Since the circles do not intersect, we can look at them as intervals. More specifically, we observe their intersection with the ~x~-axis.

Let us construct a relation contains such that ~A~ contains ~B~ if interval ~B~ is inside of interval ~A~, with allowed touching on the edges.

Let us construct a relation directly_contains such that ~A~ directly_contains ~B~ if ~A~ contains ~B~ and there is no other circle ~C~ for which holds that ~A~ contains ~C~ and ~C~ contains ~B~.

Intuitively, ~A~ directly_contains ~B~ if ~B~ is one of the first smaller circles in ~A~.

Because there is at most one circle that directly_contains an arbitrary circle, this relation is a tree.

Every circle will increase the number of regions by ~1~ or ~2~. In the case when a circle directly contains more other circles which touch along all its length, that circle is going to increase the solution by ~2~. In the other case, by ~1~.

The tree of circles can be built using the sweep algorithm where an event is at the beginning or end of a circle. The events are processed by their ~x~ coordinates. As the structure of the sweep algorithm, we will use a stack which keeps track of the current parent circle and whether we have lined up every directly contained circles next to each other. In the moment of popping a circle from the stack, the solution is increased by ~2~ if all the directly contained circles are lined up next to each other. In the other case, the solution is increased by ~1~. The final solution needs to be incremented by ~1~ because it represents the whole region above all the circles.

The complexity of the algorithm is ~\mathcal O(N \log N)~ where ~N~ is the number of circles.