Multicolor Combination Lemma

Abstract

We present an extension of the Combination Lemma of Guibas et al. (1983) that expresses the complexity of one or several faces in the overlay of many arrangements (as opposed to just two arrangements in (Guibas et al. 1989)), 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 first show that the complexity of a single face in an arrangement of k simple polygons with a total of n sides is Θ(nα(k)), where α(·) is the inverse of Ackermann's function. We also give a new and simpler proof of the bound O( m λ 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 λ 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+ m w )λ s+2(n)/n) , where w is the total complexity of the arrangement. Several other related results are also obtained.