Department of Computer Science | Institute of Theoretical Computer Science | CADMO

Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

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

Mittagsseminar Talk Information

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 L2p+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 Zd: computing Hamming Distance, L2p+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 L2p+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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