Efficient Speed-Up of the Smallest Enclosing Circle Algorithm (original) (raw)

Solution Methodologies for the Smallest Enclosing Circle Problem

2003

Tribute. We would like to dedicate this paper to Elijah Polak. Professor Polak has made substantial contributions to a truly broad spectrum of topics in nonlinear optimization, including optimization for engineering design centering, multi-criteria optimization, optimal control, feasible directions methods, quasi-Newton and Newton methods, non-differential optimization, semi-infinite optimization, conjugate directions methods, gradient projection and reduced gradient methods, and barrier methods, among many other topics. His many and varied contributions to our field are important today and will influence the research in our field well into the future.

Computing the smallest k-enclosing circle and related problems

Computational Geometry, 1994

We present an efficient algorithm for solving the "smallest k-enclosing circle" (RSC) problem: Given a set of n points in the plane and an integer k < n, find the smallest disk containing k of the points. We present two solutions. When using O(nk) storage, the problem can be solved in time O(nklog'n). When only O(nlogn) storage is allowed, the running time is O(nk log' n log n/k). We also extend our technique to obtain efficient solutions of several related problems (with similar time and storage bounds). These related problems include: finding the smallest homothetic copy of a given convex polygon P which contains k points from a given planar set, and finding the smallest disk intersecting k segments from a given planar set of non-intersecting segments.

A parallel circle-cover minimization algorithm

Information Processing Letters, 1989

North-Holland l. lntroduction Given a set of ,? circular arcs distributed in an arbitrary fashion, define the circle-couer minimization problem as the problem of determining a minimal set of arcs whose union covers the circle. Lee and Lee [5] give an optimal @(n log n) serial solution to the problem. Bertossi [1] gives an algorithm that is a straightforward parallelization of the algorithm in [5] and runs in @(log n) time using@(nz77og n) processors on a CREW PRAM. In this paper, we present a new CREW PRAM algorithm to, solve the circle-cover minimization problem in @(log n) time using n processors and @(n) global memory, which is optimal. Applications of the circle-cover minimization problem may be found in [1,5].

Computing the Closest Point to a Circle

Cccg, 2003

In this paper we consider the problem of computing the closest point to the boundary of a circle among a set S of n points. We present two algorithms to solve this problem. One algorithm runs in O(n 3 ) preprocessing time and space and O(log 2 n) query time. The other algorithm runs in O(n 1+ǫ ) preprocessing time and O(n log n) space and O(n 2/3+ǫ ) query time. Thus we exhibit a trade-off between preprocessing and query times For dimensions d ≥ 3 we present an algorithm with O(n ⌈d/2⌉+ǫ ′ ) preprocessing time to report an approximate closest point to the boundary of d-dimensional query sphere R in O(n 1−1/(d+1)+ǫ ) query time.

A circle packing algorithm

Computational Geometry, 2003

A circle packing is a configuration P of circles realizing a specified pattern of tangencies. Radii of packings in the euclidean and hyperbolic planes may be computed using an iterative process suggested by William Thurston. We describe an efficient implementation, discuss its performance, and illustrate recent applications. A central role is played by new and subtle monotonicity results for "flowers" of circles.

Fitting a set of points by a circle Jesus Garca-Lopez y

1998

Given a set of points S = fp 1; : : : ; p n g in Euclidean d-dimensional space, we address the problem of computing the d-dimensional annulus of smallest width containing the set. We give a complete characterization of the centers of annuli which are locally minimal in arbitrary dimension and we show that, for d = 2, a locally minimal annulus has two points on the inner circle and two points on the outer circle that interlace angle-wise as seen from the center of the annulus. Using this characterization, we show that, given a circular order of the points, there is at most one locally minimal annulus consistent with that order and it can be computed in time O(n logn) using a simple algorithm. Furthermore, when points are in convex position, the problem can be solved in optimal (n) time.

On the geometry of the smallest circle enclosing a finite set of points

Journal of the Franklin Institute, 2007

A number of numerical codes have been written for the problem of finding the circle of smallest radius in the Euclidean plane that encloses a finite set P of points, but these do not give much insight into the geometry of this circle. We investigate geometric properties of the minimal circle that may be useful in the theoretical analysis of applications. We show that a circle C enclosing P is minimal if and only if it is rigid in the sense that it cannot be translated while still enclosing P. We show that the center of the minimal circle is in the convex hull of P. We use this rigidity result and an analysis of the case of three points to find sharp estimates on the diameter of the minimal circle in terms of the diameter of P.

Fitting a Set of Points by a Circle

Discrete and Computational …, 1998

Given a set of points S = {p 1 ,. .. , p n } in Euclidean d-dimensional space, we address the problem of computing the d-dimensional annulus of smallest width containing the set. We give a complete characterization of the centers of annuli which are locally minimal in arbitrary dimension and we show that, for d = 2, a locally minimal annulus has two points on the inner circle and two points on the outer circle that interlace anglewise as seen from the center of the annulus. Using this characterization, we show that, given a circular order of the points, there is at most one locally minimal annulus consistent with that order and it can be computed in time O(n log n) using a simple algorithm. Furthermore, when points are in convex position, the problem can be solved in optimal (n) time.

A linearized circle packing algorithm

Computational Geometry

This paper presents a geometric algorithm for approximating radii and centers for a variety of univalent circle packings, including maximal circle packings on the unit disc and the sphere and certain polygonal circle packings in the plane. This method involves an iterative process which alternates between estimates of circle radii and locations of circle centers. The algorithm employs sparse linear systems and in practice achieves a consistent linear convergence rate that is far superior to traditional packing methods. It is deployed in a MATLAB R package which is freely available. This paper gives background on circle packing, a description of the linearized algorithm, illustrations of its use, sample performance data, and remaining challenges.

Reformulation descent applied to circle packing problems

Computers & Operations Research, 2005

Several years ago classical Euclidean geometry problems of densest packing of circles in the plane have been formulated as nonconvex optimization problems, allowing to ÿnd heuristic solutions by using any available NLP solver. In this paper we try to improve this procedure. The faster NLP solvers use ÿrst order information only, so stop in a stationary point. A simple switch from Cartesian coordinates to polar or vice versa, may destroy this stationarity and allow the solver to descend further. Such formulation switches may of course be iterated. For densest packing of equal circles into a unit circle, this simple feature turns out to yield results close to the best known, while beating second order methods by a time-factor well over 100.