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42
Geometric Shortest Paths and Network Optimization
 Handbook of Computational Geometry
, 1998
"... Introduction A natural and wellstudied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to be the sum of the weights of t ..."
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Cited by 149 (13 self)
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Introduction A natural and wellstudied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to be the sum of the weights of the edges that comprise it. Efficient algorithms are well known for this problem, as briefly summarized below. The shortest path problem takes on a new dimension when considered in a geometric domain. In contrast to graphs, where the encoding of edges is explicit, a geometric instance of a shortest path problem is usually specified by giving geometric objects that implicitly encode the graph and its edge weights. Our goal in devising efficient geometric algorithms is generally to avoid explicit construction of the entire underlying graph, since the full induced graph may be very large (even exponential in the input size, or infinite). Computing an optimal
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" network. ..."
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Cited by 139 (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 103 (21 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
New Sparseness Results on Graph Spanners
, 1992
"... Let G = (V, E) be an nvertex connected graph with positive edge weights. A subgraph G ’ = (V, E’) is a tspanner 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 span ..."
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Cited by 79 (8 self)
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Let G = (V, E) be an nvertex connected graph with positive edge weights. A subgraph G ’ = (V, E’) is a tspanner 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. Sparseness of spanners is measured by two criteria, the size, defined as the number of edges in the spanner, 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 edgeweighted graph G, for any t> 1, and any c>0, we show that a tspanner of G with weight O(n * ). wt(MST) can be constructed in polynomial time. We also show that (logz n)spanners of weight O(1). wt(MST) can be constructed. We then consider spanners for complete graphs induced by a set of points in ddimensional real normed space. The weight of an edge Zy is the norm of the ~y vector. We show that for these graphs, tspanners with total weight O(log n). wt(MST) can be constructed in polynomial time.
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 65 (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 . . . . . . . . . . . . . . . . . . ...
Faster Algorithms for Some Geometric Graph Problems in Higher Dimensions
, 1993
"... We show how to apply the wellseparated pair decomposition of a pointset P in ! d to significantly improve known time bounds on several geometric graph problems. We first present an algorithm to find an approximate Euclidean minimum spanning tree of P whose weight is at most 1 + ffl times the exa ..."
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Cited by 58 (2 self)
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We show how to apply the wellseparated pair decomposition of a pointset P in ! d to significantly improve known time bounds on several geometric graph problems. We first present an algorithm to find an approximate Euclidean minimum spanning tree of P whose weight is at most 1 + ffl times the exact minimum. We achieve a time complexity of O(n log n + (ffl \Gammad=2 log 1 ffl )n), improving the best known bound of O(ffl \Gammad n log n). We then show how to construct a graph with O(ffl \Gammad+1 n) edges in which the shortest path between any pair of points is within 1 + ffl of the Euclidean distance. Our time complexity is O(n log n+(ffl \Gammad log 1 ffl )n), a significant improvement over the best previous algorithm that produces a graph of this size. Finally, we show how to compute the exact Euclidean minimum spanning tree in time O(T d (n; n) log n), where T d (m; n) is the time to find the bichromatic closest pair between m red points and n blue points. The previo...
Approximate Nearest Neighbor Queries Revisited
, 1998
"... This paper proposes new methods to answer approximate nearest neighbor queries on a set of n points in ddimensional Euclidean space. For any fixed constant d, a data structure with O(" (1\Gammad)=2 n log n) preprocessing time and O(" (1\Gammad)=2 log n) query time achieves approximation factor ..."
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Cited by 55 (3 self)
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This paper proposes new methods to answer approximate nearest neighbor queries on a set of n points in ddimensional Euclidean space. For any fixed constant d, a data structure with O(" (1\Gammad)=2 n log n) preprocessing time and O(" (1\Gammad)=2 log n) query time achieves approximation factor 1 + " for any given 0 ! " ! 1; a variant reduces the "dependence by a factor of " \Gamma1=2 . For any arbitrary d, a data structure with O(d 2 n log n) preprocessing time and O(d 2 log n) query time achieves approximation factor O(d 3=2 ). Applications to various proximity problems are discussed. 1 Introduction Let P be a set of n point sites in ddimensional space IR d . In the wellknown post office problem, we want to preprocess P into a data structure so that a site closest to a given query point q (called the nearest neighbor of q) can be found efficiently. Distances are measured under the Euclidean metric. The post office problem has many applications within computational...
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 40 (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...
Algorithms for Dynamic Closest Pair and nBody Potential Fields
 In Proc. 6th ACMSIAM Sympos. Discrete Algorithms
, 1995
"... We present a general technique for dynamizing certain problems posed on point sets in Euclidean space for any fixed dimension d. This technique applies to a large class of structurally similar algorithms, presented previously by the authors, that make use of the wellseparated pair decomposition. We ..."
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Cited by 36 (1 self)
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We present a general technique for dynamizing certain problems posed on point sets in Euclidean space for any fixed dimension d. This technique applies to a large class of structurally similar algorithms, presented previously by the authors, that make use of the wellseparated pair decomposition. We prove efficient worstcase complexity for maintaining such computations under point insertions and deletions, and apply the technique to several problems posed on a set P containing n points. In particular, we show how to answer a query for any point x that returns a constantsize set of points, a subset of which consists of all points in P that have x as a nearest neighbor. We then show how to use such queries to maintain the closest pair of points in P . We also show how to dynamize the fast multipole method, a technique for approximating the potential field of a set of point charges. All our algorithms use the algebraic model that is standard in computational geometry, and have worstca...
A simple linear time algorithm for computing a (2k − 1)spanner of O(n 1+1/k ) size in weighted graphs
 In Proceedings of the 30th International Colloquium on Automata, Languages and Programming
, 2003
"... ) edges are required in the worst case for any (2k \Gamma 1)spanner, which has been proved for k = 1; 2; 3; 5. There exist polynomial time algorithms that can construct spanners with the size that matches this conjectured lower bound, and the best known algorithm takes O(mn 1=k) expected running ti ..."
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Cited by 35 (5 self)
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) edges are required in the worst case for any (2k \Gamma 1)spanner, which has been proved for k = 1; 2; 3; 5. There exist polynomial time algorithms that can construct spanners with the size that matches this conjectured lower bound, and the best known algorithm takes O(mn 1=k) expected running time. In this paper, we present an extremely simple linear time randomized algorithm that computes a (2k \Gamma 1)spanner of size matching the conjectured lower bound. An important feature of our algorithm is its local approach. Unlike all the previous algorithms which require computation of shortest paths, the new algorithm merely explores the edges in the neighborhood of a vertex or a group of vertices. This feature leads to designing simple externalmemory and parallel algorithms for computing sparse spanners, whose running times are optimal up to logarithmic factors.