On geometric optimization with few violated constraints

Discrete & Computational Geometry, Dec 1995

We investigate the problem of finding the best solution satisfying all butk of the given constraints, for an abstract class of optimization problems introduced by Sharir and Welzl—the so-calledLP-type problems. We give a general algorithm and discuss its efficient implementations for specific geometric problems. For instance for the problem of computing the smallest circle enclosing all butk of the givenn points in the plane, we obtain anO(n logn+k3nε) algorithm; this improves previous results fork small compared withn but moderately growing. We also establish some results concerning general properties ofLP-type problems.

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On geometric optimization with few violated constraints

Discrete Comput Geom 0 Smallest Enclosing Circles. lem: 1 Department of Applied Mathematics, Charles University , Malostransk6 nfim. 25, 11800 Praha 1 , Czech Republic We investigate the problem of finding the best solution satisfying all but k of the given constraints, for an abstract class of optimization problems introduced by Sharir and Welzl--the so-called LP-type problems. We give a general algorithm and discuss its efficient implementations for specific geometric problems. For instance, for the problem of computing the smallest circle enclosing all but k of the given n points in the plane, we obtain an O(n log n + k3n e) algorithm; this improves previous results for k small compared with n but moderately growing. We also establish some results concerning general properties of LP-type problems. 1. Introduction Given a set P of n points in the plane and an integer q, find the smallest circle enclosing at least q points of P. This has recently been investigated in several papers [3], [18], [12], [16], [25] and it also motivated this paper. In these previous works, algorithms were obtained with * This research was supported in part by Charles University Grant No. 351 and Czech Republic Grant GA(~R 201/93/2167. Part of this research was performed while the author was visiting the Computer Science Institute, Free University Berlin, and it was supported by the German-Israeli Foundation of Scientific Research and Development (G.I.F.), and part while visiting the Max-Planck Institute for Computer Science in Saarbriicken. roughly O ( n q ) running time. 1 There seems to be little h o p e at present o f improving these bounds substantially in the full range of values of q. In particular, as observed by D. Eppstein (private communication), for q close to n / 2 , a subquadratic solution would imply a subquadratic solution 2 for several other basic problems, for which one seems unlikely with present methods (information about this class of problems, collected and publicized under the name "n2-hard problems" by M. Overmars and his colleagues, can be found in [20]). In this paper we investigate improvements over the roughly O ( n q ) b o u n d in the situation when k = n - q is small compared with n (all but few points should be enclosed by the circle); this question was raised, e.g., by Efrat et al. [16]. One of the first methods coming to mind for solving the problem is to construct the qth-order Voronoi diagram for the point set P, and find the required circle by inspecting all its cells (this approach was pointed out by Aggarwal et al. [3]). It is known that the combinatorial complexity o f the q t h - o r d e r Voronoi diagram is | - q ) q ) , and it has b e e n shown recently that it can be constructed in expected time O ( n log 3 n + (n - q)q log n) [1] (see also [7] and [2] for previous results). In our setting, this says that the smallest circle enclosing all but k points can be found in O ( n log 3 n + n k log n) time. In this paper we show that still better can be done for small k, namely, that the problem can be solved in close to linear time with k as large as n l / 3 : Theorem 1.1. The smallest circle containing all but at most k o f the giuen n points in the plane can be computed in O ( n log n + kan ~) time. 3 A predecessor of our technique is a result of Megiddo [27], who demonstrated that the k = 0 case, the smallest enclosing circle for n points, can be solved in O ( n ) time. LP-Type Problems. The problem of finding the smallest enclosing circle belongs to a class o f optimization problems known as LP-type problems (or "generalized linear programming" problems). This class was introduced by Sharir and Welzl [32]. It captures the properties of linear programming relevant for the description and analysis of their linear programming algorithm. T h e definition and more information on LP-type problems is given later. Each LP-type problem has an associated parameter, the so-called combinatorial dimension (or dimension for short). For instance, for a feasible linear programming 1Here are the specific running times: Eppstein and Erickson [18] solve the problem in O(nq log q + n log n) time with O(n log n + nq + q2 log q) space, and Datta et al. [12] give an algorithm with the same running time and space improved to O(n + q2 log q). Eft-at et aL [16] achieve O(nq log2 n) time with O(nq) space or alternatively O(nq log2n Iog(n/q)) time with O(n log n) space, and the author [5] has O(n log n + nq) time with O(nq) space or time O(n log n + nq log q) with O(n) space. 2 Here "subquadratic" means O(n 2- 8) for a constant 8 > 0. 3 Throughout this paper, 6 in exponents stands for a positive constant which can be made arbitrarily small by adjusting the parameters of the algorithms. Multiplicative constants implicit in the O( ) notation may depend on e. problem (one with a solution satisfying all the constraints), this combinatorial dimension equals the geometric dimension of the underlying space. Ran (...truncated)


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J. Matoušek. On geometric optimization with few violated constraints, Discrete & Computational Geometry, 1995, pp. 365-384, Volume 14, Issue 4, DOI: 10.1007/BF02570713