Document

# Orthogonal Vectors Indexing

## File

LIPIcs.ISAAC.2017.40.pdf
• Filesize: 473 kB
• 12 pages

## Cite As

Isaac Goldstein, Moshe Lewenstein, and Ely Porat. Orthogonal Vectors Indexing. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 40:1-40:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.ISAAC.2017.40

## Abstract

In the recent years, intensive research work has been dedicated to prove conditional lower bounds in order to reveal the inner structure of the class P. These conditional lower bounds are based on many popular conjectures on well-studied problems. One of the most heavily used conjectures is the celebrated Strong Exponential Time Hypothesis (SETH). It turns out that conditional hardness proved based on SETH goes, in many cases, through an intermediate problem - the Orthogonal Vectors (OV) problem. Almost all research work regarding conditional lower bound was concentrated on time complexity. Very little attention was directed toward space complexity. In a recent work, Goldstein et al.[WADS '17] set the stage for proving conditional lower bounds regarding space and its interplay with time. In this spirit, it is tempting to investigate the space complexity of a data structure variant of OV which is called OV indexing. In this problem n boolean vectors of size clogn are given for preprocessing. As a query, a vector v is given and we are required to verify if there is an input vector that is orthogonal to it or not. This OV indexing problem is interesting in its own, but it also likely to have strong implications on problems known to be conditionally hard, in terms of time complexity, based on OV. Having this in mind, we study OV indexing in this paper from many aspects. We give some space-efficient algorithms for the problem, show a tradeoff between space and query time, describe how to solve its reporting variant, shed light on an interesting connection between this problem and the well-studied SetDisjointness problem and demonstrate how it can be solved more efficiently on random input.
##### Keywords
• SETH
• orthogonal vectors
• space complexity

## Metrics

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

## References

1. Amir Abboud, Arturs Backurs, Thomas Dueholm Hansen, Virginia Vassilevska Williams, and Or Zamir. Subtree isomorphism revisited. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 1256-1271, 2016.
2. Amir Abboud, Arturs Backurs, and Virginia Vassilevska Williams. Tight hardness results for LCS and other sequence similarity measures. In IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, 17-20 October, 2015, pages 59-78, 2015.
3. Amir Abboud, Richard Ryan Williams, and Huacheng Yu. More applications of the polynomial method to algorithm design. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 218-230, 2015.
4. Amir Abboud and Virginia Vassilevska Williams. Popular conjectures imply strong lower bounds for dynamic problems. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 434-443, 2014.
5. Amir Abboud, Virginia Vassilevska Williams, and Joshua R. Wang. Approximation and fixed parameter subquadratic algorithms for radius and diameter in sparse graphs. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 377-391, 2016.
6. Amir Abboud, Virginia Vassilevska Williams, and Oren Weimann. Consequences of faster alignment of sequences. In Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part I, pages 39-51, 2014.
7. Arturs Backurs and Piotr Indyk. Edit distance cannot be computed in strongly subquadratic time (unless SETH is false). In Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, June 14-17, 2015, pages 51-58, 2015.
8. Karl Bringmann. Why walking the dog takes time: Frechet distance has no strongly subquadratic algorithms unless SETH fails. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 661-670, 2014.
9. Karl Bringmann and Marvin Künnemann. Quadratic conditional lower bounds for string problems and dynamic time warping. In IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, 17-20 October, 2015, pages 79-97, 2015.
10. Moses Charikar, Piotr Indyk, and Rina Panigrahy. New algorithms for subset query, partial match, orthogonal range searching, and related problems. In Automata, Languages and Programming, 29th International Colloquium, ICALP 2002, Malaga, Spain, July 8-13, 2002, Proceedings, pages 451-462, 2002.
11. Hagai Cohen and Ely Porat. Fast set intersection and two-patterns matching. Theor. Comput. Sci., 411(40-42):3795-3800, 2010.
12. Hagai Cohen and Ely Porat. On the hardness of distance oracle for sparse graph. CoRR, abs/1006.1117, 2010.
13. Richard Cole, Lee-Ad Gottlieb, and Moshe Lewenstein. Dictionary matching and indexing with errors and don't cares. In Proceedings of the 36th Annual ACM Symposium on Theory of Computing, Chicago, IL, USA, June 13-16, 2004, pages 91-100, 2004.
14. Marek Cygan, Marcin Pilipczuk, and Michal Pilipczuk. Known algorithms for EDGE CLIQUE COVER are probably optimal. In Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013, New Orleans, Louisiana, USA, January 6-8, 2013, pages 1044-1053, 2013.
15. Pooya Davoodi, Michiel H. M. Smid, and Freek van Walderveen. Two-dimensional range diameter queries. In LATIN 2012: Theoretical Informatics - 10th Latin American Symposium, Arequipa, Peru, April 16-20, 2012. Proceedings, pages 219-230, 2012.
16. Fedor V. Fomin, Petr A. Golovach, Daniel Lokshtanov, and Saket Saurabh. Almost optimal lower bounds for problems parameterized by clique-width. SIAM J. Comput., 43(5):1541-1563, 2014.
17. Isaac Goldstein, Tsvi Kopelowitz, Moshe Lewenstein, and Ely Porat. Conditional lower bounds for space/time tradeoffs. In To appear in WADS 2017, 2017.
18. Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-sat. J. Comput. Syst. Sci., 62(2):367-375, 2001.
19. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? J. Comput. Syst. Sci., 63(4):512-530, 2001.
20. Kasper Green Larsen and R. Ryan Williams. Faster online matrix-vector multiplication. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 2182-2189, 2017.
21. Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. Lower bounds based on the exponential time hypothesis. Bulletin of the EATCS, 105:41-72, 2011.
22. Daniel Moeller, Ramamohan Paturi, and Stefan Schneider. Subquadratic algorithms for succinct stable matching. In Computer Science - Theory and Applications - 11th International Computer Science Symposium in Russia, CSR 2016, St. Petersburg, Russia, June 9-13, 2016, Proceedings, pages 294-308, 2016.
23. Mihai Patrascu and Liam Roditty. Distance oracles beyond the thorup-zwick bound. SIAM J. Comput., 43(1):300-311, 2014.
24. Mihai Patrascu, Liam Roditty, and Mikkel Thorup. A new infinity of distance oracles for sparse graphs. In 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, New Brunswick, NJ, USA, October 20-23, 2012, pages 738-747, 2012.
25. Mihai Patrascu and Ryan Williams. On the possibility of faster SAT algorithms. In Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, Austin, Texas, USA, January 17-19, 2010, pages 1065-1075, 2010.
26. Ronald L. Rivest. Analysis of Associative Retrieval Algorithm. PhD thesis, Stanford University, 1974.
27. Ronald L. Rivest. Partial-match retrieval algorithms. SIAM J. Comput., 5(1):19-50, 1976.
28. Liam Roditty and Virginia Vassilevska Williams. Fast approximation algorithms for the diameter and radius of sparse graphs. In Symposium on Theory of Computing Conference, STOC'13, Palo Alto, CA, USA, June 1-4, 2013, pages 515-524, 2013.
29. Ryan Williams. A new algorithm for optimal 2-constraint satisfaction and its implications. Theor. Comput. Sci., 348(2-3):357-365, 2005.
30. Ryan Williams and Huacheng Yu. Finding orthogonal vectors in discrete structures. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 1867-1877, 2014.