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)