We survey results in geometric network design theory, including algorithms for constructing minimum spanning trees and low-dilation 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...

We prove four results on randomized incremental constructions (RICs): ffl an analysis of the expected behavior under insertion and deletions, ffl a fully dynamic data structure for convex hull maintenance in arbitrary dimensions, ffl a tail estimate for the space complexity of RICs, ffl a lower bound on the complexity of a game related to RICs. 1

Range searching is one of the central problems in computational geometry, because it arises in many applications and a wide variety of geometric problems can be formulated as a range-searching problem. A typical range-searching problem has the following form. Let S be a set of n points in R d , and let R be a family of subsets; elements of R are called ranges . We wish to preprocess S into a data structure so that for a query range R, the points in S " R can be reported or counted efficiently. Typical examples of ranges include rectangles, halfspaces, simplices, and balls. If we are only interested in answering a single query, it can be done in linear time, using linear space, by simply checking for each point p 2 S whether p lies in the query range.

In geometric range searching, algorithmic problems of the following type are considered: Given an n-point set P in the plane, build a data structure so that, given a query triangle R, the number of points of P lying in R can be determined quickly. Problems of this type are of crucial importance in computational geometry, as they can be used as subroutines in many seemingly unrelated algorithms. We present a survey of results and main techniques in this area.

this paper was published in Proceedings of the 29th Annual ACM Symposium on Theory of Computing, pages 618--625, 1997

We propose a new data structure to compute the Delaunay triangulation of a set of points in the plane. It combines good worst case complexity, fast behavior on real data, and small memory occupation.

We maintain the minimum spanning tree of a point set in the plane, subject to point insertions and deletions, in amortized time O(n 1/2 log 2 n) per update operation. We reduce the problem to maintaining bichromatic closest pairs, which we solve in time O(n # ) per update. Our algorithm uses a novel construction, the ordered nearest neighbor path of a set of points. Our results generalize to higher dimensions, and to fully dynamic algorithms for maintaining minima of binary functions, including the diameter of a point set and the bichromatic farthest pair. 1 Introduction A dynamic geometric data structure is one that maintains the solution to some problem, defined on a geometric input such as a point set, as the input undergoes update operations such as insertions or deletions of single points. Dynamic algorithms have been studied for many geometric optimization problems, including closest pairs [7, 23, 25, 26], diameter [7, 26], width [4], convex hulls [15, 22], linear ...

The Delaunay Tree is a hierarchical data structure that has been introduced in [6] and analyzed in [7,4]. For a given set of sites S in the plane and an order of insertion for these sites, the Delaunay Tree stores all the successive Delaunay triangulations. As proved before, the Delaunay Tree allows the insertion of a site in logarithmic expected time and linear expected space, when the insertion sequence is randomized.

We give a data structure that allows arbitrary insertions and deletions on a planar point set P and supports basic queries on the convex hull of P , such as membership and tangent-finding. Updates take O(log 1+" n) amortized time and queries take O(log n) time each, where n is the maximum size of P and " is any fixed positive constant. For some advanced queries such as bridge-finding, both our bounds increase to O(log 3=2 n). The only previous fully dynamic solution was by Overmars and van Leeuwen from 1981 and required O(log 2 n) time per update. 1 Introduction Although the algorithmic study of convex hulls is as old as computational geometry itself, the basic problem of optimally maintaining the planar convex hull under insertions and deletions of points [30, 34] remains unsolved and has been regarded by some as one of the foremost open problems in the area [14, 26]. Besides its natural appeal, such a dynamic data structure has a wide range of applications, as it is often us...

We propose a new data structure to compute the Delaunay triangulation of a set of points in the plane. It combines good worst case complexity, fast behavior on real data, small memory occupation and the possibility of fully dynamic insertions and deletions. The location structure is organized into several levels. The lowest level just consists of the triangulation, then each level contains the triangulation of a small sample of the level below. Point location is done by walking in a triangulation to determine the nearest neighbor of the query at that level, then the walk restarts from that neighbor at the level below. Using a small subset (3%) to sample a level allows a small memory occupation; the walk and the use of the nearest neighbor to change levels quickly locate the query.