Hamming distance completeness and sparse matrix multiplication

We investigate relations between (+, ) vector products for binary integer functions . We show that there exists a broad class of products that are equivalent under one-to-polylog reductions to the computation of the Hamming distance. Examples include, but are not exhausted by: the dominance product, the threshold product and `2p+1 distances for constant p. Our result has the following consequences: • The following problems are of the same complexity (up to polylog factors) for n given vectors in Z: computing AllPairsHammingDistances, AllPairsL2p+1Distances for constant p, AllPairsThresholdProducts and AllPairsDominanceProducts. As a consequence, Yuster’s (SODA’09) algorithm improves not only Matoušek’s (IPL’91), but also the results of Indyk, Lewenstein, Lipsky and Porat (ICALP’04) and Min, Kao and Zhu (COCOON’09). Moreover, we obtain by reduction algorithms for All Pairs `3, `5, . . . Distances with the same runtime. • The following problems are of the same complexity (up to polylog factors) for a given text of length n and a pattern of length m: HammingDistancePatternMatching, LessThanPatternMatching, ThresholdPatternMatching and L2p+1PatternMatching for constant p. Thus it is no coincidence that for all those problems the current best upper bounds are Õ(nm) time due to results of Abrahamson (SICOMP’87), Amir and Farach (Ann. Math. Artif. Intell.’91), Atallah and Duket (IPL’11), Clifford, Clifford and Iliopoulous (CPM’05) and Amir, Lipsky, Porat and Umanski (CPM’05). The algorithms for `3, `5, . . . Pattern Matchings we obtained by our reduction are new. Additionally, we show that the complexity of AllPairsHammingDistances (and thus of other aforementioned AllPairsproblems) is within a polylog factor from Sparse(n, d, n;nd, nd), where Sparse(a, b, c;m1,m2) is the time of multiplying sparse matrices of size a × b and b × c, with m1 and m2 nonzero entries, respectively. This means that the current upperbounds by Yuster cannot be improved without improving the sparse matrix multiplication algorithm by Yuster and Zwick (ACM TALG’05) and vice versa. ar X iv :1 71 1. 03 88 7v 1 [ cs .D S] 1 0 N ov 2 01 7