W. Kandinsky: Circles in a Circle, 1923


We are given a set \(\cal C\) of \(m\) circles in the plane \({\mathbb R}^2\).
  1. Design an algorithm that tests if all circles in the set \(\cal C\) lie inside one of the circles.
  2. Design an algorithm that reports all pairs of intersecting circles.
  3. Design an algorithm that outputs a directed graph \(\cal D\) on \(\cal C\), in which there is an edge from \(C_1\) to \(C_2\) if circle \(C_1\) lies inside circle \(C_2\) and there is no circle \(C_3 \in \cal C\) in between (note that \(\cal D\) is a forest).
Prove the correctness of your algorithms and find their running times.