Counting circular arc intersections

In this paper we present efficient algorithms for counting intersections in a collection of circles or circular arcs. We present a randomized algorithm to count intersections in a collection of n circles whose expected running time is O(n3/2+ϵ), for any ∈ > 0. We also develop another randomized algorithm to count intersections in a set of n circular arcs whose expected running time is O(n5/3+∈), for any ∈ > 0. If all arcs have the same radius, the (expected) running time can be improved to O(n3/2+∈), for any ∈ > 0.