Pinning Balloons with Perfect Angles and Optimal AreaHalupczok, Immanual and Schulz, André (2012) Pinning Balloons with Perfect Angles and Optimal Area. In: Graph Drawing 19th International Symposium, GD 2011, September 2123, 2011 , pp. 154165(Official URL: http://dx.doi.org/10.1007/9783642258787_16). Full text not available from this repository.
Official URL: http://dx.doi.org/10.1007/9783642258787_16
AbstractWe study the problem of arranging a set of n disks with prescribed radii on n rays emanating from the origin such that two neighboring rays are separated by an angle of 2π/n. The center of the disks have to lie on the rays, and no two disk centers are allowed to lie on the same ray. We require that the disks have disjoint interiors, and that for every ray the segment between the origin and the boundary of its associated disk avoids the interior of the disks. Let r be the sum of the disk radii. We introduce a greedy strategy that constructs such a disk arrangement that can be covered with a disk centered at the origin whose radius is at most 2r, which is best possible. The greedy strategy needs O(n) arithmetic operations. As an application of our result we present an algorithm for embedding unordered trees with straight lines and perfect angular resolution such that it can be covered with a disk of radius n3.0367 , while having no edge of length smaller than 1. The tree drawing algorithm is an enhancement of a recent result by Duncan et al. [Symp. of Graph Drawing, 2010] that exploits the heavyedge tree decomposition technique to construct a drawing of the tree that can be covered with a disk of radius 2n4.
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