2026-07-26T04:14:46Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:22131 2024-12-04T06:53:43Z ddc:004 open_access

Minimum Plane Bichromatic Spanning Trees A. Akitaya, Hugo Biniaz, Ahmad Demaine, Erik D. Kleist, Linda Stock, Frederick Tóth, Csaba D. Bichromatic Spanning Tree Minimum Spanning Tree Plane Tree For a set of red and blue points in the plane, a minimum bichromatic spanning tree (MinBST) is a shortest spanning tree of the points such that every edge has a red and a blue endpoint. A MinBST can be computed in O(n log n) time where n is the number of points. In contrast to the standard Euclidean MST, which is always plane (noncrossing), a MinBST may have edges that cross each other. However, we prove that a MinBST is quasi-plane, that is, it does not contain three pairwise crossing edges, and we determine the maximum number of crossings. Moreover, we study the problem of finding a minimum plane bichromatic spanning tree (MinPBST) which is a shortest bichromatic spanning tree with pairwise noncrossing edges. This problem is known to be NP-hard. The previous best approximation algorithm, due to Borgelt et al. (2009), has a ratio of O(√n). It is also known that the optimum solution can be computed in polynomial time in some special cases, for instance, when the points are in convex position, collinear, semi-collinear, or when one color class has constant size. We present an O(log n)-factor approximation algorithm for the general case. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Hugo A. Akitaya and Ahmad Biniaz and Erik D. Demaine and Linda Kleist and Frederick Stock and Csaba D. Tóth 2024 Is Part Of LIPIcs, Volume 322, 35th International Symposium on Algorithms and Computation (ISAAC 2024) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ISAAC.2024.4 urn:nbn:de:0030-drops-221319 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2024.4 eng https://creativecommons.org/licenses/by/4.0/legalcode