Deterministic and Las Vegas Algorithms for Sparse Nonnegative Convolution

Approximating text-to-pattern Hamming distances

“…These ideas might spawn further applications. [BFN22] is complicated, and does not address our question of speeding up O(U log U ) in the regime of U close to n 2 . Our new algorithm shares some ideas from these previous algorithms, but is arguably simpler, and more accessible, than the one of [BFN22].…”

“…[BFN22] is complicated, and does not address our question of speeding up O(U log U ) in the regime of U close to n 2 . Our new algorithm shares some ideas from these previous algorithms, but is arguably simpler, and more accessible, than the one of [BFN22]. It only requires Dietzfelbinger's standard family of almost linear hash functions [Die96] (though we need to establish some new properties).…”

“…It would be natural to build Lemma 5.2 on Bringmann, Fischer and Nakos's more efficient algorithm [BFN22] for sparse nonnegative convolution that runs in O(t) time, in order to further improve our deterministic algorithm for exact Text-to-Pattern Hamming Distances. The difficulty in this approach is that, to implement [BFN22]'s idea, we have to view C as a degree U polynomial and evaluate it on t carefully chosen points. It is unclear how to do this evaluation in o(U log U ) time (O(U log U ) is the naive bound for Lemma 5.3 via FFT).…”