Finding Pairwise Intersections Inside a Query Range

We study the following problem: preprocess a set O of objects into a data structure that allows us to efficiently report all pairs of objects from O that intersect inside an axis-aligned query range Q. We present data structures of size $O(n({\rm polylog} n))$ and with query time $O((k+1)({\rm polylog} n))$ time, where k is the number of reported pairs, for two classes of objects in the plane: axis-aligned rectangles and objects with small union complexity. For the 3-dimensional case where the objects and the query range are axis-aligned boxes in R^3, we present a data structures of size $O(n\sqrt{n}({\rm polylog} n))$ and query time $O((\sqrt{n}+k)({\rm polylog} n))$. When the objects and query are fat, we obtain $O((k+1)({\rm polylog} n))$ query time using $O(n({\rm polylog} n))$ storage

Algorithms for Stable Matching and Clustering in a Grid

We study a discrete version of a geometric stable marriage problem originally proposed in a continuous setting by Hoffman, Holroyd, and Peres, in which points in the plane are stably matched to cluster centers, as prioritized by their distances, so that each cluster center is apportioned a set of points of equal area. We show that, for a discretization of the problem to an $n\times n$ grid of pixels with $k$ centers, the problem can be solved in time $O(n^2 \log^5 n)$, and we experiment with two slower but more practical algorithms and a hybrid method that switches from one of these algorithms to the other to gain greater efficiency than either algorithm alone. We also show how to combine geometric stable matchings with a $k$-means clustering algorithm, so as to provide a geometric political-districting algorithm that views distance in economic terms, and we experiment with weighted versions of stable $k$-means in order to improve the connectivity of the resulting clusters.Comment: 23 pages, 12 figures. To appear (without the appendices) at the 18th International Workshop on Combinatorial Image Analysis, June 19-21, 2017, Plovdiv, Bulgari

A Framework for Index Bulk Loading and Dynamization

In this paper we investigate automated methods for externalizing internal memory data structures. We consider a class of balanced trees that we call weight-balanced partitioning trees (or wp-trees) for indexing a set of points in Rd. Well-known examples of wp-trees include fed-trees, BBD-trees, pseudo quad trees, and BAR trees. These trees are defined with fixed degree and are thus suited for internal memory implementations. Given an efficient wp-tree construction algorithm, we present a general framework for automatically obtaining a new dynamic external data structure. Using this framework together with a new general construction (bulk loading) technique of independent interest, we obtain data structures with guaranteed good update performance in terms of I /O transfers. Our approach gives considerably improved construction and update I/O bounds of e.g. fed-trees and BBD-trees

The Skip Quadtree: A Simple Dynamic Data Structure for Multidimensional Data

We present a new multi-dimensional data structure, which we call the skip quadtree (for point data in R^2) or the skip octree (for point data in R^d, with constant d>2). Our data structure combines the best features of two well-known data structures, in that it has the well-defined "box"-shaped regions of region quadtrees and the logarithmic-height search and update hierarchical structure of skip lists. Indeed, the bottom level of our structure is exactly a region quadtree (or octree for higher dimensional data). We describe efficient algorithms for inserting and deleting points in a skip quadtree, as well as fast methods for performing point location and approximate range queries.Comment: 12 pages, 3 figures. A preliminary version of this paper appeared in the 21st ACM Symp. Comp. Geom., Pisa, 2005, pp. 296-30

Maximum-Area Rectangles in a Simple Polygon

We study the problem of finding maximum-area rectangles contained in a polygon in the plane. There has been a fair amount of work for this problem when the rectangles have to be axis-aligned or when the polygon is convex. We consider this problem in a simple polygon with n vertices, possibly with holes, and with no restriction on the orientation of the rectangles. We present an algorithm that computes a maximum-area rectangle in O(n^3 log n) time using O(kn^2) space, where k is the number of reflex vertices of P. Our algorithm can report all maximum-area rectangles in the same time using O(n^3) space. We also present a simple algorithm that finds a maximum-area rectangle contained in a convex polygon with n vertices in O(n^3) time using O(n) space

Finding pairwise intersections inside a query range

We study the following problem: preprocess a set O of objects into a data structure that allows us to efficiently report all pairs of objects from O that intersect inside an axis-aligned query range Q . We present data structures of size O(n⋅polylogn) and with query time O((k+1)⋅polylogn) time, where k is the number of reported pairs, for two classes of objects in R2 : axis-aligned rectangles and objects with small union complexity. For the 3-dimensional case where the objects and the query range are axis-aligned boxes in R3 , we present a data structure of size O(nn−−√⋅polylogn) and query time O((n−−√+k)⋅polylogn) . When the objects and query are fat, we obtain O((k+1)⋅polylogn) query time using O(n⋅polylogn) storage

Fully Dynamic Maximum Independent Sets of Disks in Polylogarithmic Update Time

A fundamental question in computational geometry is for a dynamic collection of geometric objects in Euclidean space, whether it is possible to maintain a maximum independent set in polylogarithmic update time. Already, for a set of intervals, it is known that no dynamic algorithm can maintain an exact maximum independent set with sublinear update time. Therefore, the typical objective is to explore the trade-off between update time and solution size. Substantial efforts have been made in recent years to understand this question for various families of geometric objects, such as intervals, hypercubes, hyperrectangles, and fat objects. We present the first fully dynamic approximation algorithm for disks of arbitrary radii in the plane that maintains a constant-factor approximate maximum independent set in polylogarithmic update time. First, we show that for a fully dynamic set of $n$ unit disks in the plane, a $12$-approximate maximum independent set can be maintained with worst-case update time $O(\log^2 n)$, and optimal output-sensitive reporting. Moreover, this result generalizes to fat objects of comparable sizes in any fixed dimension $d$, where the approximation ratio depends on the dimension and the fatness parameter. Our main result is that for a fully dynamic set of disks of arbitrary radii in the plane, an $O(1)$-approximate maximum independent set can be maintained in polylogarithmic expected amortized update time.Comment: Abstract is shortened to meet Arxiv's requirement on the number of character