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.
There is also a lot of other work on facility location problems for uncertain data. For instances, Cormode and McGregor @cite_19 proved that the @math -center problem on uncertain points each associated with multiple locations in high-dimension space is NP-hard and gave approximation algorithms for different problem models. Foul @cite_0 considered the Euclidean one-center problem on uncertain points each of which has a uniform distribution in a given rectangle in the plane. de Berg. et al @cite_1 studied the Euclidean 2-center problem for a set of moving points in the plane (the moving points can be considered uncertain).
@cite_0 @cite_19 @cite_1
2303014626 2963075731 2030028156 2964179137
We consider the one-dimensional one-center problem on uncertain data. We are given a set P of n (weighted) uncertain points on a real line L and each uncertain point is specified by a probability density function that is a piecewise-uniform function (i.e., a histogram). The goal is to find a point c (the center) on L such that the maximum expected distance from c to all uncertain points of P is minimized. We present a linear-time algorithm for this problem. With the dramatic growth in the number of application domains that generate probabilistic, noisy and uncertain data, there has been an increasing interest in designing algorithms for geometric or combinatorial optimization problems over such data. In this paper, we initiate the study of constructing epsilon-kernel coresets for uncertain points. We consider uncertainty in the existential model where each point's location is fixed but only occurs with a certain probability, and the locational model where each point has a probability distribution describing its location. An epsilon-kernel coreset approximates the width of a point set in any direction. We consider approximating the expected width (an epsilon-EXP-KERNEL), as well as the probability distribution on the width (an (epsilon, tau)-QUANT-KERNEL) for any direction. We show that there exists a set of O(epsilon^ -(d-1) 2 ) deterministic points which approximate the expected width under the existential and locational models, and we provide efficient algorithms for constructing such coresets. We show, however, it is not always possible to find a subset of the original uncertain points which provides such an approximation. However, if the existential probability of each point is lower bounded by a constant, an epsilon-EXP-KERNEL is still possible. We also provide efficient algorithms for construct an (epsilon, tau)-QUANT-KERNEL coreset in nearly linear time. Our techniques utilize or connect to several important notions in probability and geometry, such as Kolmogorov distances, VC uniform convergence and Tukey depth, and may be useful in other geometric optimization problem in stochastic settings. Finally, combining with known techniques, we show a few applications to approximating the extent of uncertain functions, maintaining extent measures for stochastic moving points and some shape fitting problems under uncertainty. We consider single facility location problems (1-median and weighted 1-center) on a plane with uncertain weights and coordinates of customers (demand points). Specifically, for each customer, only interval estimates for its weight and coordinates are known. It is required to find a ''minmax regret'' location, i.e. to minimize the worst-case loss in the objective function value that may occur because the decision is made without knowing the exact values of customers' weights and coordinates that will get realized. We present an O(n^2log^2n) algorithm for the interval data minmax regret rectilinear 1-median problem and an O(nlogn) algorithm for the interval data minmax regret rectilinear weighted 1-center problem. For the case of Euclidean distances, we consider uncertainty only in customers' weights. We discuss possibilities of solving approximately the minmax regret Euclidean 1-median problem, and present an O(n^22^@a^(^n^)log^2n) algorithm for solving the minmax regret Euclidean weighted 1-center problem, where @a(n) is the inverse Ackermann function. We consider a coverage problem for uncertain points in a tree. Let T be a tree containing a set ( P ) of n (weighted) demand points, and the location of each demand point (P_i P ) is uncertain but is known to be in one of (m_i ) points on T each associated with a probability. Given a covering range ( ), the problem is to find a minimum number of points (called centers) on T to build facilities for serving (or covering) these demand points in the sense that for each uncertain point (P_i P ), the expected distance from (P_i ) to at least one center is no more than ( ). The problem has not been studied before. We present an (O(|T|+M ^2 M) ) time algorithm, where |T| is the number of vertices of T and M is the total number of locations of all uncertain points of ( P ), i.e., (M= _ P_i P m_i ).