Pinning Balloons with Perfect Angles and Optimal Area

TL;DR

This paper introduces an optimal greedy method for arranging disks with prescribed radii on rays at perfect angles, ensuring disjoint interiors and a covering disk of minimal radius, with applications to tree embedding.

Contribution

It presents a new greedy strategy for disk arrangement with optimal covering radius and applies it to improve tree embedding algorithms with better coverage bounds.

Findings

The greedy strategy constructs arrangements with a covering disk radius at most 2 times the sum of radii.

The algorithm operates in O(n) arithmetic operations.

Application to tree embedding achieves a covering disk of radius n^{3.0367}, improving previous bounds.

Abstract

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 $2\pi/n$. 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 $2\r$, 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 $n^{3.0367}$, 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 $2 n^4$.