Data Structure For Point Location Research Articles

Windowing queries using Minkowski sum and their extension to MapReduce

Given a set of n segments and a query shape Q, the windowing length query asks for finding the sum of the lengths of the parts of the segments that lie inside Q. The popular places problem of a set of curves asks for the subset of the plane where each query shape centered at a point of that region intersects with at least f distinct curves. For square queries, an optimal $$O(n^2)$$ time algorithm and a matching lower bound exist. We solve the length query problem for convex polygons and disks as query shapes, with $$O(\log n+k)$$ query time and polynomial preprocessing time that depends on the complexity of the query shape. We define a new version of the problem of finding popular places in a set of trajectories where the center of a query is a popular place if the length of the curves inside that query is at least f and use our data structure to solve the original problem as well as this new version. Other than length queries, we solve reporting queries that return the set of intersected segments. For disk queries, we design a point-location data structure for congruent disks with $$O(\log n)$$ query time and $$O(n^3\log n)$$ preprocessing. We also give algorithms for computing the length query for c-packed curves, which are a class of curves for which the length of the curve inside a disk of radius r is upper-bounded by cr, where c is a constant. Also, we use length queries for polygons to approximate the minimum value c for which a curve is c-packed, if such a c exists. Our results extend to MRC and MPC models for MapReduce, where we address these problems on a set of x-monotone curves. The round complexities of our MapReduce algorithms are constant. In addition, we also implemented our popular places algorithms on trajectories on inputs as big as 15K points to evaluate the efficiency of our algorithms in practice.

Towards an Optimal Method for Dynamic Planar Point Location

We describe a fully dynamic linear-space data structure for point location in connected planar subdivisions, or more generally vertical ray shooting among nonintersecting line segments, that suppor...

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˜ 1/2 +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.

External Memory Planar Point Location with Logarithmic Updates

Point location is an extremely well-studied problem both in internal memory models and recently also in the external memory model. In this paper, we present an I/O-efficient dynamic data structure for point location in general planar subdivisions. Our structure uses linear space to store a subdivision with N segments. Insertions and deletions of segments can be performed in amortized O(log B N) I/Os and queries can be answered in $O(\log_{B}^{2} N)$ I/Os in the worst-case. The previous best known linear space dynamic structure also answers queries in $O(\log_{B}^{2} N)$ I/Os, but only supports insertions in amortized $O(\log_{B}^{2} N)$ I/Os. Our structure is also considerably simpler than previous structures.

I/O-efficient dynamic planar point location

We present an I/O-efficient dynamic data structure for point location in a general planar subdivision. Our structure uses O( N/ B) disk blocks of size B to store a subdivision of size N. Queries can be answered in O(log B 2 N) I/Os in the worst-case, and insertions and deletions can be performed in O(log 2 B N) and O(log B N) I/Os amortized, respectively. Part of our data structure is based on an external version of the so-called logarithmic method that allows for efficient dynamization of static external-memory data structures with certain characteristics. Another important part of our structure is an external data structure for vertical ray-shooting among line segments in the plane with endpoints on B +1 vertical lines, developed using an external version of dynamic fractional cascading. We believe that these methods could prove helpful in the development of other dynamic external memory data structures.

I/O-efficient point location using persistent B-trees

We present an external planar point location data structure that is I/O-efficient both in theory and practice.The developed structure uses linear space and answers a query in optimal O (log B N ) I/Os, where B is the disk block size. It is based on a persistent B-tree, and all previously developed such structures assume a total order on the elements in the structure. As a theoretical result of independent interest, we show how to remove this assumption.Most previous theoretical I/O-efficient planar point location structures are relatively complicated and have not been implemented. Based on a bucket approach, Vahrenhold and Hinrichs therefore developed a simple and practical, but theoretically non-optimal, heuristic structure. We present an extensive experimental evaluation that shows that, on a range of real-world Geographic Information Systems (GIS) data, our structure uses a similar number of I/Os as the structure of Vahrenhold and Hinrichs to answer a query. On a synthetically generated worst-case dataset our structure uses significantly fewer I/Os.

Linear Size Binary Space Partitions for Uncluttered Scenes

We describe a new and simple method for constructing binary space partitions (BSPs) in arbitrary dimensions. We also introduce the concept of uncluttered scenes, which are scenes with a certain property that we suspect many realistic scenes exhibit, and we show that our method constructs a BSP of size O(n) for an uncluttered scene consisting of n objects. The construction time is O(n log n) . Because any set of disjoint fat objects is uncluttered, our result implies an efficient method to construct a linear size BSP for fat objects. We use our BSP to develop a data structure for point location in uncluttered scenes. The query time of our structure is O( log n) , and the amount of storage is O(n) . This result can in turn be used to perform range queries with not-too-small ranges in scenes consisting of disjoint fat objects or, more generally, in so-called low-density scenes.

Dynamic Trees and Dynamic Point Location

This paper describes new methods for maintaining a point-location data structure for a dynamically changing monotone subdivision $\cal S$. The main approach is based on the maintenance of two interlaced spanning trees, one for $\cal S$ and one for the graph-theoretic planar dual of $\cal S$. Queries are answered by using a centroid decomposition of the dual tree to drive searches in the primal tree. These trees are maintained via the link-cut trees structure of Sleator and Tarjan [J. Comput. System Sci., 26 (1983), pp. 362--381], leading to a scheme that achieves vertex insertion/deletion in O(log n) time, insertion/deletion of k-edge monotone chains in O(log n + k) time, and answers queries in O(log2n ) time, with O(n) space, where n is the current size of subdivision $\cal S$. The techniques described also allow for the dual operations expand and contract to be implemented in O(log n) time, leading to an improved method for spatial point location in a 3-dimensional convex subdivision. In addition, the interlaced-tree approach is applied to on-line point location (where one builds $\cal S$ incrementally), improving the query bound to $O(\log n\log\log n)$ time and the update bounds to O(1)amortized time in this case. This appears to be the first on-line method to achieve a polylogarithmic query time and constant update time.

Range searching in low-density environments

Abstract We define a set of arbitrarily-shaped objects in R d to be a low-density environment if any axis-parallel hypercube intersects only few objects of comparable or larger size. Generalizing and simplifying previous results for fat objects, we present a data structure for point location in a low-density environment, and we show how this data structure can be extended to perform range search queries with query ranges of size comparable to the smallest object.

Point location in zones of k-flats in arrangements

Let A( H) be the arrangement of a set H of n hyperplanes in d-space. A k-flat is a k-dimensional affine subspace of d-space. The zone of a k-flat f with respect to H is the set of all faces in A( H) that intersect f. this paper we study some problems on zones of k-flats. Our most important result is a data structure for point location in the zone of a k-flat. This structure uses O(n ⌊ d 2 ⌋+ε +n k+ε) preprocessing time and space and has a query time of O(log 2 n). We also show how to test efficiently whether two flats are visible from each other with respect to a set of hyperplanes. Then point location in m faces in arrangements is studied. Our data structure for this problem has size O(n ⌊ d 2 ⌋+ε m ⌈ d 2 ⌉ d ) and the query time is O(log 2 n).

A Unified Approach to Dynamic Point Location, Ray shooting, and Shortest Paths in Planar Maps

We describe a new technique for dynamically maintaining the trapezoidal decomposition of a connected planar map $\cal M$ with $n$ vertices and apply it to the development of a unified dynamic data structure that supports point-location, ray-shooting, and shortest-path queries in $\cal M$. The space requirement is $O(n\log n)$. Point-location queries take time $O(\log n)$. Ray-shooting and shortest-path queries take time $O(\log^3 n)$ (plus $O(k)$ time if the $k$ edges of the shortest path are reported in addition to its length). Updates consist of insertions and deletions of vertices and edges, and take $O(\log^3 n)$ time (amortized for vertex updates). This is the first polylog-time dynamic data structure for shortest-path and ray-shooting queries. It is also the first dynamic point-location data structure for connected planar maps that achieves optimal query time.

DYNAMIZATION OF THE TRAPEZOID METHOD FOR PLANAR POINT LOCATION IN MONOTONE SUBDIVISIONS

We present a fully dynamic data structure for point location in a monotone subdivision, based on the trapezoid method. The operations supported are insertion and deletion of vertices and edges, and horizontal translation of vertices. Let n be the current number of vertices of the subdivision. Point location queries take O( log n) time, while updates take O ( log 2 n) time (amortized for vertex insertion/deletion and worst-case for the other updates). The space requirement is O(n log n). This is the first fully dynamic point location data structure for monotone subdivisions that achieves optimal query time.

Efficient Point Location in a Convex Spatial Cell-Complex

In this paper a new approach is proposed to point-location in a three-dimensional cell-complex $\mathcal{P}$, which may be viewed as a nontrivial generalization of a corresponding two-dimensional technique due to Sarnak and Tarjan. Specifically, in a space-sweep of $\mathcal{P}$, the intersections of the sweep-plane with $\mathcal{P}$ occurring in a given slab, i.e., between two consecutive vertices, are topologically conformal planar subdivisions. If the sweep direction is viewed as time, the descriptions of the various slabs are distinct “versions” of a two-dimensional point-location data structure, dynamically updated each time a vertex is swept. Combining the persistence-addition technique of Driscoll, Sarnak, Sleator, and Tarjan [J. Comput. System. Sci., 38 (1989), pp. 86–124] with the recently discovered dynamic structure for planar point-location in monotone subdivisions, a method with query time $O(\log ^2 N)$ and space $O(N\log ^2 N)$ for point-location in a convex cell-complex with N facets is obtained.

An incremental reconstruction method for dynamic planar point location

We present a fully dynamic technique for point location in triangulations that allows a tradeoff between query and update time, and can be used in conjunction with any of the known static point location data structures. Let S be a triangulation whose current number of vertices is n. We show that for any smooth nondecreasing integer function b(n) with 2 ⩽b(n)⩽ n , there exists a dynamic point location data structure for S with space O( n), query time O( ( log 2n) log b(n) ) , and update time O( ( log 2n)b(n) log b(n) ) O((log 2 n) b( n)/log b( n)).