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I have been wondering over the following problem:

Given a set $S$ of intervals on the number line. We can do two operations on them:

  1. Add a new interval $[l_i,r_i]$ to $S$
  2. Given an interval $[l_j, r_j]$, which is possibly not in $S$, find the longest interval from $S$ which is contained entirely in $[l_j, r_j]$

If I didn't have operation $1$ I have found a solution using a persistent segment tree.

For the full version, there is the trivial solution which runs in $O(QN)$ ($Q$ - number of queries, $N$ - total number of intervals added). I also found that using interval tree, I can directly find which intervals intersect the query interval and then I can check them manually. However, is there an even better solution? Maybe $O(Q\log N)$ or $O(Q\log^2N)$?

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  • $\begingroup$ @thefunkyjunky Please add a reference to the original problem. $\endgroup$
    – John L.
    Commented Jul 4, 2019 at 14:35
  • $\begingroup$ @Apass.Jack sorry, I'll add explanations for Q and N. There is no original problem(you can say I made it up), that's why I am asking if there is a better solution. $\endgroup$ Commented Jul 4, 2019 at 16:18
  • $\begingroup$ I'm a bit confused about your notation. Your first bound is $O(QN)$, which makes me think that what you're measuring includes the time to process all queries. Your second bounds, however, don't include $Q$ at all. But if you're processing all queries, the answer must be at least $\Omega(Q)$, since you have to read every query. $\endgroup$
    – jbapple
    Commented Jul 7, 2019 at 22:00
  • $\begingroup$ @jbapple, fixed. $\endgroup$ Commented Jul 7, 2019 at 22:27
  • $\begingroup$ Can you describe your solution using a persistent segment tree? $\endgroup$
    – jbapple
    Commented Jul 7, 2019 at 22:32

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