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DOI: 10.4230/LIPIcs.CPM.2019.14

Hamming Distance Completeness

Authors: Karim Labib, Przemysław Uznański, and Daniel Wolleb-Graf

Published in: LIPIcs, Volume 128, 30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019)

Abstract

We show, given a binary integer function diamond that is piecewise polynomial, that (+,diamond) vector products are equivalent under one-to-polylog reductions to the computation of the Hamming distance. Examples include the dominance and l_{2p+1} distances for constant p. Our results imply equivalence (up to polylog factors) between the complexity of computing All Pairs Hamming Distance, All Pairs l_{2p+1} Distance and Dominance Matrix Product, and equivalence between Hamming Distance Pattern Matching, l_{2p+1} Pattern Matching and Less-Than Pattern Matching. The resulting algorithms for l_{2p+1} Pattern Matching and All Pairs l_{2p+1}, for 2p+1 = 3,5,7,... are likely to be optimal, given lack of progress in improving upper bounds for Hamming distance in the past 30 years. While reductions between selected pairs of products were presented in the past, our work is the first to generalize them to a general class of functions, showing that a wide class of "intermediate" complexity problems are in fact equivalent.

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Karim Labib, Przemysław Uznański, and Daniel Wolleb-Graf. Hamming Distance Completeness. In 30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 128, pp. 14:1-14:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{labib_et_al:LIPIcs.CPM.2019.14,
  author =	{Labib, Karim and Uzna\'{n}ski, Przemys{\l}aw and Wolleb-Graf, Daniel},
  title =	{{Hamming Distance Completeness}},
  booktitle =	{30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019)},
  pages =	{14:1--14:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-103-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{128},
  editor =	{Pisanti, Nadia and P. Pissis, Solon},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2019.14},
  URN =		{urn:nbn:de:0030-drops-104853},
  doi =		{10.4230/LIPIcs.CPM.2019.14},
  annote =	{Keywords: fine grained complexity, approximate pattern matching, matrix products}
}

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Brief Announcement

DOI: 10.4230/LIPIcs.ICALP.2018.109

Brief Announcement: Hamming Distance Completeness and Sparse Matrix Multiplication

Authors: Daniel Graf, Karim Labib, and Przemyslaw Uznanski

Published in: LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

Abstract

We show that a broad class of (+, diamond) vector products (for binary integer functions diamond) are 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 results imply equivalence (up to poly log n factors) between complexity of computation of All Pairs: Hamming Distances, l_{2p+1} Distances, Dominance Products and Threshold Products. As a consequence, Yuster's (SODA'09) algorithm improves not only Matousek's (IPL'91), but also the results of Indyk, Lewenstein, Lipsky and Porat (ICALP'04) and Min, Kao and Zhu (COCOON'09). Furthermore, our reductions apply to the pattern matching setting, showing equivalence (up to poly log n factors) between pattern matching under Hamming Distance, l_{2p+1} Distance, Dominance Product and Threshold Product, with current best upperbounds 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 resulting algorithms for l_{2p+1} Pattern Matching and All Pairs l_{2p+1}, for 2p+1 = 3,5,7,... are new. Additionally, we show that the complexity of AllPairsHammingDistances (and thus of other aforementioned AllPairs- problems) is within poly log n from the time it takes to multiply matrices n x (n * d) and (n * d) x n, each with (n * d) non-zero entries. This means that the current upperbounds by Yuster (SODA'09) cannot be improved without improving the sparse matrix multiplication algorithm by Yuster and Zwick (ACM TALG'05) and vice versa.

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Daniel Graf, Karim Labib, and Przemyslaw Uznanski. Brief Announcement: Hamming Distance Completeness and Sparse Matrix Multiplication. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 109:1-109:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{graf_et_al:LIPIcs.ICALP.2018.109,
  author =	{Graf, Daniel and Labib, Karim and Uznanski, Przemyslaw},
  title =	{{Brief Announcement: Hamming Distance Completeness and Sparse Matrix Multiplication}},
  booktitle =	{45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)},
  pages =	{109:1--109:4},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-076-7},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{107},
  editor =	{Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.109},
  URN =		{urn:nbn:de:0030-drops-91134},
  doi =		{10.4230/LIPIcs.ICALP.2018.109},
  annote =	{Keywords: fine-grained complexity, matrix multiplication, high dimensional geometry, pattern matching}
}

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Hamming Distance Completeness

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Brief Announcement: Hamming Distance Completeness and Sparse Matrix Multiplication

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