The Complexity of Many Cells in Arrangements of Planes And Related Problems
The Complexity of Many Cells in Arrangements of Planes And Related Problems
Herbert Edelsbrunner
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For problem (a) we store, for each node v and for each point p, whose dual plane reaches v, the current candidate for the plane ttj lying immediately below p, -. The recursive algorithm can then proceed as follows. (i) At a node v lying at the bottom of the recursion, we construct the full arrange- ment of the set Ilv of the n^ planes t7, - associated with v, and locate in it each of the wiv corresponding points pj, using the point location algorithm of [EOS] or [Ch]. This can be accomplished i...n time Oim^ log^/iv), and yields initial candidates for the planes lying below each of these points. (ii) At each internal node v we process the set B^ of all points whose dual planes have reached the parent of v but have missed v, as follows. Let p, be a point in B^. Then the plane pf passes either above all the points ttj € Wt or below all of them. By duality, the point p. - lies either above all the planes tTj ^ Ilv or below all of them. In the latter case, no plane in Ilv can be the plane lying directly below p, -, so these points need not be further processed at v.
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