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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 572 (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.
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 ..."
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Cited by 152 (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
ON THE COMPLEXITY OF SOME COMMON GEOMETRIC LOCATION PROBLEMS
 SIAM J. COMPUTING
, 1984
"... Given n demand points in the plane, the pcenter problem is to find p supply points (anywhere in the plane) so as to minimize the maximum distance from a demo & point to its respective nearest supply point. The pmedian problem is to minimize the sum of distances from demand points to their resp ..."
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Cited by 118 (1 self)
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Given n demand points in the plane, the pcenter problem is to find p supply points (anywhere in the plane) so as to minimize the maximum distance from a demo & point to its respective nearest supply point. The pmedian problem is to minimize the sum of distances from demand points to their respective nearest supply points. We prove that the pcenter and the pmedia problems relative to both the Euclidean and the rectilinear metrics are NPhard. In fact, we prove that it is NPhard even to approximate the pcenter problems sufficiently closely. The reductions are from 3satisfiability.
Message ferrying: Proactive routing in highlypartitioned wireless ad hoc networks
, 2003
"... An ad hoc network allows devices with wireless interfaces to communicate with each other without any preinstalled infrastructure. Due to node mobility, limited radio power, node failure and wide deployment area, ad hoc networks are often vulnerable to network partitioning. A number of examples are i ..."
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Cited by 91 (5 self)
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An ad hoc network allows devices with wireless interfaces to communicate with each other without any preinstalled infrastructure. Due to node mobility, limited radio power, node failure and wide deployment area, ad hoc networks are often vulnerable to network partitioning. A number of examples are in battlefield, disaster recovery and wide area surveillance. Unfortunately, most existing ad hoc routing protocols will fail to deliver messages under these circumstances since no route to the destination exists. In this work, we propose the Message Ferrying or MF scheme that provides efficient data delivery in disconnected ad hoc networks. In the MF scheme, nodes move proactively to send or receive messages. By introducing nonrandomness in a node’s proactive movement and exploiting such nonrandomness to deliver messages, the MF scheme improves data delivery performance in a disconnected network. In this paper, we propose the basic design of the MF scheme and develop a general framework to classify variations of MF systems. We also study ferry route design problem in stationary node case which is shown to be NPhard and provide an efficient algorithm to compute ferry route. 1
Some NPcomplete Geometric Problems
"... We show that the STEINER TREE problem and TRAVELING SALESMAN problem for points in the plane are NPcomplete when distances are measured either by the rectilinear (Manhattan) metric or by a natural discretized version of the Euclidean metric. Our proofs also indicate that the problems are NPhard i ..."
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Cited by 86 (2 self)
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We show that the STEINER TREE problem and TRAVELING SALESMAN problem for points in the plane are NPcomplete when distances are measured either by the rectilinear (Manhattan) metric or by a natural discretized version of the Euclidean metric. Our proofs also indicate that the problems are NPhard if the distance I~asure is the (unmodified) Euclidean metric. However, for reasons we discuss, there is some question as to whether these problems, or even the wellsolved MINIMUM SPANNING TREE problem, are in NP when the distance measure is the Euclidean metric.
paths in grid graphs
 SIAM J. Comput
, 1982
"... Abstract. A grid graph is a nodeinduced finite subgraph of the infinite grid. It is rectangular if its set of nodes is the product of two intervals. Given a rectangular grid graph and two of its nodes, we give necessary and sufficient conditions for the graph to have a Hamilton path between these t ..."
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Cited by 79 (0 self)
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Abstract. A grid graph is a nodeinduced finite subgraph of the infinite grid. It is rectangular if its set of nodes is the product of two intervals. Given a rectangular grid graph and two of its nodes, we give necessary and sufficient conditions for the graph to have a Hamilton path between these two nodes. In contrast, the Hamilton path (and circuit) problem for general grid graphs is shown to be NPcomplete. This provides a new, relatively simple, proof of the result that the Euclidean traveling salesman problem is NPcomplete.
Approximation Algorithms For The Geometric Covering Salesman Problem
 Discrete Applied Mathematics
, 1995
"... We introduce a geometric version of the Covering Salesman Problem: Each of the n salesman's clients specifies a neighborhood in which they are willing to meet the salesman. Identifying a tour of minimum length that visits all neighborhoods is an NPhard problem, since it is a generalization of ..."
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Cited by 62 (3 self)
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We introduce a geometric version of the Covering Salesman Problem: Each of the n salesman's clients specifies a neighborhood in which they are willing to meet the salesman. Identifying a tour of minimum length that visits all neighborhoods is an NPhard problem, since it is a generalization of the Traveling Salesman Problem. We present simple heuristic procedures for constructing tours, for a variety of neighborhood types, whose length is guaranteed to be within a constant factor of the length of an optimal tour. The neighborhoods we consider include, parallel unit segments, translates of a polygonal region, and circles. y Partially supported by NSF Grants DMS 8903304 and ECSE8857642. 1 Introduction A salesman wants to meet a set of potential buyers. Each buyer specifies a compact set in the plane, his neighborhood, within which he is willing to meet. For example, the neighborhoods may be disks centered at the buyers locations, and each disk's radius specifies the distance that a ...
Approximation algorithms for TSP with neighborhoods in the plane
 J. ALGORITHMS
, 2001
"... In the Euclidean TSP with neighborhoods (TSPN), we are given a collection of n regions (neighborhoods) and we seek a shortest tour that visits each region. As a generalization of the classical Euclidean TSP, TSPN is also NPhard. In this paper, we present new approximation results for the TSPN, incl ..."
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Cited by 62 (8 self)
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In the Euclidean TSP with neighborhoods (TSPN), we are given a collection of n regions (neighborhoods) and we seek a shortest tour that visits each region. As a generalization of the classical Euclidean TSP, TSPN is also NPhard. In this paper, we present new approximation results for the TSPN, including (1) a constantfactor approximation algorithm for the case of arbitrary connected neighborhoods having comparable diameters; and (2) a PTAS for the important special case of disjoint unit disk neighborhoods (or nearly disjoint, nearlyunit disks). Our methods also yield improved approximation ratios for various special classes of neighborhoods, which have previously been studied. Further, we give a lineartime O(1) approximation algorithm for the case of neighborhoods that are (innite) straight lines.
On the Complexity of Nash Equilibria and Other Fixed Points (Extended Abstract)
 IN PROC. FOCS
, 2007
"... We reexamine what it means to compute Nash equilibria and, more generally, what it means to compute a fixed point of a given Brouwer function, and we investigate the complexity of the associated problems. Specifically, we study the complexity of the following problem: given a finite game, Γ, with 3 ..."
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Cited by 37 (4 self)
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We reexamine what it means to compute Nash equilibria and, more generally, what it means to compute a fixed point of a given Brouwer function, and we investigate the complexity of the associated problems. Specifically, we study the complexity of the following problem: given a finite game, Γ, with 3 or more players, and given ɛ> 0, compute an approximation within ɛ of some (actual) Nash equilibrium. We show that approximation of an actual Nash Equilibrium, even to within any nontrivial constant additive factor ɛ < 1/2 in just one desired coordinate, is at least as hard as the long standing squareroot sum problem, as well as a more general arithmetic circuit decision problem that characterizes Ptime in a unitcost model of computation with arbitrary precision rational arithmetic; thus placing the approximation problem in P, or even NP, would resolve major open problems in the complexity of numerical computation. We show similar results for market equilibria: it is hard to estimate with any nontrivial accuracy the equilibrium prices in an exchange economy with a unique equilibrium, where the economy is given by explicit algebraic formulas for the excess demand functions. We define a class, FIXP, which captures search problems that can be cast as fixed point
Computational geometry  a survey
 IEEE TRANSACTIONS ON COMPUTERS
, 1984
"... We survey the state of the art of computational geometry, a discipline that deals with the complexity of geometric problems within the framework of the analysis ofalgorithms. This newly emerged area of activities has found numerous applications in various other disciplines, such as computeraided de ..."
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Cited by 19 (3 self)
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We survey the state of the art of computational geometry, a discipline that deals with the complexity of geometric problems within the framework of the analysis ofalgorithms. This newly emerged area of activities has found numerous applications in various other disciplines, such as computeraided design, computer graphics, operations research, pattern recognition, robotics, and statistics. Five major problem areasconvex hulls, intersections, searching, proximity, and combinatorial optimizationsare discussed. Seven algorithmic techniques incremental construction, planesweep, locus, divideandconquer, geometric transformation, pruneandsearch, and dynamizationare each illustrated with an example.Acollection of problem transformations to establish lower bounds for geometric problems in the algebraic computation/decision model is also included.