Output-sensitive well-separated pair decompositions for dynamic point sets

We present a near linear time algorithm for constructing hierarchical nets in finite metric spaces with constant doubling dimension. This data-structure is then applied to obtain improved algorithms for the following problems: Approximate nearest neighbor search, well-separated pair decomposition, spanner construction, compact representation scheme, doubling measure, and computation of the (approximate) Lipschitz constant of a function. In all cases, the running (preprocessing) time is near-linear and the space being used is linear.

A new method is proposed in order to reconstruct the geometrical configuration of a large points set using minimal information. The paper develops algorithms based on graph and kinematics theories to determine the minimum number of distances, needed to uniquely represent n points in d-dimensional Euclidean space. Therefore, it is found that this theoretical minimum is d(n-2)+1 interpoint distances. The method is evaluated, on the basis of basic parameters, by means of Monte Carlo simulation using genetic algorithms for better optimization procedures. This evaluation takes into account the real case where the metric informations are interpoint dissimilarities instead of exact Euclidean distances. Two applications on real data successfully illustrate the efficiency of the method. Finally, on the basis of Monte Carlo results, the authors provide some practical recommendations to experimenters who wish to use the method in order to scale a many-objects set.

We introduce and study a network resource management problem that is a special case of non-metric k-median, naturally arising in cross platform scheduling and cloud computing. In the continuous d-dimensional container selection problem, we are given a set C of input points in d-dimensional Euclidean space, for some d >= 2, and a budget k. An input point p can be assigned to a "container point" c only if c dominates p in every dimension. The assignment cost is then equal to the L1-norm of the container point. The goal is to find k container points in the d-dimensional space, such that the total assignment cost for all input points is minimized. The discrete variant of the problem has one key distinction, namely, the container points must be chosen from a given set F of points. For the continuous version, we obtain a polynomial time approximation scheme for any fixed dimension d>= 2. On the negative side, we show that the problem is NP-hard for any d>=3. We further show that the discrete version is significantly harder, as it is NP-hard to approximate without violating the budget k in any dimension d>=3. Thus, we focus on obtaining bi-approximation algorithms. For d=2, the bi-approximation guarantee is (1+epsilon,3), i.e., for any epsilon>0, our scheme outputs a solution of size 3k and cost at most (1+epsilon) times the optimum. For fixed d>2, we present a (1+epsilon,O((1/epsilon)log k)) bi-approximation algorithm.

We study SET COVER for orthants: Given a set of points in a d-dimensional Euclidean space and a set of orthants of the form (-infty,p_1] x ... x (-infty,p_d], select a minimum number of orthants so that every point is contained in at least one selected orthant. This problem draws its motivation from applications in multi-objective optimization problems. While for d=2 the problem can be solved in polynomial time, for d>2 no algorithm is known that avoids the enumeration of all size-k subsets of the input to test whether there is a set cover of size k. Our contribution is a precise understanding of the complexity of this problem in any dimension d >= 3, when k is considered a parameter: - For d=3, we give an algorithm with runtime n^O(sqrt{k}), thus avoiding exhaustive enumeration. - For d=3, we prove a tight lower bound of n^Omega(sqrt{k}) (assuming ETH). - For d >=slant 4, we prove a tight lower bound of n^Omega(k) (assuming ETH). Here n is the size of the set of points plus the size of the set of orthants. The first statement comes as a corollary of a more general result: an algorithm for SET COVER for half-spaces in dimension 3. In particular, we show that given a set of points U in R^3, a set of half-spaces D in R^3, and an integer k, one can decide whether U can be covered by the union of at most k half-spaces from D in time |D|^O(sqrt{k})* |U|^O(1). We also study approximation for SET COVER for orthants. While in dimension 3 a PTAS can be inferred from existing results, we show that in dimension 4 and larger, there is no 1.05-approximation algorithm with runtime f(k)* n^o(k) for any computable f, where k is the optimum.

This study examines various algorithms for computing the convex hull of a set of n points in a d-dimensional space. Convex hulls are fundamental in computational geometry and are applied in computer graphics, pattern recognition, and computational biology. Such convex hulls can also be useful in symmetry problems. For instance, when points are arranged symmetrically, the convex hull is also likely to be symmetrically shaped, which can be useful for object recognition in computer vision or pattern recognition. The focus is primarily on two-dimensional algorithms, including well-known methods like Gift Wrapping, Graham Scan, Divide and Conquer, QuickHull, TORCH, Kirkpatrick–Sediel, and Chan’s algorithms. These algorithms vary in terms of time complexity and scalability to higher dimensions. This study is extended to three-dimensional convex hull algorithms, such as NAW, randomized insertion, and parallelized versions, such as CudaHull and CudaChain. This study aimed to elucidate the operational principles, step-by-step procedures, and comparative time complexities of each algorithm. The implementation in Python facilitates a detailed comparison of the algorithmic performance through stepwise analysis and graphical outputs. The ultimate goal is to provide insights into the strengths and weaknesses of each algorithm under various scenarios, thereby offering a comprehensive guide for practical implementation.

The Tukey (or halfspace) depth extends nonparametric methods toward multivariate data. The multivariate analogues of the quantiles are the central regions of the Tukey depth, defined as sets of points in the d-dimensional space whose Tukey depth exceeds given thresholds k. We address the problem of fast and exact computation of those central regions. First, we analyze an efficient Algorithm (A) from Liu, Mosler, and Mozharovskyi, and prove that it yields exact results in dimension d = 2, or for a low threshold k in arbitrary dimension. We provide examples where Algorithm (A) fails to recover the exact Tukey depth region for d > 2, and propose a modification that is guaranteed to be exact. We express the problem of computing the exact central region in its dual formulation, and use that viewpoint to demonstrate that further substantial improvements to our algorithm are unlikely. An efficient C++ implementation of our exact algorithm is freely available in the R package TukeyRegion.

Arrangements of planar curves are fundamental structures in computational geometry. We describe the recent developments in the arrangement package of Cgal, the Computational Geometry Algorithms Library, making it easier to use, to extend and to adapt to a variety of applications. This improved flexibility of the code does not come at the expense of efficiency as we mainly use generic-programming techniques, which make dexterous use of the compilation process. To the contrary, we expedited key operations as we demonstrate by experiments.

Lower envelopes are fundamental structures in computational geometry, which have many applications, such as computing general Voronoi diagrams and performing hidden surface removal in computer graphics. We present a generic, robust and efficient implementation of the divide-and-conquer algorithm for computing the envelopes of surfaces in IR3. To the best of our knowledge, this is the first exact implementation that computes envelopes in three-dimensional space. Our implementation is based on CGAL (the Computational Geometry Algorithms Library) and is designated as a CGAL package. The separation of topology and geometry in our solution allows for the reuse of the algorithm with different families of surfaces, provided that a small set of geometric objects and operations on them is supplied. We used our algorithm to compute the lower and upper envelope for several types of surfaces. Exact arithmetic is typically slower than floating-point arithmetic, especially when higher order surfaces are involved. Since our implementation follows the exact geometric computation paradigm, we minimize the number of geometric operations, and by that significantly improve the performance of the algorithm in practice. Our experiments show interesting phenomena in the behavior of the divide-and-conquer algorithm and the combinatorics of lower envelopes of random surfaces.

Geometric spanner is a fundamental structure in computational geometry and plays an important role in many geometric networks design applications. In this paper, we consider a generalization of the classical geometric spanner problem (called segment spanner): Given a set S of disjoint 2-D segments, find a spanning network G with minimum size so that for any pair of points in S, there exists a path in G with length no more than t times their Euclidean distance. Based on a number of interesting techniques (such as weakly dominating set, strongly dominating set, and interval cover), we present an efficient algorithm to construct the segment spanner. Our approach first identifies a set of Steiner points in S, then construct a point spanner for them. Our algorithm runs in O(|Q| + n2 log n) time, where Q is the set of Steiner points. We show that Q is an O(1)-approximation in terms of its size when S is relatively "well" separated by a constant. For arbitrary rectilinear segments under L1 distance, the approximation ratio improves to 2.

Detecting approximate symmetries of discrete point subsets

A new method for the mapping of the relative locations of organisms as points in a 2—dimensional space is presented. The proposed method is based on a least—squares estimate of the coordinates using observed distances between the objects being mapped (rather than triangulation based on horizontal angles). Only a small subset of the possible interpoint distances need be measured in order to obtain a very accurate map. The method is applied to 3 problems in mapping of trees in a forest.

This paper presents new parallel algorithms for generating Euclidean minimum spanning trees and spatial clustering hierarchies (known as HDBSCAN*). Our approach is based on generating a well-separated pair decomposition followed by using Kruskal's minimum spanning tree algorithm and bichromatic closest pair computations. We introduce a new notion of well-separation to reduce the work and space of our algorithm for HDBSCAN*. We also give a new parallel divide-and-conquer algorithm for computing the dendrogram and reachability plots, which are used in visualizing clusters of different scale that arise for both EMST and HDBSCAN*. We show that our algorithms are theoretically efficient: they have work (number of operations) matching their sequential counterparts, and polylogarithmic depth (parallel time). We implement our algorithms and propose a memory optimization that requires only a subset of well-separated pairs to be computed and materialized, leading to savings in both space (up to 10x) and time (up to 8x). Our experiments on large real-world and synthetic data sets using a 48-core machine show that our fastest algorithms outperform the best serial algorithms for the problems by 11.13--55.89x, and existing parallel algorithms by at least an order of magnitude.

In \em monotone classification, the input is a set P of n points in d-dimensional space, where each point carries a label 0 or 1. A point p \em dominates another point q if the coordinate of p is at least that of q on every dimension. A \em monotone classifier is a function h mapping each d-dimensional point to $\0, 1 $, subject to the condition that $h(p) \ge h(q)$ holds whenever p dominates q. The classifier h \em mis-classifies a point $p \in P$ if $h(p)$ is different from the label of p. The \em error of h is the number of points in P mis-classified by h. The objective is to find a monotone classifier with a small error. The problem is fundamental to numerous database applications in entity matching, record linkage, and duplicate detection. This paper studies two variants of the problem. In the first \em active version, all the labels are hidden in the beginning; an algorithm must pay a unit cost to \em probe (i.e., reveal) the label of a point in P. We prove that $Omega(n)$ probes are necessary to find an optimal classifier even in one-dimensional space ($d=1$). On the other hand, given an arbitrary $\eps > 0$, we show how to obtain (with high probability) a monotone classifier whose error is worse than the optimum by at most a $1 + \eps$ factor, while probing $\tO(w/\eps^2)$ labels, where w is the dominance width of P and $\tO(.)$ hides a polylogarithmic factor. For constant $\eps$, the probing cost matches an existing lower bound up to an $\tO(1)$ factor. In the second \em passive version, the point labels in P are explicitly given; the goal is to minimize CPU computation in finding an optimal classifier. We show that the problem can be settled in time polynomial to both d and n.

Let G=(V,E) be an n-vertex connected graph with positive edge weights. A subgraph G′ = (V,E′) is a t-spanner of G if for all u, v e V,the weighted distance between u and v in G′ is at most t times the weighted distance between u and v in G. We consider the problem of constructing sparse spanners, and the weight, defined as the sum of the edge weights in the spanner. In this paper, we concentrate on constructing spanners of small weight.For an arbitrary positive edge-weighted graph G, for any t> 1, and any e>0, we show that a t-spanner of G with weight O(n(2+e)/(t-1>) •wt(MST) can be constructed in polynomial time. We also show that (log2n)-spanners of weight O(log n)•wt(MST) can be constructed.We then consider spanners for complete graphs induced by a set of points in d-dimensional real normed space. The weight of an edge xy is the norm of the xy vector. We show that for these graphs, t-spanners with total weight O(log n)•wt(MST) can be constructed in polynomial time.

The constrained orthogonal Procrustes problem is the least-squares problem that calls for a rotation matrix that optimally aligns two corresponding sets of points in d-dimensional Euclidean space. This problem generalizes to the so-called Wahba's problem which is the same problem with nonnegative weights. Given a dxd matrix M, solutions to these problems are intimately related to the problem of finding a dxd rotation matrix U that maximizes the trace of UM, i.e., that makes UM a matrix of maximal trace over rotations, and it is well known this can be achieved with a method based on the computation of the singular value decomposition (SVD) of M. As the main goal of this paper, we characterize dxd matrices of maximal trace over rotation matrices in terms of their eigenvalues, and for d = 2, 3, we show how this characterization can be used to determine whether a matrix is of maximal trace over rotation matrices. Finally, although depending only slightly on the characterization, as a secondary goal of the paper, for d = 2, 3, we identify alternative ways, other than the SVD, of obtaining solutions to the aforementioned problems.