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 computer-aided design, computer graphics, operations research, pattern recognition, robotics, and statistics. Five major problem areas-convex hulls, intersections, searching, proximity, and combinatorial optimizations-are discussed. Seven algorithmic techniques incremental construction, plane-sweep, locus, divide-andconquer, geometric transformation, prune-and-search, and dynamization-are 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.

We introduce a data-structure 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 NP-complete 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 minimum-cost 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 ...

Finding the degree-constrained minimum spanning tree (d-MST) of a graph is a wellstudied NP-hard problem of importance in communications network design and other network-related 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 degree-constrained trees of low cost from solution vectors taken as input. RPM is applied in three stochastic iterative search methods: simulated annealing, multi-start 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 d-MST problems. On these and other problems we find that the genetic algorithm employing RPM outperforms simulated annealing and multi-start hillclimbing. Our experimental ...