TY - GEN
T1 - A nearly quadratic bound for the decision tree complexity of k-SUM
AU - Ezra, Esther
AU - Sharir, Micha
N1 - Publisher Copyright: © Esther Ezra and Micha Sharir.
PY - 2017/6/1
Y1 - 2017/6/1
N2 - We show that the k-SUM problem can be solved by a linear decision tree of depth O(n2 log2 n), improving the recent bound O(n3 log3 n) of Cardinal et al. [7]. Our bound depends linearly on k, and allows us to conclude that the number of linear queries required to decide the n-dimensional KNAPSACK or SUBSETSUM problems is only O(n3 log n), improving the currently best known bounds by a factor of n [28, 29]. Our algorithm extends to the RAM model, showing that the k-SUM problem can be solved in expected polynomial time, for any fixed k, with the above bound on the number of linear queries. Our approach relies on a new point-location mechanism, exploiting "ϵ-cuttings" that are based on vertical decompositions in hyperplane arrangements in high dimensions. A major side result of the analysis in this paper is a sharper bound on the complexity of the vertical decomposition of such an arrangement (in terms of its dependence on the dimension). We hope that this study will reveal further structural properties of vertical decompositions in hyperplane arrangements.
AB - We show that the k-SUM problem can be solved by a linear decision tree of depth O(n2 log2 n), improving the recent bound O(n3 log3 n) of Cardinal et al. [7]. Our bound depends linearly on k, and allows us to conclude that the number of linear queries required to decide the n-dimensional KNAPSACK or SUBSETSUM problems is only O(n3 log n), improving the currently best known bounds by a factor of n [28, 29]. Our algorithm extends to the RAM model, showing that the k-SUM problem can be solved in expected polynomial time, for any fixed k, with the above bound on the number of linear queries. Our approach relies on a new point-location mechanism, exploiting "ϵ-cuttings" that are based on vertical decompositions in hyperplane arrangements in high dimensions. A major side result of the analysis in this paper is a sharper bound on the complexity of the vertical decomposition of such an arrangement (in terms of its dependence on the dimension). We hope that this study will reveal further structural properties of vertical decompositions in hyperplane arrangements.
KW - Hyperplane arrangements
KW - K-SUM and k-LDT
KW - Linear decision tree
KW - Point-location
KW - Vertical decompositions
KW - ϵ-cuttings
UR - https://www.scopus.com/pages/publications/85029935140
U2 - 10.4230/LIPIcs.SoCG.2017.41
DO - 10.4230/LIPIcs.SoCG.2017.41
M3 - Conference contribution
T3 - Leibniz International Proceedings in Informatics, LIPIcs
SP - 411
EP - 4115
BT - 33rd International Symposium on Computational Geometry, SoCG 2017
A2 - Katz, Matthew J.
A2 - Aronov, Boris
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 33rd International Symposium on Computational Geometry, SoCG 2017
Y2 - 4 July 2017 through 7 July 2017
ER -