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Given a set of pairwise disjoint polygonal obstacles in the plane, finding an obstacle-avoiding Euclidean shortest path between two points is a classical problem in computational geometry and has been studied extensively. Previously, Hershberger and Suri (in SIAM Journal on Computing , 1999) gave an algorithm of O(n log n ) time and O(n log n ) space, where n is the total number of vertices of all obstacles. Recently, by modifying Hershberger and Suri’s algorithm, Wang (in SODA’21) reduced the space to O(n) while the runtime of the algorithm is still O(n log n ). In this article, we present a new algorithm of O(n+h log h ) time and O(n) space, provided that a triangulation of the free space is given, where h is the number of obstacles. The algorithm is better than the previous work when h is relatively small. Our algorithm builds a shortest path map for a source point s so that given any query point t , the shortest path length from s to t can be computed in O (log n ) time and a shortest s - t path can be produced in additional time linear in the number of edges of the path.
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Minimum Plane Bichromatic Spanning Trees
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.
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- PAR ID:
- 10563319
- Editor(s):
- Mestre, Julián; Wirth, Anthony
- Publisher / Repository:
- Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- Date Published:
- Volume:
- 322
- ISSN:
- 1868-8969
- ISBN:
- 978-3-95977-354-6
- Page Range / eLocation ID:
- 322-322
- Subject(s) / Keyword(s):
- Bichromatic Spanning Tree Minimum Spanning Tree Plane Tree Theory of computation → Computational geometry
- Format(s):
- Medium: X Size: 14 pages; 896649 bytes Other: application/pdf
- Size(s):
- 14 pages 896649 bytes
- Right(s):
- Creative Commons Attribution 4.0 International license; info:eu-repo/semantics/openAccess
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.4230/LIPIcs.ISAAC.2024.4
