A Dynamic I/O-Efficient Structure for One-Dimensional Top-k Range Reporting

We present a structure in external memory for "top-k range reporting", which uses linear space, answers a query in O(lg_B n + k/B) I/Os, and supports an update in O(lg_B n) amortized I/Os, where n is the input size, and B is the block size. This improves the state of the art which incurs O(lg^2_B n) amortized I/Os per update.Comment: In PODS'1

An Efficient Sum Query Algorithm for Distance-based Locally Dominating Functions

In this paper, we consider the following sum query problem: Given a point set P in R^d, and a distance-based function f(p,q) (i.e. a function of the distance between p and q) satisfying some general properties, the goal is to develop a data structure and a query algorithm for efficiently computing a (1+epsilon)-approximate solution to the sum sum_{p in P} f(p,q) for any query point q in R^d and any small constant epsilon>0. Existing techniques for this problem are mainly based on some core-set techniques which often have difficulties to deal with functions with local domination property. Based on several new insights to this problem, we develop in this paper a novel technique to overcome these encountered difficulties. Our algorithm is capable of answering queries with high success probability in time no more than ~O_{epsilon,d}(n^{0.5 + c}), and the underlying data structure can be constructed in ~O_{epsilon,d}(n^{1+c}) time for any c>0, where the hidden constant has only polynomial dependence on 1/epsilon and d. Our technique is simple and can be easily implemented for practical purpose

External Memory Three-Sided Range Reporting and Top-k Queries with Sublogarithmic Updates

An external memory data structure is presented for maintaining a dynamic set of N two-dimensional points under the insertion and deletion of points, and supporting unsorted 3-sided range reporting queries and top-k queries, where top-k queries report the k points with highest y-value within a given x-range. For any constant 0 < epsilon <= 1/2, a data structure is constructed that supports updates in amortized O(1/(epsilon * B^{1-epsilon}) * log_B(N)) IOs and queries in amortized O(1/epsilon * log_B(N+K/B)) IOs, where B is the external memory block size, and K is the size of the output to the query (for top-k queries K is the minimum of k and the number of points in the query interval). The data structure uses linear space. The update bound is a significant factor B^{1-epsilon} improvement over the previous best update bounds for these two query problems, while staying within the same query and space bounds

Orthogonal Point Location and Rectangle Stabbing Queries in 3-d

In this work, we present a collection of new results on two fundamental problems in geometric data structures: orthogonal point location and rectangle stabbing. - Orthogonal point location. We give the first linear-space data structure that supports 3-d point location queries on n disjoint axis-aligned boxes with optimal O(log n) query time in the (arithmetic) pointer machine model. This improves the previous O(log^{3/2} n) bound of Rahul [SODA 2015]. We similarly obtain the first linear-space data structure in the I/O model with optimal query cost, and also the first linear-space data structure in the word RAM model with sub-logarithmic query time. - Rectangle stabbing. We give the first linear-space data structure that supports 3-d 4-sided and 5-sided rectangle stabbing queries in optimal O(log_wn+k) time in the word RAM model. We similarly obtain the first optimal data structure for the closely related problem of 2-d top-k rectangle stabbing in the word RAM model, and also improved results for 3-d 6-sided rectangle stabbing. For point location, our solution is simpler than previous methods, and is based on an interesting variant of the van Emde Boas recursion, applied in a round-robin fashion over the dimensions, combined with bit-packing techniques. For rectangle stabbing, our solution is a variant of Alstrup, Brodal, and Rauhe\u27s grid-based recursive technique (FOCS 2000), combined with a number of new ideas

Algorithms and Data Structures for Geometric Intersection Query Problems

University of Minnesota Ph.D. dissertation. September 2017. Major: Computer Science. Advisor: Ravi Janardan. 1 computer file (PDF); xi, 126 pages.The focus of this thesis is the topic of geometric intersection queries (GIQ) which has been very well studied by the computational geometry community and the database community. In a GIQ problem, the user is not interested in the entire input geometric dataset, but only in a small subset of it and requests an informative summary of that small subset of data. Formally, the goal is to preprocess a set A of n geometric objects into a data structure so that given a query geometric object q, a certain aggregation function can be applied efficiently on the objects of A intersecting q. The classical aggregation functions studied in the literature are reporting or counting the objects of A intersecting q. In many applications, the same set A is queried several times, in which case one would like to answer a query faster by preprocessing A into a data structure. The goal is to organize the data into a data structure which occupies a small amount of space and yet responds to any user query in real-time. In this thesis the study of the GIQ problems was conducted from the point-of-view of a computational geometry researcher. Given a model of computation and a GIQ problem, what are the best possible upper bounds (resp., lower bounds) on the space and the query time that can be achieved by a data structure? Also, what is the relative hardness of various GIQ problems and aggregate functions. Here relative hardness means that given two GIQ problems A and B (or, two aggregate functions f(A, q) and g(A, q)), which of them can be answered faster by a computer (assuming data structures for both of them occupy asymptotically the same amount of space)? This thesis presents results which increase our understanding of the above questions. For many GIQ problems, data structures with optimal (or near-optimal) space and query time bounds have been achieved. The geometric settings studied are primarily orthogonal range searching where the input is points and the query is an axes-aligned rectangle, and the dual setting of rectangle stabbing where the input is a set of axes-aligned rectangles and the query is a point. The aggregation functions studied are primarily reporting, top-k, and approximate counting. Most of the data structures are built for the internal memory model (word-RAM or pointer machine model), but in some settings they are generic enough to be efficient in the I/O-model as well