of $A\star B$. This problem is referred to as sparse nonnegative convolution, and has received considerable attention in the literature; the fastest algorithms to date run in time $O(k\log^2 n)$. The main result of this paper is the first $O(k\log k)$time algorithm for sparse nonnegative convolution. Our algorithm is randomized and assumes that the length $n$ and the largest entry of $A$ and $B$ are subexponential in $k$. Surprisingly, we can phrase our algorithm as a reduction from the sparse case to the dense case of nonnegative convolution, showing that, under some mild assumptions, sparse nonnegative convolution is equivalent to dense nonnegative convolution for constanterror randomized algorithms. Specifically, if $D(n)$ is the time to convolve two nonnegative length$n$ vectors with success probability $2/3$, and $S(k)$ is the time to convolve two nonnegative vectors with output size $k$ with success probability $2/3$, then $S(k)=O(D(k)+k(\log\log k)^2)$. Our approach uses a variety of new techniques in combination with some old machinery from linear sketching and structured linear algebra, as well as new insights on linear hashing, the most classical hash function.
 Publication:

arXiv eprints
 Pub Date:
 May 2021
 arXiv:
 arXiv:2105.05984
 Bibcode:
 2021arXiv210505984B
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Computer Science  Computational Complexity
 EPrint:
 44 pages, appears in STOC 2021