Growing fat graphs

Efrat, Alon; Kobourov, Stephen G.; Stepp, Michael; Wenk, Carola

doi:10.1145/513400.513434

Cited by 7 publications

(8 citation statements)

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“…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.…”

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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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confidence: 99%

Computing Homotopic Shortest Paths Efficiently

Efrat

Kobourov

Lubiw

2002

Algorithms — ESA 2002

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“…Maximizing the distance between wires is equivalent to finding the drawing in which the edges are drawn as thick as possible, i.e., allowing the graph to grow as fat as possible. The two types of stopping conditions for fat graphs are collision of two vertices and collision of two elbows [20].…”

Section: Related Workmentioning

confidence: 99%

Visualization of clustered directed acyclic graphs with node interleaving

Kumar

Zhang

2009

Proceedings of the 2009 ACM Symposium on Applied Computing

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Graph drawing and visualization represent structural information as diagrams of abstract graphs and networks. An important subset of graphs is directed acyclic graphs (DAGs). E-Spring algorithm, extended from the popular spring embedder model, eliminates node overlaps in clustered DAGs by modeling nodes as charged particles whose repulsion is controlled by edges modeled as springs. The drawing process needs to reach a stable state when the average distances of separation between nodes are near optimal. This paper presents an enhancement to E-Spring to introduce a stopping condition, which reduces equilibrium distances between nodes and therefore results in a significantly reduced area for DAG visualization. It imposes an upper bound on the repulsive forces between nodes based on graph geometry. The algorithm employs node interleaving to eliminate any residual node overlaps. These new techniques have been validated by visualizing eBay buyer-seller relationships and resulted in overall area reductions in the range of 45% to 79%.

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“…Another example arises in VLSI routing [46,86,129], map simplification [53,54,81], and graph drawing [70,75]. Given a rough sketch of one or more paths in a planar environment with fixed obstacles-possibly representing roads or rivers near cities or other geographic features, or wires between components on a chipwe want to produce a topologically equivalent set of paths that are as short or as simple as possible, perhaps subject to some tolerance constraints.…”

Section: Introductionmentioning

confidence: 99%