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

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.

In the plane, the post-ofice problem, which asks for the closest site to a query site, and retraction motion planning, which asks for a one-dimensional retract of the free space of a robot, are both classt-cally solved by computing a Voronoi diagram. When the sites are k disjoint convex sets, we give a com-pact representation of the Voronoi diagram, using O(k) line segments, that is suficient for logarithmic time post-ofice location queries and motion plan-ning. If these sets are polygons with n total vertices, we compute this diagram optimally in O ( k log n) de-terministic time for the Euclidean metric and in O(k logn logm) deterministic time for the convex distance function defined by a convex m-gon. 1

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.

This short note considers the problem of point location in a Delaunay triangulation of n random points, using no additional preprocessing or storage other than a standard data structure representing the triangulation. A simple and easy-to-implement (but, of course, worst-case suboptimal) heuristic is shown to take expected time O(n ).

The Delaunay Tree is a hierarchical data structure that was introduced in [BT86]. It is defined from the Delaunay triangulation and, roughly speaking, represents a triangulation as a hierarchy of balls. It allows a semi-dynamic construction of the Delaunay triangulation of a finite set of n points in any dimension. In this paper, we prove that a randomized construction of the Delaunay Tree (and thus, of the Delaunay triangulation) can be done in O(n log n) expected time in the plane and in O i n d d 2 e j expected time in d-dimensional space. These results are optimal for fixed d. The algorithm is extremely simple and experimental results are given.

We present a fully dynamic randomized data structure that can answer queries about the convex hull of a set of n points in three dimensions, where insertions take O(log 3 n) expected amortized time, deletions take O(log 6 n) expected amortized time, and extreme-point queries take O(log 2 n) worst-case time. This is the first method that guarantees polylogarithmic update and query cost for arbitrary sequences of insertions and deletions, and improves the previous O(n ε)-time method by Agarwal and Matouˇsek a decade ago. As a consequence, we obtain similar results for nearest neighbor queries in two dimensions and improved results for numerous fundamental geometric problems (such as levels in three dimensions and dynamic Euclidean minimum spanning trees in the plane). 1

Brute-force dense matching is usually not satisfactory because the same search range is used for the entire image, yielding potentially many false matches. In this paper, we propose a progressive scheme for stereo matching which uses two fundamental concepts: the disparity gradient limit principle and the least commitment strategy. The first states that the disparity should vary smoothly almost everywhere, and the disparity gradient should not exceed a certain limit. The second states that we should first select only the most reliable matches and therefore postpone unreliable decisions until enough confidence is accumulated. Our technique starts with a few reliable point matches obtained automatically via feature correspondence or through user input. New matches are progressively added during an iterative matching process. At each stage, the current reliable matches constrain the search range for their neighbors according to the disparity gradient limit, thereby reducing potential matching ambiguities of those neighbors. Only unambiguous matches are selected and added to the set of reliable matches in accordance with the least commitment strategy. In addition, a correlation match measure that allows rotation of the match template is used to provide a more robust estimate. The entire process is cast within a Bayesian inference framework. Experimental results illustrate the robustness of our proposed dense stereo matching approach.

We use here the results on the influence graph[1] to adapt them for particular cases where additional information is available. In some cases, it is possible to improve the expected randomized complexity of algorithms from O(nlog n) to O(n log ⋆ n). This technique applies in the following applications: triangulation of a simple polygon, skeleton of a simple polygon, Delaunay triangulation of points knowing the EMST (euclidean minimum spanning tree).