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64
Spanning Trees and Spanners
, 1996
"... We survey results in geometric network design theory, including algorithms for constructing minimum spanning trees and lowdilation graphs. 1 Introduction This survey covers topics in geometric network design theory. The problem is easy to state: connect a collection of sites by a "good" ..."
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Cited by 148 (2 self)
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We survey results in geometric network design theory, including algorithms for constructing minimum spanning trees and lowdilation graphs. 1 Introduction This survey covers topics in geometric network design theory. The problem is easy to state: connect a collection of sites by a "good" network. For instance, one may wish to connect components of a VLSI circuit by networks of wires, in a way that uses little surface area on the chip, draws little power, and propagates signals quickly. Similar problems come up in other applications such as telecommunications, road network design, and medical imaging [1]. One network design problem, the Traveling Salesman problem, is sufficiently important to have whole books devoted to it [79]. Problems involving some form of geometric minimum or maximum spanning tree also arise in the solution of other geometric problems such as clustering [12], mesh generation [56], and robot motion planning [93]. One can vary the network design problem in many w...
Euclidean spanners: short, thin, and lanky
 IN: 27TH ACM SYMPOSIUM ON THEORY OF COMPUTING
, 1995
"... 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 d ..."
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Cited by 112 (22 self)
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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
ClosestPoint Problems in Computational Geometry
, 1997
"... This is the preliminary version of a chapter that will appear in the Handbook on Computational Geometry, edited by J.R. Sack and J. Urrutia. A comprehensive overview is given of algorithms and data structures for proximity problems on point sets in IR D . In particular, the closest pair problem, th ..."
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Cited by 73 (14 self)
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This is the preliminary version of a chapter that will appear in the Handbook on Computational Geometry, edited by J.R. Sack and J. Urrutia. A comprehensive overview is given of algorithms and data structures for proximity problems on point sets in IR D . In particular, the closest pair problem, the exact and approximate postoffice problem, and the problem of constructing spanners are discussed in detail. Contents 1 Introduction 1 2 The static closest pair problem 4 2.1 Preliminary remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Algorithms that are optimal in the algebraic computation tree model . 5 2.2.1 An algorithm based on the Voronoi diagram . . . . . . . . . . . 5 2.2.2 A divideandconquer algorithm . . . . . . . . . . . . . . . . . . 5 2.2.3 A plane sweep algorithm . . . . . . . . . . . . . . . . . . . . . . 6 2.3 A deterministic algorithm that uses indirect addressing . . . . . . . . . 7 2.3.1 The degraded grid . . . . . . . . . . . . . . . . . . ...
Approximating the stretch factor of Euclidean paths, cycles and trees
 SIAM J. Comput
, 1999
"... Given a set S of n points in R d , and a graph G having the points of S as its vertices, the stretch factor t of G is dened as the maximal value jpqj G =jpqj, where p; q 2 S, p 6= q, jpqj G is the length of a shortest path in G between p and q, and jpqj is the Euclidean distance between p and ..."
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Cited by 50 (7 self)
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Given a set S of n points in R d , and a graph G having the points of S as its vertices, the stretch factor t of G is dened as the maximal value jpqj G =jpqj, where p; q 2 S, p 6= q, jpqj G is the length of a shortest path in G between p and q, and jpqj is the Euclidean distance between p and q. We consider the problem of designing algorithms that, for an arbitrary constant > 0, compute an approximation to this stretch factor, i.e., a value t such that t t (1 + )t. We give eÆcient solutions for the cases when G is a path, cycle, or tree. The main idea used in all the algorithms is to use wellseparated pair decompositions to speed up the computations. 1 Introduction Let S be a set of n points in R d , where d 1 is a small constant, and let G be an undirected connected graph having the points of S as its vertices. The length of any edge (p; q) of G is dened as the Euclidean distance jpqj between the two vertices p and q. The length of a path in G is dened a...
Randomized and Deterministic Algorithms for Geometric Spanners of Small Diameter
, 1994
"... Let S be a set of n points in IR d and let t ? 1 be a real number. A tspanner 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 q. Such a path is c ..."
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Cited by 39 (7 self)
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Let S be a set of n points in IR d and let t ? 1 be a real number. A tspanner 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 q. Such a path is called a tspanner 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 tspanner path from p to q containing at most D edges. Randomized and deterministic algorithms are given for constructing tspanners consisting of O(n) edges and having O(logn) diameter. Also, it is shown how to maintain the randomized tspanner under random insertions and deletions. Previously, no results were known for spanners with low spanner diameter and for maintaining spanners under insertions and deletions. 1 Introduction Given a set S of n points in IR d and a real number t ? 1, a tspanner for S is a directed graph on S such th...
Approximate Distance Oracles for Geometric Graphs
, 2002
"... Given a geometric tspanner graph G in E d with n points and m edges, with edge lengths that lie within a polynomial (in n) factor of each other. Then, after O(m+n log n) preprocessing, we present an approximation scheme to answer (1+") approximate shortest path queries in O(1) time. The dat ..."
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Cited by 35 (11 self)
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Given a geometric tspanner graph G in E d with n points and m edges, with edge lengths that lie within a polynomial (in n) factor of each other. Then, after O(m+n log n) preprocessing, we present an approximation scheme to answer (1+") approximate shortest path queries in O(1) time. The data structure uses O(n log n) space.
Efficient Construction of a Bounded Degree Spanner with Low Weight
 IN PROC. 2ND ANNU. EUROPEAN SYMPOS. ALGORITHMS (ESA
, 1994
"... Let S be a set of n points in IR d and let t ? 1 be a real number. A tspanner for S is a graph having the points of S as its vertices such that for any pair p; q of points there is a path between them of length at most t times the Euclidean distance between p and q. An efficient implementatio ..."
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Cited by 32 (3 self)
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Let S be a set of n points in IR d and let t ? 1 be a real number. A tspanner for S is a graph having the points of S as its vertices such that for any pair p; q of points there is a path between them of length at most t times the Euclidean distance between p and q. An efficient implementation of a greedy algorithm is given that constructs a tspanner having bounded degree such that the total length of all its edges is bounded by O(log n) times the length of a minimum spanning tree for S. The algorithm has running time O(n log d n). Applying recent results of Das, Narasimhan and Salowe to this tspanner gives an O(n log d n) time algorithm for constructing a tspanner having bounded degree and whose total edge length is proportional to the length of a minimum spanning tree for S. Previously, no o(n 2 ) time algorithms were known for constructing a tspanner of bounded degree. In the final part of the paper, an application to the problem of distance enumeration is...
geometric and graphs issues in wireless networks
 Wirel. Commun. Mob. Comput
, 2003
"... We present an overview of the recent progress of applying computational geometry techniques to solve some questions, such as topology construction and broadcasting, in wireless ad hoc networks. Treating each wireless device as a node in a two dimensional plane, we model the wireless networks by un ..."
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Cited by 29 (2 self)
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We present an overview of the recent progress of applying computational geometry techniques to solve some questions, such as topology construction and broadcasting, in wireless ad hoc networks. Treating each wireless device as a node in a two dimensional plane, we model the wireless networks by unit disk graphs in which two nodes are connected if their Euclidean distance is no more than one. We rst summarize the current status of constructing sparse spanners for unit disk graphs with various combinations of the following properties: bounded stretch factor, bounded node degree, planar, and bounded total edges weight (compared with the minimum spanning tree). Instead of constructing subgraphs by removing links, we then review the algorithms for constructing a sparse backbone (connected dominating set), i.e., subgraph from the subset of nodes. We then review some eÆcient methods for broadcasting and multicasting with theoretic guaranteed performance.
Dynamic algorithms for geometric spanners of small diameter: Randomized solutions
 Computational Geometry: Theory and Applications
, 1994
"... Let S be a set of n points in IR d and let t ? 1 be a real number. A tspanner 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 q. Such a path is ca ..."
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Cited by 26 (4 self)
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Let S be a set of n points in IR d and let t ? 1 be a real number. A tspanner 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 q. Such a path is called a tspanner 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 tspanner path from p to q containing at most D edges. A randomized algorithm is given for constructing a tspanner that, with high probability, contains O(n) edges and has spanner diameter O(log n). A data structure of size O(n log d n) is given that maintains this tspanner in O(log d n log log n) expected amortized time per insertion and deletion, in the model of random updates, as introduced by Mulmuley. Key words: Computational geometry, proximity problems, skip lists, randomization, dynamic data structures. Preprint submitted to El...
Improved Algorithms for Constructing FaultTolerant Spanners
 Algorithmica
, 1998
"... Let S be a set of n points in a metric space, and k a positive integer. Algorithms are given that construct kfaulttolerant spanners for S. If in such a spanner at most k vertices and/or edges are removed, then each pair of points in the remaining graph is still connected by a "short" pa ..."
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Cited by 23 (5 self)
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Let S be a set of n points in a metric space, and k a positive integer. Algorithms are given that construct kfaulttolerant spanners for S. If in such a spanner at most k vertices and/or edges are removed, then each pair of points in the remaining graph is still connected by a "short" path. First, an algorithm is given that transforms an arbitrary spanner into a kfaulttolerant spanner. For the Euclidean metric in R d , this leads to an O(n log n + c k n)time algorithm that constructs a kfaulttolerant spanner of degree O(c k ), whose total edge length is bounded by O(c k ) times the weight of a minimum spanning tree of S, for some constant c. For constant values of k, this result is optimal. In the second part of the paper, an algorithm is presented for the Euclidean metric in R d . This algorithm constructs in O(n log n+k 2 n) time a kfaulttolerant spanner with O(k 2 n) edges. 1 Introduction Spanners have applications in the design of networks. Consider a se...