clms-thpp-04 - Discrete Comput Geom 31:6181(2004 DOI...

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DOI: 10.1007 / s00454-003-2949-y Discrete Comput Geom 31:61–81 (2004) Discrete & Computational Geometry © 2003 Springer-Verlag New York Inc. Testing Homotopy for Paths in the Plane Sergio Cabello, 1 Yuanxin Liu, 2 Andrea Mantler, 2 and Jack Snoeyink 2 1 Institute of Information and Computer Science, Utrecht University, 3508 TB Utrecht, The Netherlands [email protected] 2 Department of Computer Science, University of North Carolina, Chapel Hill, NC 27599-3175, USA { liuy,mantler,snoeyink } Abstract. In this paper we present an efFcient algorithm to test if two given paths are homotopic; that is, whether they wind around obstacles in the plane in the same way. ±or paths speciFed by n line segments with obstacles described by n points, several standard ways achieve quadratic running time. ±or simple paths, our algorithm runs in O ( n log n ) time, which we show is tight. ±or self-intersecting paths the problem is related to Hopcroft’s problem; our algorithm runs in O ( n 3 / 2 log n ) time. 1. Introduction A basic topological question is determining if two paths are homotopic, so that one can be deformed into another without leaving the containing space. SpeciFcally, suppose that the input consists of a set P of up to n points in the plane, and two paths, α and β , that start and end at the same points and are represented as polygonal lines of at most n segments each. The goal is to determine whether α is deformable to β without passing over any points of P (±ig. 1). Equivalently, we determine whether the closed loop αβ R , which concatenates α with the reverse of β , is contractible in the plane minus P .We assume (or simulate) general position, so that no three points are collinear and no two points are on the same vertical line. We are primarily concerned with paths that have no self-intersections. The path homotopy question arises in several application areas: In circuit board design, river routing , where the homotopy class of each wire is speciFed, is one of the few The Frst author was partially supported by the Cornenlis Lely Stichting, and the rest by NS± Grants 9988742 and 0076984.
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62 S. Cabello, Y. Liu, A. Mantler, and J. Snoeyink α β Fig. 1. Are α and β homotopic? polynomial-time solvable variations of the wire routing problem [21], [17]. In motion path planning, one might check to see that two ways of getting from point A to point B are equivalent [19]. In geographic information systems (GIS), one may wish to simplify a linear feature (road or river) while respecting the way in which the feature winds around points [5], [8]. Michael Goodchild, in an invited lecture at the 11th ACM Symposium on Computational Geometry, pointed out that even on a road map that has features with 60 m accuracy, you will still Fnd all the houses on the proper side of the road. In such a case the operators entering data have used topological constraints to make sure that the road winds properly when creating the road layer or building layer. Efrat et al. [15] recently looked at computing shortest paths among obstacles by identifying and bundling homotopic fragments. They independently developed a sweep algorithm that uses similar
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clms-thpp-04 - Discrete Comput Geom 31:6181(2004 DOI...

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