Approximating the diameter of a set of points in the Euclidean space (original) (raw)

Fast Approximation and Randomized Algorithms for Diameter

We consider approximation of diameter of a set S of n points in dimension m. Egecioglu and Kalantari [6] have shown that given any p ∈ S, by computing its farthest in S, say q, and in turn the farthest point of q, say q ′ , we have diam(S) ≤ √ 3 d(q, q ′). Furthermore, iteratively replacing p with an appropriately selected point on the line segment pq, in at most t ≤ n additional iterations, the constant bound factor is improved to c * = 5 − 2 √ 3 ≈ 1.24. Here we prove when m = 2, t = 1. This suggests in practice a few iterations may produce good solutions in any dimension. Here we also propose a randomized version and present large scale computational results with these algorithm for arbitrary m. The algorithms outperform many existing algorithms. On sets of data as large as 1, 000, 000 points, the proposed algorithms compute solutions to within an absolute error of 10 −4 .

An Optimal Deterministic Algorithm for Computing the Diameter of a Three-Dimensional Point Set

Discrete & Computational Geometry, 2001

We describe a deterministic algorithm for computing the diameter of a finite set of points in R 3 , that is, the maximum distance between any pair of points in the set. The algorithm runs in optimal time O(n log n) for a set of n points. The first optimal, but randomized, algorithm for this problem was proposed more than 10 years ago by Clarkson and Shor [11] in their groundbreaking paper on geometric applications of random sampling. Our algorithm is relatively simple except for a procedure by Matoušek [25] for the efficient deterministic construction of epsilon-nets. This work improves previous deterministic algorithms by Ramos [31] and Bespamyatnikh [7], both with running time O(n log 2 n). The diameter algorithm appears to be the last one in Clarkson and Shor's paper that up to now had no deterministic counterpart with a matching running time.

A note on diameters of point sets

Optimization Letters, 2010

Relationships between the diameter of a set of n points in the plane at mutual distance at least one, the diameter of an equilateral n-gon and the radius of a circle including n unit disks are explored. Upper bounds on the minimal diameter of a point set at mutual distance at least one are presented for up to 30 points.

Better Approximation Algorithms for the Graph Diameter

Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, 2013

Vassilevska W. [STOC 13] show that inÕ (m √ n) time, one can compute for each v ∈ V in an undirected graph, an estimate e (v) for the eccentricity (v) such that max {R, 2 /3 • (v)} ≤ e (v) ≤ min {D, 3 /2 • (v)} where R = minv (v) is the radius of the graph. Here we improve the approximation guarantee by showing that a variant of the same algorithm can achieve estimates (v) with 3 /5 • (v) ≤ (v) ≤ (v).

Finding k Points with Minimum Diameter and Related Problems

Journal of Algorithms, 1991

Let S be a set consisting of n points in the plane. We consider the problem of finding k points of S that form a "small" set under some given measure, and present efficient algorithms for several natural measures including the diameter and the variance. ej 1991 Academic Press, 1nc.

Convergent bounds on the euclidean distance

2011

Given a set V of n vectors in d-dimensional space, we provide an efficient method for computing quality upper and lower bounds of the Euclidean distances between a pair of vectors in V. For this purpose, we define a distance measure, called the MS-distance, by using the mean and the standard deviation values of vectors in V. Once we compute the mean and the standard deviation values of vectors in V in O(dn) time, the MS-distance provides upper and lower bounds of Euclidean distance between any pair of vectors in V in constant time. Furthermore, these bounds can be refined further in such a way to converge monotonically to the exact Euclidean distance within d refinement steps. An analysis on a random sequence of refinement steps shows that the MS-distance provides very tight bounds in only a few refinement steps. The MS-distance can be used to various applications where the Euclidean distance is used to measure the proximity or similarity between objects. We provide experimental results on the nearest and the farthest neighbor searches.

Approximating Euclidean distances by small degree graphs

Discrete & Computational Geometry, 1994

Given an undirected edge-weighted graph G = (V, E), a subgraph G' = (IT, E') is a t-spanner of G if, for every u, v ~ 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 approximating the distances among points of a Euclidean metric space: given a finite set V of points in ~a, we want to construct a sparse t-spanner of the complete weighted graph induced by V. The weight of an edge in these graphs is the Euclidean distance between the endpoints of the edge. We show by a simple greedy argument that, for any t > 1 and any V c R a, a t-spanner G of V exists such that G has degree bounded by a function of d and r The analysis of our bounded degree spanners improves over previously known upper bounds on the minimum number of edges of Euclidean t-spanners, even compared with spanners of bounded average degree. Our results answer two open problems, one proposed by Vaidya and the other by Keil and Gutwin. The main result of the paper concerns the case of dimension d = 2. It is fairly easy to see that, for some t (t > 7.6), t-spanners of maximum degree 6 exist for any set of points in the Euclidean plane, but it was not known that degree 5 would suffice. We prove that for some (fixed) t, t-spanners of degree 5 exist for any set of points in the plane. We do not know if 5 is the best possible upper bound on the degree. * This research was supported by Conselho Nacional de Desenvolvimento Cientifico e Tecnol6gico, Proc 203039/87.4 (Brazil). 214 J. Soares shortest path between x and y. We say that a subgraph G' = (V, E') (with the same weights on E') is a t-spanner of G if, for every x, y 6 V, dG,(x, y) < t" da(x, y). The number t is a measure of how well G' approximates G with respect to the distances. The construction of t-spanners has received recent attention in several works: [2], [3], [5], [8], [9], [11], and [18], among others. Given a set V ~ •a the complete Euclidean graph on V is the complete graph on V where each edge weight is the Euclidean distance I[x-Y]I. In this paper we consider the problem of constructing bounded degree spanners of complete Euclidean graphs. For brevity we write t-spanner of V instead of t-spanner of the complete Euclidean graph on V. Let A(G) denote the maximum degree of a graph G. Dobkin et al. [5] mention that Feder and others had shown that, for some fixed t and for any set V of points in the Euclidean plane, a t-spanner G of V exists such that A(G) < 7. Then they ask what would be the minimum A for which such a result is possible? This paper has a partial answer to this question. Our main result (Section 4) is that, for some fixed t, t-spanners with A < 5 exist. Nisan [10] has proved the same for A < 6. Section 2 contains the basic algorithm used to construct bounded degree t-spanners. Although the algorithm has been used before by Althrfer et al. [1] and Soares [16] to construct t-spanners for arbitrary graphs, it was not known that the algorithm also constructs bounded degree spanners for complete Euclidean graphs. Section 3 contains a brief analysis of the problem when V is in d-dimensional Euclidean space. We show that, for any t > 1 and any V c ~d, a t-spanner G of V exists where A(G) is bounded by a function that depends only on d and t. This answers a question proposed by Keil and Gutwin in [8]. This bound on the maximum degree implies an improvement on the previously known upper bounds on the number of edges sufficient to build Euclidean spanners. Then we show that, for each dimension d, the least A(G) for which our algorithm constructs Od(1)spanners coincides with the kissing number in dimension d. (Od(1) denotes some function of d, i.e., a constant for each d.) Section 4 contains our main result, the construction of O(1)-spanners of degree 5 for any set of points in the Euclidean plane.

Towards Sub-Quadratic Diameter Computation in Geometric Intersection Graphs

arXiv (Cornell University), 2022

We initiate the study of diameter computation in geometric intersection graphs from the fine-grained complexity perspective. A geometric intersection graph is a graph whose vertices correspond to some shapes in d-dimensional Euclidean space, such as balls, segments, or hypercubes, and whose edges correspond to pairs of intersecting shapes. The diameter of a graph is the largest distance realized by a pair of vertices in the graph. Computing the diameter in near-quadratic time is possible in several classes of intersection graphs [Chan and Skrepetos 2019], but it is not at all clear if these algorithms are optimal, especially since in the related class of planar graphs the diameter can be computed in O(n 5/3) time [Cabello 2019, Gawrychowski et al. 2021]. In this work we (conditionally) rule out sub-quadratic algorithms in several classes of intersection graphs, i.e., algorithms of running time O(n 2−δ) for some δ > 0. In particular, there are no sub-quadratic algorithms already for fat objects in small dimensions: unit balls in R 3 or congruent equilateral triangles in R 2. For unit segments and congruent equilateral triangles, we can even rule out strong sub-quadratic approximations already in R 2. It seems that the hardness of approximation may also depend on dimensionality: for axis-parallel unit hypercubes in R 12 , distinguishing between diameter 2 and 3 needs quadratic time (ruling out (3/2−ε)-approximations), whereas for axis-parallel unit squares, we give an algorithm that distinguishes between diameter 2 and 3 in near-linear time. Note that many of our lower bounds match the best known algorithms up to sub-polynomial factors. Ultimately, this fine-grained perspective may enable us to determine for which shapes we can have efficient algorithms and approximation schemes for diameter computation.

Maintaining the Approximate Width of a Set of Points in the Plane

The width of a set of n points in the plane is the smallest distance between two parallel lines that enclose the set. We maintain the set of points under insertions and deletions of points and we are able to report an approximation of the width of this dynamic point set. Our data structure takes linear space and allows for reporting the approximation with relative accuracy in O(p 1= log n) time; and the update time is O(log 2 n). The method uses the tentative prune-and-search strategy of Kirkpatrick and Snoeyink.

On approximating the maximum diameter ratio of graphs

Discrete Mathematics, 2002

It is proved that computing the maximum diameter ratio (also known as the local density) of a graph is APX-complete. The related problem of nding a maximum subgraph of a xed diameter d 1 i s proved to be even harder to approximate.