Sunil Arya

Euclidean spanners: short, thin, and lanky

Euclidean spanners are important data structures in geometric algorithm design, because they provide a means of approximating the complete Euclidean graph with only O(n) edges, so that the shortest path length between each pair of points is not more than a constant factor longer than the Euclidean distance between the points. In many applications of spanners, it is important that the spanner possess a number of additional properties: low tot al edge weight, bounded degree, and low diameter. Existing research on spanners has considered one property or the other. We show that it is possible to build spanners in optimal O (n log n) time and O(n) space that achieve optimal or near optimal tradeoffs between all combinations of these *Max-Planck-Institut fiir Informatik, D-66123 Saarbrucken, Germany. Email: {arya, michiel}@mpi-sb. mpg. de. Supported by the ESPRIT Basic Research Actions Program, under contract No. 7141 (project ALCOM 11). t Math Sciences Dept., The University of Memphis, Memphis, TN 38152. Supported in part by NSF Grant CCR9306822. E-mail: dasg@next 1.msci .memst . edu. i Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park, Maryland. Partially supported by NSF Grant CCR-93107O5. This work was done while visiting the Max-Planck-Institut fiir Informatik, Saarbriicken. E-mail: mount @cs. umd. edu. SQue~Tech, IIIC., 7600A Leesburg Pike, Falls Church, VA 22043. This work was done while visiting the Max-Planck-Institut fiir Informatik, Saarbriicken. E-mail: jsalowet!nvl, army .mil. Permission to copy without fee all or part of thk material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyri ht notice and the title of thq publication and, is date appear, a#notice is given that copyt~isby~n,sslon of the Ass@ationof Computing Machinery. o cop otherwise, or to republish, requires a fee ancf/or speci ic permission. STOC’ 95, Las Vegas, Nevada, USA @ 1995 ACM 0-89791 -718-9/95/0005..

Space-time tradeoffs for approximate nearest neighbor searching

3.50 properties. We achieve these results in large part because of a new structure, called the dumbbell tree which provides a method of decomposing a spanner into a constant number of trees, so that each of the O(n2) spanner paths is mapped entirely to a path in one of these trees.

Algorithms for fast vector quantization

Nearest neighbor searching is the problem of preprocessing a set of <i>n</i> point points in <i>d</i>-dimensional space so that, given any query point <i>q</i>, it is possible to report the closest point to <i>q</i> rapidly. In approximate nearest neighbor searching, a parameter ϵ > 0 is given, and a multiplicative error of (1 + ϵ) is allowed. We assume that the dimension <i>d</i> is a constant and treat <i>n</i> and ϵ as asymptotic quantities. Numerous solutions have been proposed, ranging from low-space solutions having space <i>O</i>(<i>n</i>) and query time <i>O</i>(log <i>n</i> + 1/ϵ^<i>d</i>−1) to high-space solutions having space roughly <i>O</i>((<i>n</i> log <i>n</i>)/ϵ^<i>d</i>) and query time <i>O</i>(log (<i>n</i>/ϵ)). We show that there is a single approach to this fundamental problem, which both improves upon existing results and spans the spectrum of space-time tradeoffs. Given a tradeoff parameter γ, where 2 ≤ γ ≤ 1/ϵ, we show that there exists a data structure of space <i>O</i>(<i>n</i>γ^<i>d</i>−1 log(1/ϵ)) that can answer queries in time <i>O</i>(log(<i>n</i>γ) + 1/(ϵγ)^{(<i>d</i>−1)/2}. When γ = 2, this yields a data structure of space <i>O</i>(<i>n</i> log (1/ϵ)) that can answer queries in time <i>O</i>(log <i>n</i> + 1/ϵ^{(<i>d</i>−1)/2}). When γ = 1/ϵ, it provides a data structure of space <i>O</i>((<i>n</i>/ϵ^<i>d</i>−1)log(1/ϵ)) that can answer queries in time <i>O</i>(log(<i>n</i>/ϵ)). Our results are based on a data structure called a (<i>t</i>,ϵ)-AVD, which is a hierarchical quadtree-based subdivision of space into cells. Each cell stores up to <i>t</i> representative points of the set, such that for any query point <i>q</i> in the cell at least one of these points is an approximate nearest neighbor of <i>q</i>. We provide new algorithms for constructing AVDs and tools for analyzing their total space requirements. We also establish lower bounds on the space complexity of AVDs, and show that, up to a factor of <i>O</i>(log (1/ϵ)), our space bounds are asymptotically tight in the two extremes, γ = 2 and γ = 1/ϵ.

Randomized and deterministic algorithms for geometric spanners of small diameter

This paper shows that if one is willing to relax the requirement of finding the true nearest neighbor, it is possible to achieve significant improvements in running time and at only a very small loss in the performance of the vector quantizer. The authors present three algorithms for nearest neighbor searching: standard and priority k-d tree search algorithms and a neighborhood graph search algorithm in which a directed graph is constructed for the point set and edges join neighboring points.<<ETX>>

Efficient construction of a bounded-degree spanner with low weight

Let S be a set of n points in IR/sup d/ and let t>1 be a real number. A t-spanner for S is a directed graph having the points of S as its vertices, such that for any pair p and q of points there is a path from p to q of length at most t times the Euclidean distance between p and p. Such a path is called a t-spanner path. The spanner diameter of such a spanner is defined as the smallest integer D such that for any pair p and q of points there is a t-spanner path from p to q containing at most D edges. Randomized and deterministic algorithms are given for constructing t-spanners consisting of O(n) edges and having O(log n) diameter. Also, it is shown how to maintain the randomized t-spanner under random insertions and deletions. Previously, no results were known for spanners with low spanner diameter and for maintaining spanners under insertions and deletions.<<ETX>>

Space-efficient approximate Voronoi diagrams

LetS be a set ofn points in ℝd and lett>1 be a real number. At-spanner forS is a graph having the points ofS as its vertices such that for any pairp, q of points there is a path between them of length at mostt times the Euclidean distance betweenp andq.An efficient implementation of a greedy algorithm is given that constructs at-spanner having bounded degree such that the total length of all its edges is bounded byO (logn) times the length of a minimum spanning tree forS. The algorithm has running timeO (n logdn).Applying recent results of Das, Narasimhan, and Salowe to thist-spanner gives anO(n logdn)-time algorithm for constructing at-spanner having bounded degree and whose total edge length is proportional to the length of a minimum spanning tree forS. Previously, noo(n2)-time algorithms were known for constructing at-spanner of bounded degree.In the final part of the paper, an application to the problem of distance enumeration is given.

Accounting for boundary effects in nearest neighbor searching

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A simple entropy-based algorithm for planar point location

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Dynamic algorithms for geometric spanners of small diameter: randomized solutions

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APPROXIMATION ALGORITHM FOR MULTIPLE-TOOL MILLING ∗

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