Minimal Enclosing Circle and Two and Three Point Partition of a Plane

The 2006 International Conference on Scientific Computing (CSC’06) Abstract Partitions of a plane, based on two or three of its points, are introduced. The study of these partitions is applied to finding the minimal enclosing circle (MEC) for a set S of n planar points. MEC(S), the MEC of n points of S, is defined by either a pair of S points with the largest distance (tight two-tuple) or by a triplet of S points spread on more than half of its circumference (tight three-tuple) with the largest radius. An extension for an existing MEC by an outside point P∈S is a MEC for point P and the points of the tight tuple for the existing MEC. It has a larger radius than existing MEC. The MEC problem is dual to a problem of finding an optimal partition of S-plane by two or three points of S defined with the largest circular region. A two point partition divides the S-plane in 4 regions, a three point – in 7 regions, one region is a circle in either partition. The MEC algorithm is based on this duality. It begins with a MEC of two arbitrary points of S and corresponding two point partition of S-plane. Next, each point P of S is examined in a separate step of the algorithm. If it is outside of the current MEC, its extension by this point is obtained. The tight tuple for the extension is formed by replacing either none or one or two points of current MEC’s tight tuple by point P. Which points of the tuple are to be replaced by point P depends on the region to which P belongs in a plane partition by the points of current tight tuple. The next circle has a larger diameter and it retains at least one set of the defining points of a previous circle, thus limiting a possible loss of its Spoints during an extension. A n-step iteration is completed once each point of S is examined. It is repeated until no point of S is found outside of a current MEC during an entire iteration. Observed number of steps in the algorithm has rarely reached 5n and never exceeded 6n in an experiment over several point distributions with n in range from 10 to 28,000,000 . Commonly considered as the fastest, Cärtner’s modification of Welzl’s algorithm [18], [6] has a proved expected performance of O(n). In the experiment MEC algorithm outperformed it in more than 7 times in average. At this point no satisfying theoretical bound, matching this remarkable performance of MEC algorithm, has been found. This incremental algorithm is an on-line algorithm: if set S gets new points during its execution, the current and following iterations continue with an updated set S without a loss of the progress achieved before the update. The algorithm has been already extended to R and this will be reported elsewhere. This paper is also about the two and three point partitions. They provide the basis for the MEC algorithm.