On Computing the Centroid of the Vertices of an Arrangement and Related Problems

Abstract

We consider the problem of computing the centroid of all the vertices in a non-degenerate arrangement of n lines. The trivial approach requires the enumeration of all $n \choose 2$ vertices. We present an $\O(n \log^2{n})$ algorithm for computing this centroid. For arrangements of n segments we give an $\O(n^{\frac{4}{3}+\epsilon})$ algorithm for computing the centroid of its vertices. For the special case that all the segments of the arrangement are chords of a simply connected planar region we achieve an $\O(n \log^5{n})$ time bound. Our bounds also generalize to certain natural weighted versions of those problems.