A note on optimal area algorithms for upward drawings of binary trees
Computational Geometry: Theory and Applications
On the Angular Resolution of Planar Graphs
SIAM Journal on Discrete Mathematics
RINGS: A Technique for Visualizing Large Hierarchies
GD '02 Revised Papers from the 10th International Symposium on Graph Drawing
Tree Drawings on the Hexagonal Grid
Graph Drawing
Journal of Computer and System Sciences
Straight-line orthogonal drawings of binary and ternary trees
GD'07 Proceedings of the 15th international conference on Graph drawing
Complexity analysis of balloon drawing for rooted trees
Theoretical Computer Science
Drawing trees with perfect angular resolution and polynomial area
GD'10 Proceedings of the 18th international conference on Graph drawing
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We 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 $\widetilde r$ 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 $2\widetilde r$ , 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 heavy-edge tree decomposition technique to construct a drawing of the tree that can be covered with a disk of radius 2 n4.