TY - GEN
T1 - Near-Optimal linear decision trees for k-SUM and related problems
AU - Kane, Daniel M.
AU - Lovett, Shachar
AU - Moran, Shay
N1 - Publisher Copyright: © 2018 Copyright held by the owner/author(s).
PY - 2018/6/20
Y1 - 2018/6/20
N2 - 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.
AB - 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.
KW - Hyperplane arrangement
KW - Inference dimension
KW - Linear decision tree
UR - http://www.scopus.com/inward/record.url?scp=85049906431&partnerID=8YFLogxK
U2 - 10.1145/3188745.3188770
DO - 10.1145/3188745.3188770
M3 - منشور من مؤتمر
T3 - Proceedings of the Annual ACM Symposium on Theory of Computing
SP - 1249
EP - 1259
BT - STOC 2018 - Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing
A2 - Henzinger, Monika
A2 - Kempe, David
A2 - Diakonikolas, Ilias
T2 - 50th Annual ACM Symposium on Theory of Computing, STOC 2018
Y2 - 25 June 2018 through 29 June 2018
ER -