The Voronoi diagram is a fundamental structure in computationalgeometry and arises naturally in many different fields. This chapter surveys properties of the Voronoi diagram and its geometric dual, the Delaunay triangulation. The emphasis is on practical algorithms for the construction of Voronoi diagrams. 1 Introduction Let S be a set of n points in d-dimensional euclidean space E d . The points of S are called sites. The Voronoi diagram of S splits E d into regions with one region for each site, so that the points in the region for site s2S are closer to s than to any other site in S. The Delaunay triangulation of S is the unique triangulation of S so that there are no elements of S inside the circumsphere of any triangle. Here `triangulation' is extended from the planar usage to arbitrary dimension: a triangulation decomposes the convex hull of S into simplices using elements of S as vertices. The existence and uniqueness of the Delaunay triangulation are perhaps not obvio...

The reverse search technique has been recently introduced by the authors for efficient enumeration of vertices of polyhedra and arrangements. In this paper, we develop this idea in a general framework and show its broader applications to various problems in operations research, combinatorics, and geometry. In particular, we propose new algorithms for listing (i) all triangulations of a set of n points in the plane, (ii) all cells in a hyperplane arrangement in R d , (iii) all spanning trees of a graph, (iv) all Euclidean (non-crossing) trees spanning a set of n points in the plane, (v) all connected induced subgraphs of a graph, and (vi) all topological orderings of an acyclic graph. Finally we propose a new algorithm for the 0-1 integer programming problem which can be considered as an alternative to the branch-and-bound algorithm. 1 Introduction The listing of all objects that satisfy a specified property is a fundamental problem in combinatorics, computational geometr...

We consider the problem of enumerating triangulations of n points in the plane in general position. We introduce a tree of triangulations and present an algorithm for enumerating triangulations in O(log log n) time per triangulation. It improves the previous bound by almost linear factor. Keywords: Triangulations; Enumeration; Reverse Search 1

This thesis contains new results on the subject of polygonal structure reconfiguration. Specifically, the types of structures considered here are polygons, polygonal chains, triangulations, and polyhedral surfaces. A sequence of vertices (points), successively joined by straight edges, is a polygonal chain. If the sequence is cyclic, then the object is a polygon. A planar triangulation is a set of vertices with a maximal number of non-crossing straight edges joining them. A polyhedral surface is a three-dimensional structure consisting of flat polygonal faces that are joined by common edges. For each of these structures there exist several methods of reconfiguration. Any such method must provide a well-defined way of transforming one instance of a struc-ture to any other. Several types of reconfigurations are reviewed in the introduction, which is followed by new results. We begin with efficient algorithms for comparing monotone chains. Next, we prove that flat chains with unit-length edges and an-gles within a wide range always admit reconfigurations, under the dihedral model of motion. In this model, angles and edge lengths are preserved. For the universal