Hamming distance completeness and sparse matrix multiplication
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We investigate relations between (+,) vector products for binary integer functions . We show that there exists a broad class of products equivalent under one-to-polylog reductions to the computation of the Hamming distance. Examples include: the dominance product, the threshold product and ℓ_2p+1 distances for constant p. Our result has the following consequences: The following All Pairs- problems are of the same complexity (up to polylog factors) for n vectors in Z^d: computing Hamming Distance, ℓ_2p+1 Distance, Threshold Products and Dominance Products. 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). Obtain by reduction algorithms for All Pairs ℓ_3,ℓ_5,... Distances are new. The following Pattern Matching problems are of the same complexity (up to polylog factors) for a text of length n and a pattern of length m: Hamming Distance, Less-than, Threshold and ℓ_2p+1. For all of them the current best upper bounds are O(n√(m m)) 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 obtained algorithms for ℓ_3,ℓ_5,... Pattern Matchings are new. We also show that the complexity of All Pairs Hamming Distances is within a polylog factor from sparse(n,d^2,n;nd,nd), where sparse(a,b,c;m_1,m_2) is the time of multiplying sparse matrices of size a× b and b× c, with m_1 and m_2 nonzero entries. 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.
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