Computing Homotopic Shortest Paths Efficiently

Abstract

This paper addresses the problem of finding shortest paths homotopic to a given disjoint set of paths that wind amongst point obstacles in the plane. We present a faster algorithm than previously known. IntroductionFinding Euclidean shortest paths in simple polygons is a well-studied problem. The funnel algorithm of Chazelle [3] and Lee and Preparata [10] finds the shortest path between two points in a simple polygon. Hershberger and Snoeyink [9] unify earlier results for computing shortest paths in polygons. They optimize a given path among obstacles in the plane under the Euclidean and link metrics and under polygonal convex distance functions. Related work has been done in addressing a classic VLSI problem, the continuous homotopic routing problem [4,11]. For this problem it is required to route wires with fixed terminals among fixed obstacles when a sketch of the wires is given, i.e., each wire is given a specified homotopy class. If the wiring sketch is not given or the terminals are not fixed, the problem is NP-hard [12,15,16].Some of the early work on continuous homotopic routing was done by Cole and Siegel [4] and Leiserson and Maley [11]. They show that in the L ∞ norm a solution can be found in O(k 3 log n) time and O(k 3 ) space, where n is the number of wires and k is the maximum of the input and output complexities of the wiring. Maley [13] shows how to extend the distance metric to arbitrary polygonal distance functions (including Euclidean distance) and presents a O(k 4 log n) time and O(k 4 ) space algorithm. The best result so far is due to Gao et al. [8] who present a O(kn 2 log(kn)) time and O(kn 2 ) space algorithm. Duncan et al. [6] and Efrat et al. [7] present an O(kn + n 3 ) algorithm for the related fat edge drawing problem: given a planar weighted graph G with maximum degree 1 and an embedding for G, find a planar drawing such that all the edges are drawn as thick as possible and proportional to the corresponding edge weights.The topological notion of homotopy formally captures the notion of deforming paths. Let α, β : [0, 1] −→ R 2 be two continuous curves parameterized by arc-length. Then α and β are homotopic with respect to a set of obstacles V ⊆ R 2 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] → R 2 with the following three properties: 1

Homotopic Rectilinear Routing with Few Links and Thick Edges

“…To compute them efficiently, we first compute rectilowest and rectihighest paths. For that we modify the algorithm by Efrat et al [6], which also bundles homotopic y-monotone chains. The rectilowest and rectihighest paths are represented by O(n) bundles with at most O(n) links each.…”

“…We consider general input paths, albeit with fixed homotopy classes, and study minimum-link rectilinear instead of shortest paths. There are several papers [2,6] that find shortest paths homotopic to a given collection of input paths. However, while a set of shortest paths homotopic to a set of non-crossing input paths is necessarily non-crossing, the same does not hold for minimumlink rectilinear paths.…”

“…Intuitively, any incremental approach to this problem should be doomed due to a cascading number of links, but we show how to move already inserted paths without increasing the number of links or creating crossings to make room for each new path. To efficiently move homotopic (sub-)paths we use a bundling technique as described in [6]. Furthermore, we "grow" the thick paths by extending the results of [5] to rectilinear paths and rectangular obstacles.…”

Testing Homotopy for Paths in the Plane

“…This does cause extra complications in handling self-intersections, especially in Section 3.6. (The work of Efrat et al [14] uses the pushpin definition. )…”

“…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 [14] recently looked at computing shortest paths among obstacles by identifying and bundling homotopic fragments. They independently developed a sweep algorithm that uses has similar ideas to ours.…”