Brief Announcement: Hamming Distance Completeness and Sparse Matrix Multiplication

Authors Daniel Graf, Karim Labib, Przemyslaw Uznanski



PDF
Thumbnail PDF

File

LIPIcs.ICALP.2018.109.pdf
  • Filesize: 416 kB
  • 4 pages

Document Identifiers

Author Details

Daniel Graf
  • Department of Computer Science, ETH Zürich, Switzerland
Karim Labib
  • Department of Computer Science, ETH Zürich, Switzerland
Przemyslaw Uznanski
  • Department of Computer Science, ETH Zürich, Switzerland

Cite AsGet BibTex

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)
https://doi.org/10.4230/LIPIcs.ICALP.2018.109

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • fine-grained complexity
  • matrix multiplication
  • high dimensional geometry
  • pattern matching

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Amihood Amir, Ohad Lipsky, Ely Porat, and Julia Umanski. Approximate matching in the L₁ metric. In CPM, pages 91-103, 2005. URL: http://dx.doi.org/10.1007/11496656_9.
  2. Mikhail J. Atallah and Timothy W. Duket. Pattern matching in the hamming distance with thresholds. Inf. Process. Lett., 111(14):674-677, 2011. URL: http://dx.doi.org/10.1016/j.ipl.2011.04.004.
  3. Peter Clifford, Raphaël Clifford, and Costas S. Iliopoulos. Faster algorithms for δ,γ-matching and related problems. In CPM, pages 68-78, 2005. URL: http://dx.doi.org/10.1007/11496656_7.
  4. Ran Duan and Seth Pettie. Fast algorithms for (max, min)-matrix multiplication and bottleneck shortest paths. In SODA, pages 384-391, 2009. URL: http://dl.acm.org/citation.cfm?id=1496770.1496813.
  5. Omer Gold and Micha Sharir. Dominance product and high-dimensional closest pair under L_∞. In ISAAC, pages 39:1-39:12, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.ISAAC.2017.39.
  6. Piotr Indyk, Moshe Lewenstein, Ohad Lipsky, and Ely Porat. Closest pair problems in very high dimensions. In ICALP, pages 782-792, 2004. URL: http://dx.doi.org/10.1007/978-3-540-27836-8_66.
  7. Ohad Lipsky and Ely Porat. L₁ pattern matching lower bound. Inf. Process. Lett., 105(4):141-143, 2008. URL: http://dx.doi.org/10.1016/j.ipl.2007.08.011.
  8. Kerui Min, Ming-Yang Kao, and Hong Zhu. The closest pair problem under the Hamming metric. In COCOON, pages 205-214, 2009. URL: http://dx.doi.org/10.1007/978-3-642-02882-3_21.
  9. Virginia Vassilevska. Efficient algorithms for path problems in weighted graphs. PhD thesis, Carnegie Mellon University, 2008. Google Scholar
  10. Virginia Vassilevska, Ryan Williams, and Raphael Yuster. All pairs bottleneck paths and max-min matrix products in truly subcubic time. Theory of Computing, 5(1):173-189, 2009. URL: http://dx.doi.org/10.4086/toc.2009.v005a009.
  11. Raphael Yuster. Efficient algorithms on sets of permutations, dominance, and real-weighted APSP. In SODA, pages 950-957, 2009. URL: http://dl.acm.org/citation.cfm?id=1496770.1496873.
  12. Raphael Yuster and Uri Zwick. Fast sparse matrix multiplication. ACM Trans. Algorithms, 1(1):2-13, 2005. URL: http://dx.doi.org/10.1145/1077464.1077466.
  13. Peng Zhang and Mikhail J. Atallah. On approximate pattern matching with thresholds. Inf. Process. Lett., 123:21-26, 2017. URL: http://dx.doi.org/10.1016/j.ipl.2017.03.001.