The power of non-rectilinear holes

Lingas, Andrzej

doi:10.1007/bfb0012784

Cited by 89 publications

(43 citation statements)

References 5 publications

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“…An illustration is given in Figure 2(b), where the object consists of two approximate convex parts p 1 and p 2 , connected by a non-convex junction q 12 . Since exact convex decomposition is NP-hard for shapes with holes [16], there are many approximate solutions proposed in the literature (eg. [17]).…”

Section: Approximate Convex Decompositionmentioning

confidence: 99%

Articulation-Invariant Representation of Non-planar Shapes

Gopalan

Turaga

Chellappa

2010

Computer Vision – ECCV 2010

135

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Section: Approximate Convex Decompositionmentioning

confidence: 99%

Articulation-Invariant Representation of Non-planar Shapes

Gopalan

Turaga

Chellappa

2010

Computer Vision – ECCV 2010

135

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“…In the case that Steiner points are allowed, the problem can be solved in O(n + r 3 ) time [5]. For polygons with holes, the problem is NP-hard in both cases [16].…”

Section: Related Workmentioning

confidence: 99%

Partitioning Polygons via Graph Augmentation

Haunert

Meulemans

2016

Geographic Information Science

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This is the accepted version of the paper.This version of the publication may differ from the final published version. wouter.meulemans@city.ac.uk Permanent repository linkAbstract. We study graph augmentation under the dilation criterion. In our case, we consider a plane geometric graph G = (V, E) and a set C of edges. We aim to add to G a minimal number of nonintersecting edges from C to bound the ratio between the graph-based distance and the Euclidean distance for all pairs of vertices described by C. Motivated by the problem of decomposing a polygon into natural subregions, we present an optimal linear-time algorithm for the case that P is a simple polygon and C models an internal triangulation of P . The algorithm admits some straightforward extensions. Most importantly, in pseudopolynomial time, it can approximate a solution of minimum total length or, if C is weighted, compute a solution of minimum total weight. We show that minimizing the total length or the total weight is weakly NP-hard. Finally, we show how our algorithm can be used for two well-known problems in GIS: generating variable-scale maps and area aggregation.

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“…Restricted to simple polygons, these problems have efficient solutions. Keil and Snoeyink [22] gave an O(n 3 ) time algorithm for computing the minimum number convex partition of a simple polygon with n vertices; this problem is NP-hard for polygons with holes [32]. Chazelle and Dobkin [9] gave an O(n 3 ) time algorithm for the minimum number convex Steiner partition of a simple polygon with n vertices.…”

Section: Our Contribution (I)mentioning

confidence: 99%

Minimum Weight Convex Steiner Partitions

Dumitrescu

Tóth

2009

Algorithmica

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New tight bounds are presented on the minimum length of planar straight line graphs connecting n given points in the plane and having convex faces. Specifically, we show that the minimum length of a convex Steiner partition for n points in the plane is at most O(log n/ log log n) times longer than a Euclidean minimum spanning tree (EMST), and this bound is the best possible. Without Steiner points, the corresponding bound is known to be Θ(log n), attained for n vertices of a pseudo-triangle. We also show that the minimum length convex Steiner partition of n points along a pseudo-triangle is at most O(log log n) times longer than an EMST, and this bound is also the best possible. Our methods are constructive and lead to O(n log n) time algorithms for computing convex Steiner partitions having O(n) Steiner points and weight within the above worst-case bounds in both cases.

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The power of non-rectilinear holes

Cited by 89 publications

References 5 publications

Articulation-Invariant Representation of Non-planar Shapes

Articulation-Invariant Representation of Non-planar Shapes

Partitioning Polygons via Graph Augmentation

Minimum Weight Convex Steiner Partitions

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