We study the problem of finding small trees. Classical network design problems are considered with the additional constraint that only a specified number k of nodes are required to be connected in the solution. A prototypical example is the kMST problem in which we require a tree of minimum weight spanning at least k nodes in an edge-weighted graph. We show that the kMST problem is NP-hard even for points in the Euclidean plane. We provide approximation algorithms with performance ratio 2 p k for the general edge-weighted case and O(k 1=4 ) for the case of points in the plane. Polynomial-time exact solutions are also presented for the class of treewidth-bounded graphs which includes trees, series-parallel graphs, and bounded bandwidth graphs, and for points on the boundary of a convex region in the Euclidean plane. We also investigate the problem of finding short trees, and more generally, that of finding networks with minimum diameter. A simple technique is used to prov...

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 post-office 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 divide-and-conquer 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 . . . . . . . . . . . . . . . . . . ...

Weintroduce a new method for finding several types of optimal k-point sets, minimizing perimeter, diameter, circumradius, and related measures, by testing sets of the O(k) nearest neighbors to each point. We argue that this is better in a number of ways than previous algorithms, whichwere based on high order Voronoi diagrams. Our technique allows us for the first time to efficiently maintain minimal sets as new points are inserted, to generalize our algorithms to higher dimensions, to find minimal convex k-vertex polygons and polytopes, and to improvemany previous results. Weachievemany of our results via a new algorithm for finding rectilinear nearest neighbors in the plane in time O(n log n+kn). We also demonstrate a related technique for finding minimum area k-point sets in the plane, based on testing sets of nearest vertical neighbors to each line segment determined by a pair of points. A generalization of this technique also allows us to find minimum volume and boundary measure sets in arbitrary dimensions.

Given a set P of n points in the plane and a number k, we want to find a polygon ~ with vertices in P of minimum area that satisfies one of the following properties: (1) cK is a convex k-gon, (2) ~ is an empty convex k-gon, or (3) ~ is the convex hull of exactly k points of P. We give algorithms for solving each of these three problems in time O(kn3). The space complexity is O(n) for k = 4 and O(kn 2) for k> 5. The algorithms are based on a dynamic ptogramming approach. We generalize this approach to polygons with minimum perimeter, polygons with maximum perimeter or area, polygons containing the maximum or minimum number of points, polygons with minimum weight (for some weights added to vertices), etc., in similar time bounds.

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.

Given a set P of n points in the plane, we wish to find a set Q P of k points for which the convex hull conv(Q) has the minimum area.

Motivated by questions in location planning, we show for a set of colored point sites in the plane how to compute the smallest— by perimeter or area—axis-parallel rectangle and the narrowest strip enclosing at least one site of each color.

We consider the problem of removing c points from a set S of n points so that the resulting point set has the smallest possible convex hull. Our main result is an O � n � � � � � 4c c 2c (3c) + log n time algorithm that solves this problem when “smallest ” is taken to mean least area or least perimeter. 1

Manuel Abellanas # Ferran Hurtado ## Christian Icking ** * Rolf Klein ** ** Elmar Langetepe ** ** Lihong Ma ** ** Belen Palop ** * ** Vera Sacristan ## Suppose there are k types of facilities, e. g. schools, post o#ces, supermarkets, modeled by n colored points in the plane, each type by its own color. One basic goal in choosing a residence location is in having at least one representative of each facility type in the neighborhood. In this paper we provide algorithms that may help to achieve this goal for various specifications of the term "neighborhood". Several problems on multicolored point sets have been previously considered, such as the bichromatic closest pair, see e. g. Preparata and Shamos [14, Section 5.7], Agarwal et al. [1], and Graf and Hinrichs [8], the group Steiner tree, see Mitchell [11, Section 7.1], or the chromatic nearest neighbor search,seeMountetal.[12]. Let us call a set color-spanning if it contains at least one point of each color. A natural app...

We present efficient parallel algorithms for some geometric clustering and partitioning problems. Our algorithms run in the CREW PRAM model of parallel computation. Given a point set P of n points in two dimensions, the clustering problems are to find a k-point subset such that some measure for this subset is minimized. We consider the problems of finding a k-point subset with minimum L1 perimeter and minimum L1 diameter. For the L1 perimeter case, our algorithm runs in O(log 2 n) time and O(n log 2 n + nk 2 log 2 k) work. For the L1 diameter case, our algorithm runs in O(log 2 n+ log 2 k log log k log k) time and O(n log 2 n) work. We consider partitioning problems of the following nature. Given a planar point set S (jSj = n), a measure acting on S and a pair of values 1 and 2 , does there exist a bipartition S = S 1 [S 2 such that (S 1 ) i for i = 1; 2? We consider several measures like diameter under L1 and L 1 metric; area, perimeter of the smallest...