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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 21 (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.
A Data Structure for Bicategories With Application to Speeding Up an Approximation Algorithm
, 1993
"... We introduce a datastructure problem on graphs we call the bicategory problem, and a data structure that solves it in O( p m log m) amortized time, where m is the number of edges. We show how this data structure can be used in quickly computing approximately minimumcost networks. The resulting t ..."
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Cited by 11 (1 self)
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We introduce a datastructure problem on graphs we call the bicategory problem, and a data structure that solves it in O( p m log m) amortized time, where m is the number of edges. We show how this data structure can be used in quickly computing approximately minimumcost networks. The resulting time bound for the approximation algorithm is O(n p m log m) where n and m are the number of nodes and edges in the input graph, an improvement over the previously known bound of O(n 2 p log log n) when the input graph is sparse . 1 Introduction The Steiner tree problem in networks is a classic problem in optimization, proved NPcomplete by Karp in his original paper [5]. Given a graph with costs on its edges, and given a subset of the nodes called terminals, the goal is to select a minimumcost connected subgraph spanning all the terminals. Several approximation algorithms have been designed for this problem (a few are [7, 9, 10]), but it was not until 1988 that Mehlhorn [8] devised a ...
A New Evolutionary Approach to the DegreeConstrained Minimum Spanning Tree Problem
 IEEE Transactions on Evolutionary Computation
, 1999
"... Finding the degreeconstrained minimum spanning tree (dMST) of a graph is a wellstudied NPhard problem of importance in communications network design and other networkrelated problems. In this paper we describe some previously proposed algorithms for solving the problem, and then introduce a nove ..."
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Cited by 10 (2 self)
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Finding the degreeconstrained minimum spanning tree (dMST) of a graph is a wellstudied NPhard problem of importance in communications network design and other networkrelated problems. In this paper we describe some previously proposed algorithms for solving the problem, and then introduce a novel tree construction algorithm called the Randomised Primal Method (RPM) which builds degreeconstrained trees of low cost from solution vectors taken as input. RPM is applied in three stochastic iterative search methods: simulated annealing, multistart hillclimbing, and a genetic algorithm. While other researchers have mainly concentrated on finding spanning trees in Euclidean graphs, we consider the more general case of random graph problems. We describe two random graph generators which produce particularly challenging dMST problems. On these and other problems we find that the genetic algorithm employing RPM outperforms simulated annealing and multistart hillclimbing. Our experimental ...