On continuous Homotopic one layer routing

Gao, Shaodi; Jerrum, Mark; Kaufman, Marina; Mehlhorn, Kurt; Rülling, Wolfgang

doi:10.1145/73393.73433

Cited by 29 publications

(4 citation statements)

References 3 publications

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“…A similar observation can be made regarding Theorem 1 of Angelier and Pocchiola [3]. While the use of universal coverings, or portions of universal coverings, in the design of geometric algorithms had already appeared in the early days of the computational geometry literature, e.g., [15,22], it seems to be the first time that branched coverings are used in the design of a geometric algorithm. (Branched coverings are used in [43] to define the dual Voronoi diagram of a constrained Delaunay triangulation in the plane, but apparently without algorithmic consequences-see also the discussion in [12, page 30].)…”

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confidence: 55%

Computing Pseudotriangulations via Branched Coverings

Habert

Pocchiola

2012

Discrete Comput Geom

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We describe an efficient algorithm to compute a pseudotriangulation of a finite planar family of pairwise disjoint convex bodies presented by its chirotope. The design of the algorithm relies on a deepening of the theory of visibility complexes and on the extension of that theory to the setting of branched coverings. The problem of computing a pseudotriangulation that contains a given set of bitangent line segments is also examined.

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supporting

confidence: 55%

Computing Pseudotriangulations via Branched Coverings

Habert

Pocchiola

2012

Discrete Comput Geom

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“…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.…”

Section: Introductionmentioning

confidence: 99%

Computing Homotopic Shortest Paths Efficiently

Efrat

Kobourov

Lubiw

2002

Algorithms — ESA 2002

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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

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“…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 minimum-link paths. Our problem is also related to wire routing in VLSI design [4,7,9,10]. None of these papers strives to minimize the number of links.…”

Section: Fig 1 Input and Outputmentioning

confidence: 99%

Homotopic $\mathcal{C}$ -Oriented Routing

Verbeek

2013

Graph Drawing

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We study the problem of finding non-crossing minimum-link C-oriented paths that are homotopic to a set of input paths in an environment with C-oriented obstacles. We introduce a special type of C-oriented paths-smooth paths-and present a 2-approximation algorithm that runs in O(n 2 (n + log κ) + kin log n) time, where n is the total number of paths and obstacle vertices, kin is the total number of links in the input, and κ = |C|. The algorithm also computes an O(κ)-approximation for general C-oriented paths. As a related result we show that, given a set of C-oriented paths with L links in total, non-crossing C-oriented paths homotopic to the input paths can require a total of Ω(L log κ) links. K. Verbeek is supported by the Netherlands Organisation for Scientific Research (NWO) under project no. 639.022.707.

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On continuous Homotopic one layer routing

Abstract: We give an O(n s-log n) time and O(n s) space algorithm for the continuous homotopic one layer routing problem. The main contribution is an extension of the sweep paradigm to a universal cover space of the plane.

Cited by 29 publications

References 3 publications

Computing Pseudotriangulations via Branched Coverings

Computing Pseudotriangulations via Branched Coverings

Computing Homotopic Shortest Paths Efficiently

Homotopic $\mathcal{C}$ -Oriented Routing

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