Concrete and Abstract Voronoi Diagrams by Rolf Klein

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 airplane which are towards this web site than to the opposite ones. Such walls are of significant significance to computing device technological know-how and plenty of different fields. The problem is to compute Voronoi diagrams speedy. the matter is that their constitution relies on the idea of distance and one of these website. during this booklet the writer proposes a unifying process via introducing summary Voronoi diagrams. those are in line with the concept that of bisecting curves, that are required to have a few easy homes which are truly possessed by means of such a lot bisectors of concrete Voronoi diagrams. summary Voronoi diagrams might be computed successfully and there exists a worst-case effective set of rules of divide-and-conquer style that applies to all summary Voronoi diagrams gratifying a definite constraint. the writer indicates that this constraint is fulfilled by way of the concrete diagrams in accordance with huge periods of metrics within the plane.

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Concrete and Abstract Voronoi Diagrams

The Voronoi diagram of a suite of web sites is a partition of the airplane into areas, one to every web site, such that the area of every website comprises all issues of the aircraft which are in the direction of this website than to the opposite ones. Such walls are of significant value to computing device technological know-how and lots of different fields. The problem is to compute Voronoi diagrams speedy.

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P r o o f : If v is a (possibly not included) endpoint of/3 t h e n (/3,p,q) belongs to exactly one point induced by v. If/3 doesn't end at v then it gives rise to two borderlines, b' = (/3',p, q) and b" = (/3t! ,p, q), radiating from v. 5; this piece must be bordered by b~ and b'. Furthermore, p = p~ = min S ~ must hold. Thus, v induces the point (v, B), where B = {b~, b"}. 13. Conversely, if the set B of an arbitrary induced node (v, B) consists of two elements only, t h e n these must be of the form (/3', p,, q), (/3", p,, q), and the maximal (p~, q)-borderline containing/3' or/3" contains/3' U {v} U/3" and passes (v, B)o o Next, we turn to a close examination of the shape of the Voronoi regions.

If p,q e ~2 and p -< q t h e n J(p,q) := C(p,q) ND(p,q) is a curve homeomorphic to (0,1). T h e intersection of two such curves, J(p,q) and J(v, w), consists of finitely m a n y connected components. 3. By definition, J(p, q) is closed, as a set. e. one-to-one and bicontinuous) then limt_~±oo If(t)[ = oc. (For, if t, --* cc and [f(t~)l < C for all n, then the infinite set { f ( t . ) ; n >_ 1} would be bounded and would, by the BolzanoWeierstrass Theorem, have an accumulation point, z, such t h a t for a subsequence (t,~) of (t,) f (t,,) --* z holds.

P r o p e r t y 2) means t h a t the metric d is continuous at infinity with respect to the Euclidean metric. 3 would be hindered). Nice metrics have nice Voronoi diagrams; we summarize as follows. 13 Let d be nice, and let V(S) be the Voronoi diagram of a finite set S C ~2 with respect to d. Then each region R(p, S) is a path-connected set with a non-empty interior. It contains all points of the plane that are closer (with respect to d) to p than to any other point in S. Its closure consists of points that have p as a nearest neighbor in S.

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