Searching for the Closest-Pair in a Query Translate

We consider a range-search variant of the closest-pair problem. Let Gamma be a fixed shape in the plane. We are interested in storing a given set of n points in the plane in some data structure such that for any specified translate of Gamma, the closest pair of points contained in the translate can be reported efficiently. We present results on this problem for two important settings: when Gamma is a polygon (possibly with holes) and when Gamma is a general convex body whose boundary is smooth. When Gamma is a polygon, we present a data structure using O(n) space and O(log n) query time, which is asymptotically optimal. When Gamma is a general convex body with a smooth boundary, we give a near-optimal data structure using O(n log n) space and O(log^2 n) query time. Our results settle some open questions posed by Xue et al. at SoCG 2018

On the power of the semi-separated pair decomposition

A Semi-Separated Pair Decomposition (SSPD), with parameter s > 1, of a set is a set {(A i ,B i )} of pairs of subsets of S such that for each i, there are balls and containing A i and B i respectively such that min ( radius ) , radius ), and for any two points p, q S there is a unique index i such that p A i and q B i or vice-versa. In this paper, we use the SSPD to obtain the following results: First, we consider the construction of geometric t-spanners in the context of imprecise points and we prove that any set of n imprecise points, modeled as pairwise disjoint balls, admits a t-spanner with edges which can be computed in time. If all balls have the same radius, the number of edges reduces to . Secondly, for a set of n points in the plane, we design a query data structure for half-plane closest-pair queries that can be built in time using space and answers a query in time, for any ε> 0. By reducing the preprocessing time to and using space, the query can be answered in time. Moreover, we improve the preprocessing time of an existing axis-parallel rectangle closest-pair query data structure from quadratic to near-linear. Finally, we revisit some previously studied problems, namely spanners for complete k-partite graphs and l

Efficient geometric algorithms for preference top-k queries, stochastic line arrangements, and proximity problems

University of Minnesota Ph.D. dissertation. June 2017. Major: Computer Science. Advisor: Ravi Janardan. 1 computer file (PDF); x, 150 pages.Problems arising in diverse real-world applications can often be modeled by geometric objects such as points, lines, and polygons. The goal of this dissertation research is to design efficient algorithms for such geometric problems and provide guarantees on their performance via rigorous theoretical analysis. Three related problems are discussed in this thesis. The first problem revisits the well-known problem of answering preference top-k queries, which arise in a wide range of applications in databases and computational geometry. Given a set of n points, each with d real-valued attributes, the goal is to organize the points into a suitable data structure so that user preference queries can be answered efficiently. A query consists of a d-dimensional vector w, representing a user's preference for each attribute, and an integer k, representing the number of data points to be retrieved. The answer to a query is the k highest-scoring points relative to w, where the score of a point, p, is designed to reflect how well it captures, in aggregate, the user's preferences for the different attributes. This thesis contributes efficient exact solutions in low dimensions (2D and 3D), and a new sampling-based approximation algorithm in higher dimensions. The second problem extends the fundamental geometric concept of a line arrangement to stochastic data. A line arrangement in the plane is a partition of the plane into vertices, edges, and faces. Surprisingly, diverse problems, including the preference top-k query and k-order Voronoi Diagram, essentially boil down to answering questions about the set of k-topmost lines at some abscissa. This thesis considers line arrangements in a new setting, where each line has an associated existence probability representing uncertainty that is inherent in real-world data. An upper-bound is derived on the expected number of changes in the set of k-topmost lines, taken over the entire x-axis, and a worst-case upper bound is given for k = 1. Also, given is an efficient algorithm to compute the most likely k-topmost lines in the arrangement. Applications of this problem including the most likely Voronoi Diagram in R^1 and stochastic preference top-k query are discussed. The third problem discussed is geometric proximity search in both the stochastic setting and the query-retrieval setting. Under the stochastic setting, the thesis considers two fundamental problems, namely, the stochastic closest pair problem and the k most likely nearest neighbor search. In both problems, the data points are assumed to lie on a tree embedded in R^2 and distances are measured along the tree (a so-called tree space). For the former, efficient solutions are given to compute the probability that the closest pair distance of a realization of the input is at least l and to compute the expected closest pair distance. For the latter, the thesis generalizes the concept of most likely Voronoi Diagram from R^1 to tree space and bounds its combinatorial complexity. A data structure for the diagram and an algorithm to construct it are also given. For the query-retrieval version which is considered in R^2, the goal is to retrieve the closest pair within a user-specified query range. The contributions here include efficient data structures and algorithms that have fast query time while using linear or near-linear space for a variety of query shapes. In addition, a generic framework is presented, which returns a closest pair that is no farther apart than the closest pair in a suitably shrunken version of the query range

Data structures for range-aggregate extent queries

A fundamental and well-studied problem in computational geometry is range searching, where the goal is to preprocess a set, S, of geometric objects (e.g., points in the plane) so that the subset S′⊆S that is contained in a query range (e.g., an axes-parallel rectangle) can be reported efficiently. However, in many situations, what is of interest is to generate a more informative "summary" of the output, obtained by applying a suitable aggregation function on S′. Examples of such aggregation functions include count, sum, min, max, mean, median, mode, and top-k that are usually computed on a set of weights defined suitably on the objects. Such range-aggregate query problems have been the subject of much recent research in both the database and the computational geometry communities. In this paper, we further generalize this line of work by considering aggregation func