Download Concrete and Abstract Voronoi Diagrams by Rolf Klein PDF

By Rolf Klein

The Voronoi diagram of a collection of websites is a partition of the airplane into areas, one to every website, such that the quarter of every website comprises all issues of the aircraft which are towards this web site than to the opposite ones. Such walls are of significant value to computing device technology and plenty of different fields. The problem is to compute Voronoi diagrams fast. the matter is that their constitution is determined by the idea of distance and this type of web site. during this e-book the writer proposes a unifying technique via introducing summary Voronoi diagrams. those are in keeping with the idea that of bisecting curves, that are required to have a few easy homes which are truly possessed by way of such a lot bisectors of concrete Voronoi diagrams. summary Voronoi diagrams should be computed successfully and there exists a worst-case effective set of rules of divide-and-conquer sort that applies to all summary Voronoi diagrams pleasurable a definite constraint. the writer indicates that this constraint is fulfilled via the concrete diagrams in line with huge periods of metrics within the plane.

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Additional info for Concrete and Abstract Voronoi Diagrams

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1 yields the same regions and the same diagram; but the meaning of Dip , q) can be different ifp < q). We will be concerned only with such systems of bisecting curves that have the following properties. 2 A system of bisecting curves (S, {J(p,q);p,q 6 S,p ¢ q}) is called admissible iff the following conditions are fulfilled. 1. The intersection of any two bisecting curves consists of finitely many connected components. 2. For each non-empty subset S' of S A) R(p~ S') is path-connected and has a non-empty interior, for each p E S r.

To each edge e of M the DCEL of M contains one edge record that holds the following information. For each of the two vertices, vl and v2, incident with e the edge record of e stores • a pointer to the record of that edge e~ which is encountered first on a terclockwise march around vi, when starting from e. • the name of that face of M which is bordered by e and by ei.

The part of i(V(S)) t h a t is connected to an edge out of e l , . . , e~ can be encircled by a simple closed curve, C, that is contained in F U {~}, for some bounded face F. 4. 21: No cut-vertex is contained in i(V(S)). then the same argumentation can be applied to those incident edges that are not part of F, cf. 11. [] For the following, let P be a simple closed curve. An embedded planar graph M is called a F-map iff M has at least 3 faces, one of t h e m being the closure of the outer domain of F.

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