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Voronoi diagrams  a survey of a fundamental geometric data structure
 ACM COMPUTING SURVEYS
, 1991
"... This paper presents a survey of the Voronoi diagram, one of the most fundamental data structures in computational geometry. It demonstrates the importance and usefulness of the Voronoi diagram in a wide variety of fields inside and outside computer science and surveys the history of its development. ..."
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Cited by 560 (5 self)
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This paper presents a survey of the Voronoi diagram, one of the most fundamental data structures in computational geometry. It demonstrates the importance and usefulness of the Voronoi diagram in a wide variety of fields inside and outside computer science and surveys the history of its development. The paper puts particular emphasis on the unified exposition of its mathematical and algorithmic properties. Finally, the paper provides the first comprehensive bibliography on Voronoi diagrams and related structures.
Voronoi Diagrams and Delaunay Triangulations
 Computing in Euclidean Geometry
, 1992
"... The Voronoi diagram is a fundamental structure in computationalgeometry and arises naturally in many different fields. This chapter surveys properties of the Voronoi diagram and its geometric dual, the Delaunay triangulation. The emphasis is on practical algorithms for the construction of Voronoi ..."
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Cited by 198 (3 self)
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The Voronoi diagram is a fundamental structure in computationalgeometry and arises naturally in many different fields. This chapter surveys properties of the Voronoi diagram and its geometric dual, the Delaunay triangulation. The emphasis is on practical algorithms for the construction of Voronoi diagrams. 1 Introduction Let S be a set of n points in ddimensional euclidean space E d . The points of S are called sites. The Voronoi diagram of S splits E d into regions with one region for each site, so that the points in the region for site s2S are closer to s than to any other site in S. The Delaunay triangulation of S is the unique triangulation of S so that there are no elements of S inside the circumsphere of any triangle. Here `triangulation' is extended from the planar usage to arbitrary dimension: a triangulation decomposes the convex hull of S into simplices using elements of S as vertices. The existence and uniqueness of the Delaunay triangulation are perhaps not obvio...
Mesh Generation And Optimal Triangulation
, 1992
"... We survey the computational geometry relevant to finite element mesh generation. We especially focus on optimal triangulations of geometric domains in two and threedimensions. An optimal triangulation is a partition of the domain into triangles or tetrahedra, that is best according to some cri ..."
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Cited by 180 (8 self)
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We survey the computational geometry relevant to finite element mesh generation. We especially focus on optimal triangulations of geometric domains in two and threedimensions. An optimal triangulation is a partition of the domain into triangles or tetrahedra, that is best according to some criterion that measures the size, shape, or number of triangles. We discuss algorithms both for the optimization of triangulations on a fixed set of vertices and for the placement of new vertices (Steiner points). We briefly survey the heuristic algorithms used in some practical mesh generators.
Geometric Spanner for Routing in Mobile Networks
, 2001
"... Abstract—We propose a new routing graph, the restricted Delaunay graph (RDG), for mobile ad hoc networks. Combined with a node clustering algorithm, the RDG can be used as an underlying graph for geographic routing protocols. This graph has the following attractive properties: 1) it is planar; 2) be ..."
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Cited by 153 (19 self)
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Abstract—We propose a new routing graph, the restricted Delaunay graph (RDG), for mobile ad hoc networks. Combined with a node clustering algorithm, the RDG can be used as an underlying graph for geographic routing protocols. This graph has the following attractive properties: 1) it is planar; 2) between any two graph nodes there exists a path whose length, whether measured in terms of topological or Euclidean distance, is only a constant times the minimum length possible; and 3) the graph can be maintained efficiently in a distributed manner when the nodes move around. Furthermore, each node only needs constant time to make routing decisions. We show by simulation that the RDG outperforms previously proposed routing graphs in the context of the Greedy perimeter stateless routing (GPSR) protocol. Finally, we investigate theoretical bounds on the quality of paths discovered using GPSR. Index Terms—Geographical routing, spanners, wireless ad hoc networks. I.
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 147 (12 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
Voronoi Diagrams
 Handbook of Computational Geometry
"... Voronoi diagrams can also be thought of as lower envelopes, in the sense mentioned at the beginning of this subsection. Namely, for each point x not situated on a bisecting curve, the relation p x q defines a total ordering on S. If we construct a set of surfaces H p , p S,in3space such t ..."
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Cited by 143 (19 self)
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Voronoi diagrams can also be thought of as lower envelopes, in the sense mentioned at the beginning of this subsection. Namely, for each point x not situated on a bisecting curve, the relation p x q defines a total ordering on S. If we construct a set of surfaces H p , p S,in3space such that H p is below H q i# p x q holds, then the projection of their lower envelope equals the abstract Voronoi diagram.
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 143 (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...
A polylogarithmic approximation algorithm for the group Steiner tree problem
 Journal of Algorithms
, 2000
"... The group Steiner tree problem is a generalization of the Steiner tree problem where we ae given several subsets (groups) of vertices in a weighted graph, and the goal is to find a minimumweight connected subgraph containing at least one vertex from each group. The problem was introduced by Reich a ..."
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Cited by 134 (9 self)
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The group Steiner tree problem is a generalization of the Steiner tree problem where we ae given several subsets (groups) of vertices in a weighted graph, and the goal is to find a minimumweight connected subgraph containing at least one vertex from each group. The problem was introduced by Reich and Widmayer and finds applications in VLSI design. The group Steiner tree problem generalizes the set covering problem, and is therefore at least as had. We give a randomized O(log 3 n log k)approximation algorithm for the group Steiner tree problem on an nnode graph, where k is the number of groups. The best previous ink)v/ (Bateman, Helvig, performance guarantee was (1 +  Robins and Zelikovsky).
An Optimal Algorithm for Euclidean Shortest Paths in the Plane
 SIAM J. Comput
, 1997
"... We propose an optimaltime algorithm for a classical problem in plane computational geometry: computing a shortest path between two points in the presence of polygonal obstacles. Our algorithm runs in worstcase time O(n log n) and requires O(n log n) space, where n is the total number of vertice ..."
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Cited by 86 (1 self)
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We propose an optimaltime algorithm for a classical problem in plane computational geometry: computing a shortest path between two points in the presence of polygonal obstacles. Our algorithm runs in worstcase time O(n log n) and requires O(n log n) space, where n is the total number of vertices in the obstacle polygons. The algorithm is based on an efficient implementation of wavefront propagation among polygonal obstacles, and it actually computes a planar map encoding shortest paths from a fixed source point to all other points of the plane; the map can be used to answer singlesource shortest path queries in O(logn) time. The time complexity of our algorithm is a significant improvement over all previously published results on the shortest path problem. Finally, we also discuss extensions to more general shortest path problems, involving nonpoint and multiple sources. 1 Introduction 1.1 The Background and Our Result The Euclidean shortest path problem is one of the o...