Matthew Dickerson

Algorithms for proximity problems in higher dimensions

Abstract We present algorithms for five interdistance enumeration problems that take as input a set S of n points in R d (for a fixed but arbitrary dimension d ) and as output enumerate pairs of points in S satisfying various conditions. We present: an O ( n log n + k ) time and O ( n ) space algorithm that takes as additional input a distance δ and outputs all k pairs of points in S separated by a distance of δ or less; an O ( n log n + k log k ) time and O ( n + k ) space algorithm that enumerates in non-decreasing order the k closest pairs of points in S ; an O ( n log n + k ) time algorithm for the same problem without any order restrictions; an O ( nk log n ) time and O ( n ) space algorithm that enumerates in nondecreasing order the nk pairs representing the k nearest neighbors of each point in S ; and an O ( n log n + kn ) time algorithm for the same problem without any order restrictions. The algorithms combine a modification of the planar approach of Dickerson, Drysdale, and Sack [11] with the method of Bern, Eppstein, and Gilbert [3] for augmenting a point set to have a linear size bounded degree Delaunay triangulation. Thus, in addition to providing new solutions to these problems, the paper also shows how the Delaunay triangulation can be used as the underlying data structure in a unified approach to proximity problems even in higher dimensions.

SIMPLE ALGORITHMS FOR ENUMERATING INTERPOINT DISTANCES AND FINDING k NEAREST NEIGHBORS

We present an O(n log n+k log k) time and O(n+k) space algorithm which takes as input a set of n points in the plane and enumerates the k smallest distances between pairs of points in nondecreasing order. We also present an O(n log n+kn log k) solution to the problem of finding the k nearest neighbors for each of n points. Both algorithms are conceptually very simple, are easy to implement, and are based on a common data structure: the Delaunay triangulation. Variants of the algorithms work for any convex distance function metric.

A (usually?) connected subgraph of the minimum weight triangulation

that computes a subgraph of the minimum weight triangulation (A4WL”) of a general point set. The algorithm works by finding a collection of edges guaranteed to be in any locally minimal triangulation. We call this subgraph the LMT-skeleton. We also give two variants of our algorithm that produce a more complete subgraph of the MWT: an extended LMT-skeleton requiring worst case 0(n6) time and O(n3) space; and a modified LMT-skeleton requiring 0(n2) time and 0( TZ1”5) space in the expected case for uniform distributions.

2-Point site Voronoi diagrams

In this paper, we define a new type of a planar distance function from a point to a pair of points. We focus on a few such distance functions, analyze the structure and complexity of the corresponding nearest- and furthest-neighbor Voronoi diagrams (in which every region is defined by a pair of point sites), and show how to compute the diagrams efficiently.

Voronoi diagrams for convex polygon-offset distance functions

In this paper we develop the concept of a convexpolygon-offset distance function. Using offset as a notion of distance, we show how to compute the corresponding nearest- and furthest-site Voronoi diagrams of point sites in the plane. We provide near-optimal deterministicO(n(logn + log2m) +m)-time algorithms, wheren is the number of points andm is the complexity of the underlying polygon, for computing compact representations of both diagrams.

Offset-polygon annulus placement problems

The @d-annulus of a polygon P is the closed region containing all points in the plane at distance at most @d from the boundary of P. An inner (resp., outer) @d-offset polygon is the polygon defined by the inner (resp., outer) boundary of its @d-annulus. In this paper we address three major problems of covering a given point set S by an offset version or a polygonal annulus of a polygon P. First, the Maximum Cover objective is, given a value of @d, to cover as many points from S as possible by the @d-offset (or by the @d-annulus) of P, allowing translation and rotation. Second, the Containment problem is to minimize the value of @d such that there is a rigid transformation of the @d-offset (or the @d-annulus) of P that covers all points from S. Third, in the Partial Containment problem we seek the minimum offset of P covering k=<|S| points. These problems arise in many applications where one needs to match a given polygonal figure (a known model) to a set of points (usually, obtained measures). We address several variants of these problems, including convex and simple polygons, as well as polygons with holes and sets of polygons, and obtain algorithms with low-degree polynomial running times in all cases.

Translating a convex polygon to contain a maximum number of points

Abstract Given a set S of n points in the plane and a convex polygon P with m vertices, we consider the problem of finding a translation of P that contains the maximum number of points in S . We present two different solutions. Our first algorithm uses standard line-sweep techniques and requires O( nk log( nm ) + m ) time where k is the maximum number of points contained. Our second algorithm requires O( nk log( mk ) + m ) time, which is the asymptotically fastest known algorithm for this problem. Both algorithms require optimal O( m + n ) space. The algorithms also solve in the same running time the bichromatic variant of the problem, where we are given two point sets A and B and the goal is to maximize the number of points covered from A while minimizing the number of points covered from B .

Optimal placement of convex polygons to maximize point containment

Given a convex polygon P with m vertices and a set S of n points in the plane, we consider the problem of finding a placement of P (allowing both translation and rotation) that contains the maximum number of points in S. We present first an algorithm requiring O(n^2km^2log(mn)) time and O(n + m) space, where k is the maximum number of points contained. We then give a refinement that makes use of bucketing to improve the running time to O(nk^2c^2m^2log(mk)), where c is the ratio of length to width of the polygon. This provides an improvement over the best previously known algorithm linear in n when k is large (@Q(n)) and a cubic when k is small. We also show that the algorithm can be extended to solve bichromatic and general weighted variants of the problem. The algorithm is self-contained and utilizes the geometric properties of the containing regions in the parameter space of transformations.

A Large Subgraph of the Minimum Weight Triangulation

Abstract. We present an O(n4)-time and O(n2)-space algorithm that computes a subgraph of the minimum weight triangulation (MWT) of a general point set. The algorithm works by finding a collection of edges guaranteed to be in any locally minimal triangulation. We call this subgraph the LMT-skeleton. We also give a variant called the modifiedLMT-skeleton that is both a more complete subgraph of the MWT and is faster to compute requiring only O(n2) time and O(n) space in the expected case for uniform distributions. Several experimental implementations of both approaches have shown that for moderate-sized point sets (up to 350 points^1) the skeletons are connected, enabling an efficient completion of the exact MWT. We are thus able to compute the MWT of substantially larger random point sets than have previously been computed. ^1Though in this paper we summarize some empirical findings for input sets of up to 350 points, a variant of the algorithm has been implemented and tested on up to 40,000 points producing connected subgraphs [2].

Fast greedy triangulation algorithms

The greedy triangulation of a set