Hello Codeforces community, This is an editorial for an 11-year-old problem in CF, and the official editorial still has it TODO, so yeah, why not?
I'll use step-by-step guides to demonstrate the path to finding the solution to this problem, and I hope it will be helpful to other people.
1. The intervals are given offline.
In this subtask, let's assume that the intervals are given to us, and the queries are asked after all the intervals are given.
1.1. $$$n \le 1000$$$
Note: this is the whole intuition behind the main solution.
Let's us define an edge from interval $$$i$$$ to $$$j$$$ bidirectional when $$$i$$$, $$$j$$$ intersect but none of which contains the other(e.g. [1, 10] and [5, 15]).
And an edge unidirectional from $$$i$$$ to $$$j$$$ when $$$i$$$ is contained within $$$j$$$.
Now we can use 2 unidirectional edges to form a bidirectional edge.
We draw a unidirectional edge from interval $$$i$$$ to $$$j$$$, when we can directly get to $$$j$$$ from $$$i$$$. Let this graph be $$$G$$$. Take an $$$Scc$$$ from $$$G$$$, $$$C$$$. What we can notice in this graph is that there is a path between all pairs of vertices in $$$C$$$. So let's condensate the $Scc$s into single vertices and create the condensation graph.
Let $$$L_C be the minimum l in all of intervals of C, R_C is similarly defined$$$
