COMPACT INTERVAL TREES: A DATA STRUCTURE FOR CONVEX HULLS

Abstract
In this paper, we investigate the problem of finding the common tangents of two convex polygons that intersect in two (unknown) points. First, we give a Θ( log 2n) bound for algorithms that store the polygons in independent arrays. Second, we show how to beat the lower bound if the vertices of the convex polygons are drawn from a fixed set of n points. We introduce a data structure called a compact interval tree that supports common tangent computations, as well as the standard binary-search-based queries, in O( log n) time apiece. Third, we apply compact interval trees to solve the subpath hull query problem: given a simple path, preprocess it so that we can find the convex hull of a query subpath quickly. With O(n log n) preprocessing, we can assemble a compact interval tree that represents the convex hull of a query subpath in O( log n) time. In order to represent arrangements of Lines implicitly, Edelsbrunner et al. used a less efficient structure, called bridge trees, to solve the subpath hull query problem. Our compact interval trees improve their results by a factor of O( log n). Thus, the present paper replaces the paper on bridge trees referred to by Edelsbrunner et al.

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