Circle Packing Theorem - Algorithmic Aspects

Algorithmic Aspects

Collins & Stephenson (2003) describe a numerical relaxation algorithm for finding circle packings, based on ideas of William Thurston. The version of the circle packing problem that they solve takes as input a planar graph, in which all the internal faces are triangles and for which the external vertices have been labeled by positive numbers. It produces as output a circle packing whose tangencies represent the given graph, and for which the circles representing the external vertices have the radii specified in the input. As they suggest, the key to the problem is to first calculate the radii of the circles in the packing; once the radii are known, the geometric positions of the circles are not difficult to calculate. They begin with a set of tentative radii that do not correspond to a valid packing, and then repeatedly perform the following steps:

1. Choose an internal vertex v of the input graph.
2. Calculate the total angle θ that its k neighboring circles would cover around the circle for v, if the neighbors were placed tangent to each other and to the central circle using their tentative radii.
3. Determine a representative radius r for the neighboring circles, such that k circles of radius r would give the same covering angle θ as the neighbors of v give.
4. Set the new radius for v to be the value for which k circles of radius r would give a covering angle of exactly 2π.

Each of these steps may be performed with simple trigonometric calculations, and as Collins and Stephenson argue, the system of radii converges rapidly to a unique fixed point for which all covering angles are exactly 2π. Once the system has converged, the circles may be placed one at a time, at each step using the positions and radii of two neighboring circles to determine the center of each successive circle.

Mohar (1993) describes a similar iterative technique for finding simultaneous packings of a polyhedral graph and its dual, in which the dual circles are at right angles to the primal circles. He proves that the method takes time polynomial in the number of circles and in log 1/ε, where ε is a bound on the distance of the centers and radii of the computed packing from those in an optimal packing.