Minimum Length Embedding of Planar Graphs at Fixed Vertex Locations

Chan, Timothy M.; Hoffmann, Hella-Franziska; Kiazyk, Stephen; Lubiw, Anna

doi:10.1007/978-3-319-03841-4_33

Cited by 14 publications

(19 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…They also bounded the number of crossings per edge pair; to 16 and 4, respectively. In that respect, Frati et al also improve upon a result of Chan et al [12] who needed 24 crossings per edge pair. While Chan et al used up to 72n bends per exclusive edge, they managed to keep their drawing on a grid of polynomial size (O(n 2 ) × O(n 2 )), which is neither the case for Frati et al [19] nor for Grilli et al [20].…”

Section: Introductionmentioning

confidence: 88%

Simultaneous Drawing of Planar Graphs with Right-Angle Crossings and Few Bends

Bekos

Dijk

Kindermann

et al. 2015

WALCOM: Algorithms and Computation

View full text Add to dashboard Cite

Abstract. Given two planar graphs that are defined on the same set of vertices, a RAC simultaneous drawing is a drawing of the two graphs where each graph is drawn planar, no two edges overlap, and edges of one graph can cross edges of the other graph only at right angles. In the geometric version of the problem, vertices are drawn as points and edges as straight-line segments. It is known, however, that even pairs of very simple classes of planar graphs (such as wheels and matchings) do not always admit a geometric RAC simultaneous drawing. In order to enlarge the class of graphs that admit RAC simultaneous drawings, we allow edges to have bends. We prove that any pair of planar graphs admits a RAC simultaneous drawing with at most six bends per edge. For more restricted classes of planar graphs (e.g., matchings, paths, cycles, outerplanar graphs, and subhamiltonian graphs), we significantly reduce the required number of bends per edge. All our drawings use quadratic area.

show abstract

Section: Introductionmentioning

confidence: 88%

Simultaneous Drawing of Planar Graphs with Right-Angle Crossings and Few Bends

Bekos

Dijk

Kindermann

et al. 2015

WALCOM: Algorithms and Computation

View full text Add to dashboard Cite

show abstract

“…For polygonal edges, Bastert and Fekete [2] proved that PSE with minimum number of bends or minimum total edge length is NP-hard, even when the graph is a matching. For minimizing the total edge length and the same graph class, Liebling et al [16] introduced heuristics and Chan et al [8] presented approximation algorithms. Chan et al also considered paths and general planar graphs.…”

Section: Introductionmentioning

confidence: 99%

Multi-sided Boundary Labeling

et al. 2015

View full text Add to dashboard Cite

In the Boundary Labeling problem, we are given a set of n points, referred to as sites, inside an axis-parallel rectangle R, and a set of n pairwise disjoint rectangular labels that are attached to R from the outside. The task is to connect the sites to the labels by non-intersecting rectilinear paths, so-called leaders, with at most one bend.In this paper, we study the Multi-Sided Boundary Labeling problem, with labels lying on at least two sides of the enclosing rectangle. We present a polynomial-time algorithm that computes a crossing-free leader layout if one exists. So far, such an algorithm has only been known for the cases in which labels lie on one side or on two opposite sides of R (here a crossing-free solution always exists). The case where labels may lie on adjacent sides is more difficult. We present efficient algorithms for testing the existence of a crossing-free leader layout that labels all sites and also for maximizing the number of labeled sites in a crossingfree leader layout. For two-sided boundary labeling with adjacent sides, we further show how to minimize the total leader length in a crossing-free layout.

show abstract

“…√ n log n) [CHKL13] Related Work. Apart from the result of Efrat et al [EHKP14], so far the only two variants of k-CESF that have been studied are those with extreme values of k. As mentioned above, 1-CESF is the same as EST, which is NP-hard [GJ79].…”

Section: O(mentioning

confidence: 99%

“…Bastert and Fekete [BF98] claimed that (n/2)-CESF is NP-hard, but their proof has not been formally published. Recently, Chan et al [CHKL13] considered (n/2)-CESF in the context of embedding planar graphs at fixed vertex locations. They gave an O( √ n log n)-approximation algorithm based on an idea of Liebling et al [LMM + 95] for computing a short non-crossing tour of all given points.…”

Section: O(mentioning

confidence: 99%

Colored Non-crossing Euclidean Steiner Forest

Bereg

Fleszar

Kindermann

et al. 2015

Algorithms and Computation

View full text Add to dashboard Cite

Given a set of k-colored points in the plane, we consider the problem of finding k trees such that each tree connects all points of one color class, no two trees cross, and the total edge length of the trees is minimized. For k = 1, this is the well-known Euclidean Steiner tree problem. For general k, a kρ-approximation algorithm is known, where ρ ≤ 1.21 is the Steiner ratio.We present a PTAS for k = 2, a (5/3 + ε)-approximation algorithm for k = 3, and two approximation algorithms for general k, with ratios O( √ n log k) and k + ε.

show abstract

Minimum Length Embedding of Planar Graphs at Fixed Vertex Locations

Cited by 14 publications

References 27 publications

Simultaneous Drawing of Planar Graphs with Right-Angle Crossings and Few Bends

Simultaneous Drawing of Planar Graphs with Right-Angle Crossings and Few Bends

Multi-sided Boundary Labeling

Colored Non-crossing Euclidean Steiner Forest

Contact Info

Product

Resources

About