Subquadratic algorithms for some 3SUM-hard geometric problems in the algebraic decision-tree model

Boris Aronov, Mark de Berg*, Jean Cardinal, Esther Ezra, John Iacono, Micha Sharir

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We present subquadratic algorithms in the algebraic decision-tree model for several 3SUM-hard geometric problems, all of which can be reduced to the following question: Given two sets A, B, each consisting of n pairwise disjoint segments in the plane, and a set C of n triangles in the plane, we want to count, for each triangle Δ∈C, the number of intersection points between the segments of A and those of B that lie in Δ. We present solutions in the algebraic decision-tree model whose cost is O(n60/31+ε), for any ε>0. Our approach is based on a primal-dual range searching mechanism, which exploits the multi-level polynomial partitioning machinery recently developed by Agarwal et al. (2021) [3]. A key step in the procedure is a variant of point location in arrangements, say of lines in the plane, which is based solely on the order type of the lines, a “handicap” that turns out to be beneficial for speeding up our algorithm.

Original languageEnglish
Article number101945
JournalComputational Geometry: Theory and Applications
StatePublished - Feb 2023


FundersFunder number
Blavatnik Research Fund in Computer Science
German-Israeli Science Foundation
National Science FoundationCCF-20-08551, 2014/170, CCF-15-40656
German-Israeli Foundation for Scientific Research and Development
United States-Israel Binational Science Foundation
Fonds De La Recherche Scientifique - FNRSJ.0146.18, 824/17, AF-1553354
Nederlandse Organisatie voor Wetenschappelijk Onderzoek024.002.003
Israel Science Foundation1367/2016, 260/18
Tel Aviv University


    • 3SUM-hard problems
    • Algebraic decision-tree model
    • Order type
    • Point location
    • Polynomial partitions


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