Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Tuesday, January 30, 2018, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Karim Labib

Hamming Distance Completeness & Sparse Matrix Multiplication

We investigate relations between (+,g) vector products for binary integer functions g. 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:

1) 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 Matousek's (IPL'91) result but also the results of Indyk, Lewenstein, Lipsky, and Porat (ICALP'04) and Min, Kao and Zhu (COCOON'09). Thus, algorithms for All Pairs L₃,L₅,… Distances, obtained by our reductions, are new.

2) 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 log 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 L₃,L₅,… Pattern Matchings are new.

We also show that the complexity of All Pairs Hamming Distances of 'n' vectors each of dimension 'd' is within a polylog factor from sparse(n,d²,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. This means that the current upper-bounds by Yuster cannot be improved without improving the sparse matrix multiplication algorithm by Yuster and Zwick~(ACM TALG'05) and vice versa.

Based on joint work with: Daniel Graf (ETH), Przemysław Uznański (ETH)

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