TY - GEN
T1 - On 3SUM-hard Problems in the Decision Tree Model
AU - Ezra, Esther
N1 - Publisher Copyright: © 2021, Springer Nature Switzerland AG.
PY - 2021
Y1 - 2021
N2 - 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).
AB - 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).
KW - 3SUM-hard problems
KW - Algebraic decision tree model
KW - Collinearity testing
KW - Segment concurrency
UR - http://www.scopus.com/inward/record.url?scp=85112182392&partnerID=8YFLogxK
U2 - 10.1007/978-3-030-80049-9_16
DO - 10.1007/978-3-030-80049-9_16
M3 - منشور من مؤتمر
SN - 9783030800482
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 178
EP - 188
BT - Connecting with Computability - 17th Conference on Computability in Europe, CiE 2021, Proceedings
A2 - De Mol, Liesbeth
A2 - Weiermann, Andreas
A2 - Manea, Florin
A2 - Fernández-Duque, David
PB - Springer Science and Business Media Deutschland GmbH
T2 - 17th Conference on Computability in Europe, CiE 2021
Y2 - 5 July 2021 through 9 July 2021
ER -