Michiel H. M. Smid

Geometric Spanner Networks

Part I. Introduction: 1. Introduction 2. Algorithms and graphs 3. The algebraic computation-tree model Part II. Spanners Based on Simplical Cones: 4. Spanners based on the Q-graph 5. Cones in higher dimensional space and Q-graphs 6. Geometric analysis: the gap property 7. The gap-greedy algorithm 8. Enumerating distances using spanners of bounded degree Part III. The Well Separated Pair Decomposition and its Applications: 9. The well-separated pair decomposition 10. Applications of well-separated pairs 11. The Dumbbell theorem 12. Shortcutting trees and spanners with low spanner diameter 13. Approximating the stretch factor of Euclidean graphs Part IV. The Path Greedy Algorithm: 14. Geometric analysis: the leapfrog property 15. The path-greedy algorithm Part V. Further Results and Applications: 16. The distance range hierarchy 17. Approximating shortest paths in spanners 18. Fault-tolerant spanners 19. Designing approximation algorithms with spanners 20. Further results and open problems.

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..

Closest point problems in computational geometry

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.

On the false-positive rate of Bloom filters

Abstract A comprehensive overview is given of algorithms and data structures for proximity problems on point sets in ℝ D . In particular, the closest pair problem, the exact and approximate post-office problem, and the problem of constructing spanners are discussed in detail.

Planar Spanners and Approximate Shortest Path Queries among Obstacles in the Plane

Bloom filters are a randomized data structure for membership queries dating back to 1970. Bloom filters sometimes give erroneous answers to queries, called false positives. Bloom analyzed the probability of such erroneous answers, called the false-positive rate, and Blooms analysis has appeared in many publications throughout the years. We show that Blooms analysis is incorrect and give a correct analysis.

Constructing plane spanners of bounded degree and low weight

We consider the problem of finding an obstacle-avoiding path between two points s and t in the plane, amidst a set of disjoint polygonal obstacles with a total of n vertices. The length of this path should be within a small constant factor c of the length of the shortest possible obstacle-avoiding s-t path measured in the L p -metric. Such an approximate shortest path is called a c-short path, or a short path with stretch factor c. The goal is to preprocess the obstacle-scattered plane by creating an efficient data structure that enables fast reporting of a c-short path (or its length). In this paper, we give a family of algorithms for the above problem that achieve an interesting trade-off between the stretch factor, the query time and the preprocessing bounds. Our main results are algorithms that achieve logarithmic length query time, after subquadratic time and space preprocessing.

On some geometric optimization problems in layered manufacturing

Abstract Given a set S of n points in the plane, we give an O(n log n)-time algorithm that constructs a plane t-spanner for S, with t ≈ 10, such that the degree of each point of S is bounded from above by 27, and the total edge length is proportional to the weight of a minimum spanning tree of S. Previously, no algorithms were known for constructing plane t-spanners of bounded degree.

Further results on generalized intersection searching problems: counting, reporting, and dynamization

Abstract Efficient geometric algorithms are given for optimization problems arising in layered manufacturing, where a 3D object is built by slicing its CAD model into layers and manufacturing the layers successively. The problems considered include minimizing the stair-step error on the surfaces of the manufactured object under various formulations, minimizing the volume of the so-called support structures used, and minimizing the contact area between the supports and the manufactured object—all of which are factors that affect the speed and accuracy of the process. The stair-step minimization algorithm is valid for any polyhedron, while the support minimization algorithms are applicable only to convex polyhedra. The techniques used to obtain these results include construction and searching of certain arrangements on the sphere, 3D convex hulls, halfplane range searching, and constrained optimization.

Range mode and range median queries on lists and trees

In a generalized intersection searching problem, a set, S, of colored geometric objects is to be preprocessed so that given some query object, q, the distinct colors of the objects intersected by q can be reported or counted efficiently. In the dynamic setting, colored objects can be inserted into or deleted from S. These problems generalize the well-studied standard intersection searching problems and are rich in applications. Unfortunately, the techniques known for the standard problems do not yield efficient solutions for the generalized problems. Moreover, previous work on generalized problems applies only to the reporting problems and that too mainly to the static case. In this paper, a uniform framework is presented to solve efficiently the counting/reporting/dynamic versions of a variety of generalized intersection searching problems, including: 1-, 2-, and 3-dimensional range searching, quadrant searching, and 2-dimensional point enclosure searching. Several other related results are also mentioned.

Static and dynamic algorithms for k -point clustering problems

We consider algorithms for preprocessing labelled lists and trees so that, for any two nodes u and v we can answer queries of the form: What is the mode or median label in the sequence of labels on the path from u to v.