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On the Hardness of Set Disjointness and Set Intersection with Bounded Universe

Authors Isaac Goldstein, Moshe Lewenstein, Ely Porat

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Author Details

Isaac Goldstein
  • Bar-Ilan University, Ramat Gan, Israel
Moshe Lewenstein
  • Bar-Ilan University, Ramat Gan, Israel
Ely Porat
  • Bar-Ilan University, Ramat Gan, Israel

Funding

  • Goldstein, Isaac: This research is supported by the Adams Foundation of the Israel Academy of Sciences and Humanities.
  • Lewenstein, Moshe: This work was partially supported by ISF grant #1278/16.
  • Porat, Ely: This work was partially supported by ISF grant #1278/16 and ERC grant MPM - 683064.

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Isaac Goldstein, Moshe Lewenstein, and Ely Porat. On the Hardness of Set Disjointness and Set Intersection with Bounded Universe. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 7:1-7:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ISAAC.2019.7

Abstract

In the SetDisjointness problem, a collection of m sets S_1,S_2,...,S_m from some universe U is preprocessed in order to answer queries on the emptiness of the intersection of some two query sets from the collection. In the SetIntersection variant, all the elements in the intersection of the query sets are required to be reported. These are two fundamental problems that were considered in several papers from both the upper bound and lower bound perspective.
Several conditional lower bounds for these problems were proven for the tradeoff between preprocessing and query time or the tradeoff between space and query time. Moreover, there are several unconditional hardness results for these problems in some specific computational models. The fundamental nature of the SetDisjointness and SetIntersection problems makes them useful for proving the conditional hardness of other problems from various areas. However, the universe of the elements in the sets may be very large, which may cause the reduction to some other problems to be inefficient and therefore it is not useful for proving their conditional hardness.
In this paper, we prove the conditional hardness of SetDisjointness and SetIntersection with bounded universe. This conditional hardness is shown for both the interplay between preprocessing and query time and the interplay between space and query time. Moreover, we present several applications of these new conditional lower bounds. These applications demonstrates the strength of our new conditional lower bounds as they exploit the limited universe size. We believe that this new framework of conditional lower bounds with bounded universe can be useful for further significant applications.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • set disjointness
  • set intersection
  • 3SUM
  • space-time tradeoff
  • conditional lower bounds

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References

  1. Amir Abboud and Kevin Lewi. Exact Weight Subgraphs and the k-Sum Conjecture. In International Colloquium on Automata, Languages and Programming, ICALP 2013, pages 1-12, 2013. Google Scholar
  2. Amir Abboud and Virginia Vassilevska Williams. Popular Conjectures Imply Strong Lower Bounds for Dynamic Problems. In Foundations of Computer Science, FOCS 2014, pages 434-443, 2014. Google Scholar
  3. Amir Abboud, Virginia Vassilevska Williams, and Oren Weimann. Consequences of Faster Alignment of Sequences. In International Colloquium on Automata, Languages and Programming, ICALP 2014, pages 39-51, 2014. Google Scholar
  4. Amir Abboud, Virginia Vassilevska Williams, and Huacheng Yu. Matching Triangles and Basing Hardness on an Extremely Popular Conjecture. In Symposium on Theory of Computing, STOC 2015, pages 41-50, 2015. Google Scholar
  5. Peyman Afshani and Jesper Sindahl Nielsen. Data Structure Lower Bounds for Document Indexing Problems. In International Colloquium on Automata, Languages, and Programming, ICALP 2016, pages 93:1-93:15, 2016. Google Scholar
  6. Rachit Agarwal. The Space-Stretch-Time Tradeoff in Distance Oracles. In Algorithms - ESA 2014 - 22th Annual European Symposium, Wroclaw, Poland, September 8-10, 2014. Proceedings, pages 49-60, 2014. Google Scholar
  7. Rachit Agarwal and Philip Brighten Godfrey. Distance Oracles for Stretch Less Than 2. In Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013, New Orleans, Louisiana, USA, January 6-8, 2013, pages 526-538, 2013. Google Scholar
  8. Rachit Agarwal, Philip Brighten Godfrey, and Sariel Har-Peled. Approximate distance queries and compact routing in sparse graphs. In INFOCOM 2011. 30th IEEE International Conference on Computer Communications, Joint Conference of the IEEE Computer and Communications Societies, 10-15 April 2011, Shanghai, China, pages 1754-1762, 2011. Google Scholar
  9. Amihood Amir, Timothy M. Chan, Moshe Lewenstein, and Noa Lewenstein. On Hardness of Jumbled Indexing. In International Colloquium on Automata, Languages and Programming, ICALP 2014, pages 114-125, 2014. Google Scholar
  10. Amihood Amir, Tsvi Kopelowitz, Avivit Levy, Seth Pettie, Ely Porat, and B. Riva Shalom. Mind the Gap: Essentially Optimal Algorithms for Online Dictionary Matching with One Gap. In International Symposium on Algorithms and Computation, ISAAC 2016, pages 12:1-12:12, 2016. Google Scholar
  11. Gill Barequet and Sariel Har-Peled. Polygon-containment and Translational min-Hausdorff-Distance between segment Sets are 3SUM-hard. In Symposium on Discrete Algorithms, SODA 1999, pages 862-863, 1999. Google Scholar
  12. Timothy M. Chan, Stephane Durocher, Kasper Green Larsen, Jason Morrison, and Bryan T. Wilkinson. Linear-Space Data Structures for Range Mode Query in Arrays. Theory Comput. Syst., 55(4):719-741, 2014. Google Scholar
  13. Shiri Chechik. Approximate distance oracles with constant query time. In Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 654-663, 2014. Google Scholar
  14. Hagai Cohen. Fast Set Intersection and Two-Patterns Matching. Master’s thesis, Bar-Ilan University, Ramat-Gan, Israel, 2010. Google Scholar
  15. Hagai Cohen and Ely Porat. Fast set intersection and two-patterns matching. Theor. Comput. Sci., 411(40-42):3795-3800, 2010. Google Scholar
  16. Hagai Cohen and Ely Porat. On the hardness of distance oracle for sparse graph. CoRR, abs/1006.1117, 2010. URL: http://arxiv.org/abs/1006.1117.
  17. 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. Google Scholar
  18. Paul F. Dietz, Kurt Mehlhorn, Rajeev Raman, and Christian Uhrig. Lower Bounds for Set Intersection Queries. Algorithmica, 14(2):154-168, 1995. Google Scholar
  19. Anka Gajentaan and Mark H. Overmars. On a Class of O(n²) Problems in Computational Geometry. Comput. Geom., 5:165-185, 1995. Google Scholar
  20. Isaac Goldstein, Tsvi Kopelowitz, Moshe Lewenstein, and Ely Porat. How Hard is it to Find (Honest) Witnesses? In European Symposium on Algorithms, ESA 2016, pages 45:1-45:16, 2016. Google Scholar
  21. Isaac Goldstein, Tsvi Kopelowitz, Moshe Lewenstein, and Ely Porat. Conditional Lower Bounds for Space/Time Tradeoffs. In Algorithms and Data Structures Symposium, WADS 2017, pages 421-436, 2017. Google Scholar
  22. Alexander Golovnev, Siyao Guo, Thibaut Horel, Sunoo Park, and Vinod Vaikuntanathan. 3SUM with Preprocessing: Algorithms, Lower Bounds and Cryptographic Applications. arXiv, 2019. URL: http://arxiv.org/abs/1907.08355.
  23. Tsvi Kopelowitz, Seth Pettie, and Ely Porat. Dynamic Set Intersection. In Algorithms and Data Structures - 14th International Symposium, WADS 2015, Victoria, BC, Canada, August 5-7, 2015. Proceedings, pages 470-481, 2015. Google Scholar
  24. Tsvi Kopelowitz, Seth Pettie, and Ely Porat. Higher Lower Bounds from the 3SUM Conjecture. In Symposium on Discrete Algorithms, SODA 2016, pages 1272-1287, 2016. Google Scholar
  25. Tsvi Kopelowitz and Ely Porat. The Strong 3SUM-INDEXING Conjecture is False. arXiv, 2019. URL: http://arxiv.org/abs/1907.11206.
  26. Danny Krizanc, Pat Morin, and Michiel H. M. Smid. Range Mode and Range Median Queries on Lists and Trees. Nord. J. Comput., 12(1):1-17, 2005. Google Scholar
  27. Mihai Patrascu. Towards polynomial lower bounds for dynamic problems. In Symposium on Theory of Computing, STOC 2010, pages 603-610, 2010. Google Scholar
  28. Mihai Patrascu and Liam Roditty. Distance Oracles beyond the Thorup-Zwick Bound. SIAM J. Comput., 43(1):300-311, 2014. Google Scholar
  29. 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. Google Scholar
  30. Mikkel Thorup and Uri Zwick. Approximate distance oracles. J. ACM, 52(1):1-24, 2005. Google Scholar
  31. Joshua R. Wang. Space-Efficient Randomized Algorithms for K-SUM. In European Symposium on Algorithms, ESA 2014, pages 810-829, 2014. Google Scholar
  32. Virginia Vassilevska Williams. On some fine-grained questions in algorithms and complexity. In International Congress of Mathematicians, ICM 2018, 2018. Google Scholar
  33. Virginia Vassilevska Williams and Ryan Williams. Finding, Minimizing, and Counting Weighted Subgraphs. SIAM J. Comput., 42(3):831-854, 2013. Google Scholar
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