Finding the number of intersections of $n$ line segments with endpoints on two parallel lines

Question

Let there be two sets of $n$ points:

$A=\{p_1, p_2, \dots, p_n\}$ on $y=0$

$B=\{q_1, q_2, \dots, q_n\}$ on $y=1$

Each point $p_i$ is connected to its corresponding point $q_i$ to form a line segment.

Example:

I need to write a divide-and-conquer algorithm which returns the number of intersection points of all $n$ line segments and runs in $O(n logn)$.

I was reading about Sweep Line Algorithm but found out its complexity depends on the number of intersections, which can be ${O}{n\choose 2} \subseteq O(n^2)$ (besides the fact that it isn't a divide-and-conquer algorithm). I believe I should use the fact that each set of points has the same $y$ value, but I'm not sure how. Any suggestions?

Answer 1

Since all points are distinct, this is a version of the Counting Inversions problem. First, sort the points $p_1, \dots, p_n$ in order of increasing $x$ coordinate to obtain an ordered list $p[1],p[2], \dots, p[n]$. We now relabel the points $q_1, \dots, q_n$ with respect to this new ordering of the $p$'s such that $p[i]$ and $q[i]$ are endpoints of a line segment, for all $1 \leq i \leq n$. Altogether, these steps take $O(n \log n)$ time.

Now, we want to count the number of inversions in $q[1], \dots, q[n]$; an inversion is a situation where $i < j$ and $q[i] > q[j]$ (we compare the points by their $x$ coordinates). This is because $i < j$ implies $p[i] < p[j]$ by construction, and $p[i] < p[j]$ and $q[i] > q[j]$ if and only if the line segments $(p[i], q[i])$ and $(p[j], q[j])$ intersect. Hence, the number of line segment intersections is precisely the number of inversions in $q[1], \dots, q[n]$.

There is a well known divide and conquer approach for counting these inversions. Can you take it from here?