In this paper, we consider the rectilinear one-center problem on uncertain points in the plane. In this problem, we are given a set @math of @math (weighted) uncertain points in the plane and each uncertain point has @math possible locations each associated with a probability for the point appearing at that location. The goal is to find a point @math in the plane which minimizes the maximum expected rectilinear distance from @math to all uncertain points of @math , and @math is called a rectilinear center. We present an algorithm that solves the problem in @math time. Since the input size of the problem is @math , our algorithm is optimal.
The @math -center problems on deterministic points are classical problems and have been studied extensively. When all points are in the plane, the problems on most distance metrics are NP-hard @cite_15 . However, some special cases can be solved in polynomial time, e.g., the one-center problem @cite_7 , the two-center problem @cite_8 , the rectilinear three-center problem @cite_20 , the line-constrained @math -center problems (where all centers are restricted to be on a given line in the plane) @cite_3 @cite_4 @cite_17 .
@cite_4 @cite_7 @cite_8 @cite_3 @cite_15 @cite_20 @cite_17
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Abstract This paper considers the planar Euclidean two-center problem: given a planar n-point set S, find two congruent circular disks of the smallest radius covering S. The main result is a deterministic algorithm with running time O(nlog2nlog2logn), improving the previous O(nlog9n) bound of Sharir and almost matching the randomized O(nlog2n) bound of Eppstein. If a point in the intersection of the two disks is given, then we can solve the problem in O(nlogn) time with high probability. We study two versions of the 2-center problem for moving points in the plane. Given a set P of n points, the Euclidean 2-center problem asks for two congruent disks of minimum size that together cover P; the rectilinear 2-center problem correspondingly asks for two congruent axis-aligned squares of minimum size that together cover P. Our methods work in the black-box KDS model, where we receive the locations of the points at regular time steps and we know an upper bound d_ max on the maximum displacement of any point within one time step. We show how to maintain the rectilinear 2-center in amortized sub-linear time per time step, under certain assumptions on the distribution of the point set P. For the Euclidean 2-center we give a similar result: we can maintain in amortized sub-linear time (again under certain assumptions on the distribution) a (1+e)-approximation of the optimal 2-center. In many cases---namely when the distance between the centers of the disks is relatively large or relatively small---the solution we maintain is actually optimal. We also present results for the case where the maximum speed of the centers is bounded. We describe a simple scheme to maintain a 2-approximation of the rectilinear 2-center, and we provide a scheme which gives a better approximation factor depending on several parameters of the point set and the maximum allowed displacement of the centers. The latter result can be used to obtain a 2.29-approximation for the Euclidean 2-center; this is an improvement over the previously best known bound of 8 π approx 2.55. These algorithms run in amortized sub-linear time per time step, as before under certain assumptions on the distribution. The @math -center problem is a classical combinatorial optimization problem which asks to find @math centers such that the maximum distance of any input point in a set @math to its assigned center is minimized. The problem allows for elegant @math -approximations. However, the situation becomes significantly more difficult when constraints are added to the problem. We raise the question whether general methods can be derived to turn an approximation algorithm for a clustering problem with some constraints into an approximation algorithm that respects one constraint more. Our constraint of choice is privacy: Here, we are asked to only open a center when at least @math clients will be assigned to it. We show how to combine privacy with several other constraints. Given n demand points in the plane, the p-center problem is to find p supply points (anywhere in the plane) so as to minimize the maximum distance from a demand point to its respective nearest supply point. The p-median problem is to minimize the sum of distances from demand points to their respective nearest supply points. We prove that the p-center and the p-median problems relative to both the Euclidean and the rectilinear metrics are NP-hard. In fact, we prove that it is NP-hard even to approximate the p-center problems sufficiently closely. The reductions are from 3-satisfiability.