Daniel M. Kane
ABSTRACT
We construct near optimal linear decision trees for a variety of decision problems in combinatorics and discrete geometry. For example, for any constant k, we construct linear decision trees that solve the k-SUM problem on n elements using O(n log2 n) linear queries. Moreover, the queries we use are comparison queries, which compare the sums of two k-subsets; when viewed as linear queries, comparison queries are 2k-sparse and have only {−1,0,1} coefficients. We give similar constructions for sorting sumsets A+B and for solving the SUBSET-SUM problem, both with optimal number of queries, up to poly-logarithmic terms.
Our constructions are based on the notion of “inference dimension”, recently introduced by the authors in the context of active classification with comparison queries. This can be viewed as another contribution to the fruitful link between machine learning and discrete geometry, which goes back to the discovery of the VC dimension.
Supplemental Material
- {AC05} Nir Ailon and Bernard Chazelle. Lower bounds for linear degeneracy testing. Journal of the ACM (JACM), 52(2):157–171, 2005. Google Scholar
Digital Library
- {CGI + 16} Marco L Carmosino, Jiawei Gao, Russell Impagliazzo, Ivan Mihajlin, Ramamohan Paturi, and Stefan Schneider. Nondeterministic extensions of the strong exponential time hypothesis and consequences for non-reducibility. In Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science, pages 261–270. ACM, 2016. Google ScholarDigital Library
- {CIO16} Jean Cardinal, John Iacono, and Aurélien Ooms. Solving k-SUM using few linear queries. In 24th Annual European Symposium on Algorithms, ESA 2016, pages 25:1–25:17, 2016.Google Scholar
- {CL15} Timothy M Chan and Moshe Lewenstein. Clustered integer 3SUM via additive combinatorics. In Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, pages 31–40. ACM, 2015. Google ScholarDigital Library
- {DL74} David Dobkin and Richard J Lipton. On some generalizations of binary search. In Proceedings of the sixth annual ACM symposium on Theory of computing, pages 310–316. ACM, 1974. Google ScholarDigital Library
- {Eri99} Jeff Erickson. Bounds for linear satisfiability problems. Chicago Journal of Theoretical Computer Science, page 8, 1999.Google Scholar
- {ES17} Esther Ezra and Micha Sharir. A nearly quadratic bound for the decision tree complexity of k-SUM. In 33rd International Symposium on Computational Geometry, SoCG 2017, pages 41:1–41:15, 2017.Google Scholar
- {Fre76a} Michael L Fredman. How good is the information theory bound in sorting? Theoretical Computer Science, 1(4):355–361, 1976.Google ScholarCross Ref
- {Fre76b} Michael L Fredman. New bounds on the complexity of the shortest path problem. SIAM Journal on Computing, 5(1):83–89, 1976.Google ScholarDigital Library
- {GO95} Anka Gajentaan and Mark H Overmars. On a class of o(n 2 ) problems in computational geometry. Computational geometry, 5(3):165–185, 1995. Google ScholarDigital Library
- {GP14} Allan Grønlund and Seth Pettie. Threesomes, degenerates, and love triangles. In Foundations of Computer Science (FOCS), 2014 IEEE 55th Annual Symposium on, pages 621–630. IEEE, 2014. Google ScholarDigital Library
- {GS17} Omer Gold and Micha Sharir. Improved bounds for 3sum, k-sum, and linear degeneracy. In 25th Annual European Symposium on Algorithms, ESA 2017, pages 42:1–42:13, 2017.Google Scholar
- {GT62} Eiichi Goto and Hidetosi Takahasi. Some theorems useful in threshold logic for enumerating boolean functions. In IFIP Congress, pages 747–752, 1962.Google Scholar
- {KLMZ17} Daniel M Kane, Shachar Lovett, Shay Moran, and Jiapeng Zhang. Active classification with comparison queries. arXiv preprint arXiv:1704.03564, 2017.Google Scholar
- {MadH84} Friedhelm Meyer auf der Heide. A polynomial linear search algorithm for the n-dimensional knapsack problem. Journal of the ACM (JACM), 31(3):668–676, 1984. Google ScholarDigital Library
- STOC’18, June 25–29, 2018, Los Angeles, CA, USA Daniel M. Kane, Shachar Lovett, and Shay Moran {Mei93} Stefan Meiser. Point location in arrangements of hyperplanes. Information and Computation, 106(2):286–303, 1993. Google ScholarDigital Library
- {Pet02} Seth Pettie. On the comparison-addition complexity of all-pairs shortest paths. In Algorithms and Computation, 13th International Symposium, ISAAC 2002 Vancouver, BC, Canada, November 21-23, 2002, Proceedings, pages 32–43, 2002. Google ScholarDigital Library
- {VC71} VN Vapnik and A Ya Chervonenkis. On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probability & Its Applications, 16(2):264–280, 1971.Google Scholar
- {VC15} Vladimir N Vapnik and A Ya Chervonenkis. On the uniform convergence of relative frequencies of events to their probabilities. In Measures of Complexity, pages 11–30. Springer, 2015.Google Scholar
- {VW15} Virginia Vassilevska Williams. Hardness of easy problems: Basing hardness on popular conjectures such as the strong exponential time hypothesis (invited talk). In LIPIcs-Leibniz International Proceedings in Informatics, volume 43. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2015.Google Scholar
- {Wil14} Ryan Williams. Faster all-pairs shortest paths via circuit complexity. In Proceedings of the forty-sixth annual ACM symposium on Theory of computing, pages 664–673. ACM, 2014. Google ScholarDigital Library
- {Yao81} Andrew Chi-Chih Yao. On the parallel computation for the knapsack problem. In Proceedings of the thirteenth annual ACM symposium on Theory of computing, pages 123–127. ACM, 1981. Google ScholarDigital Library
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- Near-optimal linear decision trees for k-SUM and related problems
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