The Complexity of One or Many Faces in the Overlay ofMany
نویسنده
چکیده
We present an extension of the Combination Lemma of GSS89] that expresses the complexity of one or several faces in the overlay of many arrangements, as a function of the number of arrangements, the number of faces, and the complexities of these faces in the separate arrangements. Several applications of the new Combination Lemma are presented: We rst show that the complexity of a single face in an arrangement of k simple polygons with a total of n sides is (nn(k)), where () is the inverse of Ackerman's function. We also give a new and simpler proof of the bound O (p mm s+2 (n)) on the total number of edges of m faces in an arrangement of n Jordan arcs, each pair of which intersect in at most s points, where s (n) is the maximum length of a Davenport-Schinzel sequence of order s with n symbols. We extend this result, showing that the total number of edges of m faces in a sparse arrangement of n Jordan arcs is O (n + p m p w) s+2 (n) n , where w is the total complexity of the arrangement. Several other applications and variants of the Combination Lemma are also presented.
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تاریخ انتشار 1995