On k-sets in arrangements of curves and surfaces

Abstract

We extend the notion of k-sets and (≤k)-sets (see [3], [12], and [19]) to arrangements of curves and surfaces. In the case of curves in the plane, we assume that each curve is simple and separates the plane. A k-point is an intersection point of a pair of the curves which is covered by exactly k interiors of (or half-planes bounded by) other curves; the k-set is the set of all k-points in such an arrangement, and the (≤k)-set is the union of all j-sets, for j≤k. Adapting the probabilistic analysis technique of Clarkson and Shor [13], we obtain bounds that relate the maximum size of the (≤k)-set to the maximum size of a 0-set of a sample of the curves. Using known bounds on the size of such 0-sets we obtain asympotically tight bounds for the maximum size of the (≤k)-set in the following special cases: (i) If each pair of curves intersect at most twice, the maximum size is Θ(nkα(nk)). (ii) If the curves are unbounded arcs and each pair of them intersect at most three times, then the maximum size is Θ(nkα(n/k)). (iii) If the curves are x-monotone arcs and each pair of them intersect in at most some fixed number s of points, then the maximum size of the (≤k)-set is Θ(k2λs (nk)), where λs (m) is the maximum length of (m,s)-Davenport-Schinzel sequences. We also obtain generalizations of these results to certain classes of surfaces in three and higher dimensions. Finally, we present various applications of these results to arrangements of segments and curves, high-order Voronoi diagrams, partial stabbing of disjoint convex sets in the plane, and more. An interesting application yields and O(n log n) bound on the expected number of vertically visible features in an arrangement of n horizontal discs when they are stacked on top of each other in random order. This in turn leads to an efficient randomized preprocessing of n discs in the plane so as to allow fast stabbing queries, in which we want to report all discs containing a query point. © 1991 Springer-Verlag New York Inc.