2026-07-25T03:07:47Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:25570 2026-03-19T13:35:10Z ddc:004 open_access

Counting Unit Circular Arc Intersections Wang, Haitao circular arc intersections unit circles arrangements cuttings segment intersections Given a set of n circular arcs of the same radius in the plane, we consider the problem of computing the number of intersections among the arcs. The problem was studied before and the previously best algorithm solves the problem in O(n^{4/3+ε}) time [Agarwal, Pellegrini, and Sharir, SIAM J. Comput., 1993], for any constant ε > 0. No progress has been made on the problem for more than 30 years. We present a new algorithm of O(n^{4/3}log^{16/3} n) time and improve it to O(n^{1+ε}+K^{1/3}n^{2/3}((n²)/(n+K))^{ε}log^{16/3}n) time for small K, where K is the number of intersections of all arcs. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Haitao Wang 2026 Is Part Of LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.STACS.2026.81 urn:nbn:de:0030-drops-255707 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.81 eng https://creativecommons.org/licenses/by/4.0/legalcode