A Static Optimality Transformation with Applications to Planar Point Location

Over the last decade, there have been several data structures that, given a planar subdivision and a probability distribution over the plane, provide a way for answering point location queries that is fine-tuned for the distribution. All these methods suffer from the requirement that the query distribution must be known in advance. We present a new data structure for point location queries in planar triangulations. Our structure is asymptotically as fast as the optimal structures, but it requires no prior information about the queries. This is a 2D analogue of the jump from Knuth's optimum binary search trees (discovered in 1971) to the splay trees of Sleator and Tarjan in 1985. While the former need to know the query distribution, the latter are statically optimal. This means that we can adapt to the query sequence and achieve the same asymptotic performance as an optimum static structure, without needing any additional information.Comment: 13 pages, 1 figure, a preliminary version appeared at SoCG 201

Entropy, Triangulation, and Point Location in Planar Subdivisions

A data structure is presented for point location in connected planar subdivisions when the distribution of queries is known in advance. The data structure has an expected query time that is within a constant factor of optimal. More specifically, an algorithm is presented that preprocesses a connected planar subdivision G of size n and a query distribution D to produce a point location data structure for G. The expected number of point-line comparisons performed by this data structure, when the queries are distributed according to D, is H + O(H^{2/3}+1) where H=H(G,D) is a lower bound on the expected number of point-line comparisons performed by any linear decision tree for point location in G under the query distribution D. The preprocessing algorithm runs in O(n log n) time and produces a data structure of size O(n). These results are obtained by creating a Steiner triangulation of G that has near-minimum entropy.Comment: 19 pages, 4 figures, lots of formula

A static optimality transformation with applications to planar point location

Proceedings of the 27th {ACM} Symposium on Computational Geometry, Paris, France, June 13-15, 2011info:eu-repo/semantics/publishe

Methods for Achieving Fast Query Times in Point Location Data Structures

Given a collection S of n line segments in the plane, the planar point location problem is to construct a data structure that can efficiently determine for a given query point p the first segment(s) in S intersected by vertical rays emanating out from p. It is well known that linear-space data structures can be constructed so as to achieve O(log n) query times. But applications, such as those common in geographic information systems, motivate a re-examination of this problem with the goal of improving query times further while also simplifying the methods needed to achieve such query times. In this paper we perform such a re-examination, focusing on the issues that arise in three different classes of point-location query sequences: ffl sequences that are reasonably uniform spatially and temporally (in which case the constant factors in the query times become critical), ffl sequences that are non-uniform spatially or temporally (in which case one desires data structures that adapt to s..

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