In this paper, 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 appear 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 @math . The problem has not been studied before. We present an O(|T|+M ^2 M) time algorithm for the problem, 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. In addition, by using this algorithm, we solve a k-center problem on T for the uncertain points of P.
Very recently, Li and Huang @cite_6 considered the same @math -center problem under the same uncertain model as ours but in the Euclidean space, and they gave an approximation algorithm. Facility location problems in other uncertain models have also been considered. For example, L "o ffler and van Kreveld @cite_39 gave algorithms for computing the smallest enclosing circle for imprecise points each of which is contained in a planar region (e.g., a circle or a square). J @cite_31 studied the problem of computing the distribution of the radius of the smallest enclosing circle for uncertain points each of which has multiple locations in the plane. de @cite_0 proposed algorithms for dynamically maintaining Euclidean @math -centers for a set of moving points in the plane (the moving points are considered uncertain). See also the problems for minimizing the maximum regret, e.g., @cite_7 @cite_9 @cite_36 .
@cite_7 @cite_36 @cite_9 @cite_6 @cite_39 @cite_0 @cite_31
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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. 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 the stochastic geometry model where the location of each node is a random point in a given metric space, or the existence of each node is uncertain. We study the problems of computing the expected lengths of several combinatorial or geometric optimization problems over stochastic points, including closest pair, minimum spanning tree, (k )-clustering, minimum perfect matching, and minimum cycle cover. We also consider the problem of estimating the probability that the length of closest pair, or the diameter, is at most, or at least, a given threshold. Most of the above problems are known to be ( # P )-hard. We obtain FPRAS (Fully Polynomial Randomized Approximation Scheme) for most of them in both the existential and locational uncertainty models. Our result for stochastic minimum spanning trees in the locational uncertain model improves upon the previously known constant factor approximation algorithm. Our results for other problems are the first known to the best of our knowledge.