A nearly quadratic bound for the decision tree complexity of k-SUM

Esther Ezra; Micha Sharir

doi:10.4230/LIPIcs.SoCG.2017.41

A nearly quadratic bound for the decision tree complexity of k-SUM

Esther Ezra, Micha Sharir

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

7 Scopus citations

Abstract

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.

Original language	English
Title of host publication	33rd International Symposium on Computational Geometry, SoCG 2017
Editors	Matthew J. Katz, Boris Aronov
Publisher	Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Pages	411-4115
Number of pages	3705
ISBN (Electronic)	9783959770385
DOIs	https://doi.org/10.4230/LIPIcs.SoCG.2017.41
State	Published - 1 Jun 2017
Externally published	Yes
Event	33rd International Symposium on Computational Geometry, SoCG 2017 - Brisbane, Australia Duration: 4 Jul 2017 → 7 Jul 2017

Publication series

Name	Leibniz International Proceedings in Informatics, LIPIcs
Volume	77
ISSN (Print)	1868-8969

Conference

Conference	33rd International Symposium on Computational Geometry, SoCG 2017
Country/Territory	Australia
City	Brisbane
Period	4/07/17 → 7/07/17

Bibliographical note

Publisher Copyright:
© Esther Ezra and Micha Sharir.

Keywords

Hyperplane arrangements
K-SUM and k-LDT
Linear decision tree
Point-location
Vertical decompositions
ϵ-cuttings

Access to Document

10.4230/LIPIcs.SoCG.2017.41

Cite this

Ezra, E., & Sharir, M. (2017). A nearly quadratic bound for the decision tree complexity of k-SUM. In M. J. Katz, & B. Aronov (Eds.), 33rd International Symposium on Computational Geometry, SoCG 2017 (pp. 411-4115). (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 77). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.SoCG.2017.41

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title = "A nearly quadratic bound for the decision tree complexity of k-SUM",

abstract = "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.",

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Ezra, E & Sharir, M 2017, A nearly quadratic bound for the decision tree complexity of k-SUM. in MJ Katz & B Aronov (eds), 33rd International Symposium on Computational Geometry, SoCG 2017. Leibniz International Proceedings in Informatics, LIPIcs, vol. 77, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, pp. 411-4115, 33rd International Symposium on Computational Geometry, SoCG 2017, Brisbane, Australia, 4/07/17. https://doi.org/10.4230/LIPIcs.SoCG.2017.41

A nearly quadratic bound for the decision tree complexity of k-SUM. / Ezra, Esther; Sharir, Micha.
33rd International Symposium on Computational Geometry, SoCG 2017. ed. / Matthew J. Katz; Boris Aronov. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2017. p. 411-4115 (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 77).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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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.

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Ezra E, Sharir M. A nearly quadratic bound for the decision tree complexity of k-SUM. In Katz MJ, Aronov B, editors, 33rd International Symposium on Computational Geometry, SoCG 2017. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. 2017. p. 411-4115. (Leibniz International Proceedings in Informatics, LIPIcs). doi: 10.4230/LIPIcs.SoCG.2017.41

A nearly quadratic bound for the decision tree complexity of k-SUM

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Bibliographical note

Keywords

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