The Unified Segment Tree and its Application to the Rectangle Intersection Problem

In this paper we introduce a variation on the multidimensional segment tree, formed by unifying different interpretations of the dimensionalities of the data structure. We give some new definitions to previously well-defined concepts that arise naturally in this variation, and we show some properties concerning the relationships between the nodes, and the regions those nodes represent. We think these properties will enable the data to be utilized in new situations, beyond those previously studied. As an example, we show that the data structure can be used to solve the Rectangle Intersection Problem in a more straightforward and natural way than had be done in the past.Comment: 14 pages, 6 figure

Reverse Nearest Neighbor Heat Maps: A Tool for Influence Exploration

We study the problem of constructing a reverse nearest neighbor (RNN) heat map by finding the RNN set of every point in a two-dimensional space. Based on the RNN set of a point, we obtain a quantitative influence (i.e., heat) for the point. The heat map provides a global view on the influence distribution in the space, and hence supports exploratory analyses in many applications such as marketing and resource management. To construct such a heat map, we first reduce it to a problem called Region Coloring (RC), which divides the space into disjoint regions within which all the points have the same RNN set. We then propose a novel algorithm named CREST that efficiently solves the RC problem by labeling each region with the heat value of its containing points. In CREST, we propose innovative techniques to avoid processing expensive RNN queries and greatly reduce the number of region labeling operations. We perform detailed analyses on the complexity of CREST and lower bounds of the RC problem, and prove that CREST is asymptotically optimal in the worst case. Extensive experiments with both real and synthetic data sets demonstrate that CREST outperforms alternative algorithms by several orders of magnitude.Comment: Accepted to appear in ICDE 201

Personalized Trust Management in Decision Making: A Dynamic Clustering Approach

This paper presents a personalized approach for distributed trust management by employing the k-means range algorithm, a combination of the partitional k-means clustering algorithm with orthogonal range search concepts. The aim of this approach is to aid the human or computer agent in organizing information from multiple sources into clusters according to its “trust features”. Thus the agent can perform complicated trust assessments in real-time situations and cooperate with decision-making software to assist in purchasing activities. We conclude by discussing the implications and advantages of this approach in trust management in traditional and mobile e-commerce applications

Rectilinear minimum link paths in two and higher dimensions

The thesis discusses algorithms for the minimum link path problem, which is a well known geometric path finding problem. The goal is to find a path that does the minimum number of turns amidst obstacles in a continuous space. We focus on the most classical variant, the rectilinear minimum link path problem, where the path and the obstacles are restricted to the directions of the coordinate axes. We study the rectilinear minimum link path problem in the plane and in the three-dimensional space, as well as in higher dimensional domains. We present several new algorithms for solving the problem in domains of varying dimension. For the planar case we develop a simple method that has the optimal O(n log n) time complexity. For three-dimensional domains we present a new algorithm with running time O(n^2 log^2 n), which is an improvement over the best previously known result O(n^2.5 log n). The algorithm can also be generalized to higher dimensions, leading to an O(n^(D-1) log^(D-1) n) time algorithm in D-dimensional domains. We describe the new algorithms as well as the data structures used. The algorithms work by maintaining a reachable region that is gradually expanded to form a shortest path map from the starting point. The algorithms rely on several efficient data structures: the reachable region is tracked by using a simple recursive space decomposition, and the region is expanded by a sweep plane method that uses a multidimensional segment tree

Computing Volumes and Convex Hulls: Variations and Extensions

Geometric techniques are frequently utilized to analyze and reason about multi-dimensional data. When confronted with large quantities of such data, simplifying geometric statistics or summaries are often a necessary first step. In this thesis, we make contributions to two such fundamental concepts of computational geometry: Klee's Measure and Convex Hulls. The former is concerned with computing the total volume occupied by a set of overlapping rectangular boxes in d-dimensional space, while the latter is concerned with identifying extreme vertices in a multi-dimensional set of points. Both problems are frequently used to analyze optimal solutions to multi-objective optimization problems: a variant of Klee's problem called the Hypervolume Indicator gives a quantitative measure for the quality of a discrete Pareto Optimal set, while the Convex Hull represents the subset of solutions that are optimal with respect to at least one linear optimization function.In the first part of the thesis, we investigate several practical and natural variations of Klee's Measure Problem. We develop a specialized algorithm for a specific case of Klee's problem called the “grounded” case, which also solves the Hypervolume Indicator problem faster than any earlier solution for certain dimensions. Next, we extend Klee's problem to an uncertainty setting where the existence of the input boxes are defined probabilistically, and study computing the expectation of the volume. Additionally, we develop efficient algorithms for a discrete version of the problem, where the volume of a box is redefined to be the cardinality of its overlap with a given point set.The second part of the thesis investigates the convex hull problem on uncertain input. To this extent, we examine two probabilistic uncertainty models for point sets. The first model incorporates uncertainty in the existence of the input points. The second model extends the first one by incorporating locational uncertainty. For both models, we study the problem of computing the probability that a given point is contained in the convex hull of the uncertain points. We also consider the problem of finding the most likely convex hull, i.e., the mode of the convex hull random variable