On 3SUM-hard Problems in the Decision Tree Model

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


We describe subquadratic algorithms, in the algebraic decision-tree model of computation, for detecting whether there exists a triple of points, belonging to three respective sets A, B, and C of points in the plane, that satisfy a pair of polynomial equations. In particular, this has an application to detect collinearity among three sets A, B, C of n points each, in the complex plane, when each of the sets A, B, C lies on some constant-degree algebraic curve. In another development, we present a subquadratic algorithm, in the algebraic decision-tree model, for the following problem: Given a pair of sets A, B each consisting of n pairwise disjoint line segments in the plane, and a third set C of arbitrary line segments in the plane, determine whether A× B× C contains a triple of concurrent segments. This is one of four 3sum-hard geometric problems recently studied by Chan (2020). The results reported in this extended abstract are based on the recent studies of the author with Aronov and Sharir (2020, 2021).

Original languageEnglish
Title of host publicationConnecting with Computability - 17th Conference on Computability in Europe, CiE 2021, Proceedings
EditorsLiesbeth De Mol, Andreas Weiermann, Florin Manea, David Fernández-Duque
PublisherSpringer Science and Business Media Deutschland GmbH
Number of pages11
ISBN (Print)9783030800482
StatePublished - 2021
Event17th Conference on Computability in Europe, CiE 2021 - Virtual, Online
Duration: 5 Jul 20219 Jul 2021

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume12813 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference17th Conference on Computability in Europe, CiE 2021
CityVirtual, Online

Bibliographical note

Publisher Copyright:
© 2021, Springer Nature Switzerland AG.


  • 3SUM-hard problems
  • Algebraic decision tree model
  • Collinearity testing
  • Segment concurrency


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