A Nearly Quadratic Bound for the Decision Tree Complexity of k-SUM

Authors Esther Ezra, Micha Sharir

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Esther Ezra
Micha Sharir

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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). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 41:1-41:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


We show that the k-SUM problem can be solved by a linear decision tree of depth O(n^2 log^2 n),improving the recent bound O(n^3 log^3 n) of Cardinal et al. 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(n^3 log n), improving the currently best known bounds by a factor of n. 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 "Epsilon-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.
  • k-SUM and k-LDT
  • linear decision tree
  • hyperplane arrangements
  • point-location
  • vertical decompositions
  • Epsilon-cuttings


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