Computing Homotopic Shortest Paths Efficiently

Computing Homotopic Shortest Paths Efficiently Alon Efrat1 , Stephen G. Kobourov1, and Anna Lubiw2 1 2 Dept. of Computer Science, Univ. of Arizona, {alon, kobourov}@cs.arizona.edu School of Computer Science, Univ. of Waterloo, alubiw@uwaterloo.ca Abstract Geometric shortest paths are a major topic in computational geometry; see the survey paper by Mitchell [12]. A shortest path between two points in a simple polygon can be found in linear time using the “funnel” algorithm of Chazelle [3] and Lee and Preparata [10]. A more general problem is to find a shortest path between two points in a polygonal domain. In this case the “rubber band” solution is not unique, or, to put it another way, different paths may have different homotopy types. When the homotopy type of the solution is not specified, there are two main approaches, the visibility graph approach, and the continuous Dijkstra (or shortest path map) approach [12]. In this paper, we address the problem of finding a shortest path when the homotopy type is specified. Colloquially, we have a “sketch” of how the path should wind its way among the obstacles, and we want to pull the path tight to shorten it. Homotopic shortest paths are used in VLSI routing [4, 8, 11]. A related problem is that of drawing graphs with “fat edges”: given a planar weighted graph G, find a planar drawing such that all the edges are drawn as thickly as possible and proportional to the corresponding edge weights. Duncan et al. [5] and Efrat et al. [7] present an O(kn + n3 ) algorithm for this problem, where n is the number of edges and k is the maximum of their input and output complexities. Hershberger and Snoeyink [9] give an algorithm for the homotopic shortest path problem. Their algorithm assumes a triangulation of size n of the polygonal domain, and finds a shortest path homotopic to a given path of k edges in time linear in k plus the number of triangles (with repetition) visited by the input path. This can be nk in the worst case, and our aim is to reduce it when output size permits. Cabello et al. [2] consider the related problem of testing if two given paths are homotopically equivalent. Our original work [6] was done independently of theirs and the idea of the first step of the algorithms is the same. The current presentation of our work incorporates their more efficient implementation of this idea. We now define homotopy, and give a precise description of our problem. Let α, β : [0, 1] −→ R2 be two continuous curves parameterized by arc-length. Then α and β are homotopic with respect to a set of obstacles V ⊆ R2 if α can be continuously deformed into β while avoiding the obstacles; more formally, if there exists a continuous function h : [0, 1] × [0, 1] → R2 such that: 1. h(0, t) = α(t) and h(1, t) = β(t), for 0 ≤ t ≤ 1 π1 t1 µ1 t2 π2 t1 t3 µ3 t2 t3 t3 µ4 t4 σ1 t1 t2 σ2 t4 t4 µ2 (a) (b) (c) Fig. 1. (a) Paths π1 and π2 joining terminals t1 to t2 and t3 to t4 , respectively. (b) After vertical shortcuts, π1 consists of 3 monotone pieces: µ1 from t1 to t3 , µ2 from t3 to t4 , and µ3 from t4 to t2 ; π2 consists of one x-monotone piece, µ4 , homotopically equivalent to µ2 ; (c) Final homotopic shortest paths, σ1 and σ2 . 2. h(λ, 0) = α(0) = β(0) and h(λ, 1) = α(1) = β(1) for 0 ≤ λ ≤ 1 3. h(λ, t) ∈ / V for 0 ≤ λ ≤ 1, 0 < t < 1 Let Π = {π1 , π2 , . . . , πn } be a set of disjoint, simple polygonal paths and let the endpoints of the paths in Π define the set T of at most 2n fixed points in the plane. Note that we allow a path to degenerate to a single point. We call the fixed points of T “terminals,” and call the interior vertices of the paths “bends,” and use “points” in a more generic sense, e.g. “a point on a path.” We assume that no two terminals/bends lie on the same vertical line. Our goal is to replace each path πi ∈ Π by a shortest path σi that is homotopic to πi with respect to the set of obstacles T ; see Fig. 1. Note that σi is unique. Let Σ = {σ1 , . . . σn } be the set of resulting paths. Observe that these output paths may [self] intersect by way of segments lying on top of each other, but will be non-crossing. Let kin be the number of edges in all the paths of Π. Let kout be the number of edges in all the paths of Σ. Note that kin and kout can be arbitrarily large compared to n, and that kout ≤ nkin . The algorithm of Hershberger and Snoeyink [9] finds homotopic shortest paths in time O(nkin ). The deterministic √ algorithm presented in this paper runs in time O(kout + kin log n + n n), and the randomized algorithm in time O(kout + kin log n + n(log n)1+ε ). These are improvements except when kin is quite small compared to n. Our algorithm relies on the algorithm of Bar-Yehuda and Chazelle [1], which uses linear time polygon triangulation and ideas of linear time Jordan sorting, and finds a trapezoidization of n disjoint polygonal chains with a total of k edges in time O(k + n(log n)1+ε ). Replacing this by plane sweep makes our algorithm implementable √ and yields a running time of O(kout + (n + kin ) log(n + kin )) with an extra O(n n) factor for the deterministic version. The algorithm of Cabello et al. [2] tests whether two paths are homotopically equivalent under the pin, pushpin, or tack models in O((n + kin ) log(n + kin )) time as they opt for the practical plane sweep approach. References 1. R. Bar-Yehuda and B. Chazelle. Triangulating disjoint Jordan chains. 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