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Lower bounds for linear degeneracy testing

Authors:

Bernard ChazelleAuthors Info & Claims

Journal of the ACM (JACM), Volume 52, Issue 2

Pages 157 - 171

https://doi.org/10.1145/1059513.1059515

Published: 01 March 2005 Publication History

Abstract

In the late nineties, Erickson proved a remarkable lower bound on the decision tree complexity of one of the central problems of computational geometry: given n numbers, do any r of them add up to 0? His lower bound of Ω(n^⌈r/2⌉), for any fixed r, is optimal if the polynomials at the nodes are linear and at most r-variate. We generalize his bound to s-variate polynomials for s > r. Erickson's bound decays quickly as r grows and never reaches above pseudo-polynomial: we provide an exponential improvement. Our arguments are based on three ideas: (i) a geometrization of Erickson's proof technique; (ii) the use of error-correcting codes; and (iii) a tensor product construction for permutation matrices.

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Cited By

Cardinal JSharir M(2024)Improved Algebraic Degeneracy TestingDiscrete & Computational Geometry10.1007/s00454-024-00673-7Online publication date: 25-Jun-2024
https://doi.org/10.1007/s00454-024-00673-7
Carmeli NTziavelis NGatterbauer WKimelfeld BRiedewald M(2023)Tractable Orders for Direct Access to Ranked Answers of Conjunctive QueriesACM Transactions on Database Systems10.1145/357851748:1(1-45)Online publication date: 13-Mar-2023
https://dl.acm.org/doi/10.1145/3578517
Aronov BCardinal J(2022)Geometric Pattern Matching Reduces to k-SUMDiscrete & Computational Geometry10.1007/s00454-021-00324-168:3(850-859)Online publication date: 1-Oct-2022
https://dl.acm.org/doi/10.1007/s00454-021-00324-1
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Index Terms

Lower bounds for linear degeneracy testing
1. Theory of computation
  1. Design and analysis of algorithms

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cover image Journal of the ACM

Journal of the ACM Volume 52, Issue 2

March 2005

189 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/1059513

Issue’s Table of Contents

Copyright © 2005 ACM.

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Publication History

Published: 01 March 2005

Published in JACM Volume 52, Issue 2

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Cited By

Cardinal JSharir M(2024)Improved Algebraic Degeneracy TestingDiscrete & Computational Geometry10.1007/s00454-024-00673-7Online publication date: 25-Jun-2024
https://doi.org/10.1007/s00454-024-00673-7
Carmeli NTziavelis NGatterbauer WKimelfeld BRiedewald M(2023)Tractable Orders for Direct Access to Ranked Answers of Conjunctive QueriesACM Transactions on Database Systems10.1145/357851748:1(1-45)Online publication date: 13-Mar-2023
https://dl.acm.org/doi/10.1145/3578517
Aronov BCardinal J(2022)Geometric Pattern Matching Reduces to k-SUMDiscrete & Computational Geometry10.1007/s00454-021-00324-168:3(850-859)Online publication date: 1-Oct-2022
https://dl.acm.org/doi/10.1007/s00454-021-00324-1
Carmeli NTziavelis NGatterbauer WKimelfeld BRiedewald MLibkin LPichler RGuagliardo P(2021)Tractable Orders for Direct Access to Ranked Answers of Conjunctive QueriesProceedings of the 40th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems10.1145/3452021.3458331(325-341)Online publication date: 20-Jun-2021
https://dl.acm.org/doi/10.1145/3452021.3458331
Ezra E(2021)On 3SUM-hard Problems in the Decision Tree ModelConnecting with Computability10.1007/978-3-030-80049-9_16(178-188)Online publication date: 5-Jul-2021
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Dudek BGawrychowski PStarikovskaya TMakarychev KMakarychev YTulsiani MKamath GChuzhoy J(2020)All non-trivial variants of 3-LDT are equivalentProceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing10.1145/3357713.3384275(974-981)Online publication date: 22-Jun-2020
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Kane DLovett SMoran S(2019)Near-optimal Linear Decision Trees for k-SUM and Related ProblemsJournal of the ACM10.1145/328595366:3(1-18)Online publication date: 12-Apr-2019
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Chen T(2019)Unmixing the Mixed Volume ComputationDiscrete & Computational Geometry10.1007/s00454-019-00078-x62:1(55-86)Online publication date: 1-Jul-2019
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