Computing the convolution $Astar B$ of two length-$n$ vectors $A,B$ is an ubiquitous computational primitive. 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 $Astar B$ uses the Fast Fourier Transform and runs in time $O(nlog n)$. However, often $A$ and $B$ satisfy sparsity conditions, and hence one could hope for significant improvements. The ideal goal is an $O(klog k)$-time algorithm, where $k$ is the number of non-zero elements in the output, i.e., the size of the support of $Astar B$. 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 $O(klog^2 n)$. The main result of this paper is the first $O(klog k)$-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 constant-error 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 $S(k)=O(D(k)+k(loglog k)^2)$. 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.
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