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.
Two models on uncertain data have been commonly considered: the model @cite_8 @cite_23 @cite_17 @cite_27 @cite_34 @cite_28 and the model @cite_35 @cite_13 @cite_33 @cite_22 . In the existential model an uncertain point has a specific location but its existence is uncertain while in the locational model an uncertain point always exists but its location is uncertain and follows a probability distribution function. Our problems belong to the locational model. In fact, the same problems under existential model are essentially the weighted case for deterministic'' points (i.e., each @math has a single certain'' location), and the center-coverage problem is solvable in linear time @cite_19 and the @math -center problem is solvable in @math time @cite_21 @cite_12 .
@cite_35 @cite_33 @cite_22 @cite_8 @cite_28 @cite_21 @cite_19 @cite_27 @cite_23 @cite_34 @cite_13 @cite_12 @cite_17
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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. Consider a set of points in d dimensions where the existence or the location of each point is determined by a probability distribution. The convex hull of this set is a random variable distributed over exponentially many choices. We are interested in finding the most likely convex hull, namely, the one with the maximum probability of occurrence. We investigate this problem under two natural models of uncertainty: the point (also called the tuple) model where each point (site) has a fixed position s i but only exists with some probability π i , for 0 < π i ≤ 1, and the multipoint model where each point has multiple possible locations or it may not appear at all. We show that the most likely hull under the point model can be computed in O(n 3) time for n points in d = 2 dimensions, but it is NP–hard for d ≥ 3 dimensions. On the other hand, we show that the problem is NP–hard under the multipoint model even for d = 2 dimensions. We also present hardness results for approximating the probability of the most likely hull. While we focus on the most likely hull for concreteness, our results hold for other natural definitions of a probabilistic hull. 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 ).