I couldn't solve this. Please do tell me any simple approach which can pass the time constraint.
Problem:
You are given n intervals which are termed as special intervals. Each interval is of a different type.
Again, you are given a set of q non-special intervals. For each non-special interval in the given q intervals, you have to find the number of different types of special intervals in that non-special interval.
Note: A special interval is inside a non-special interval if there exists a point x which belongs to both special interval and non-special interval.
Input format
First line: n denoting the number of special intervals
Next n lines: Two integers denoting lspecial[i] and rspecial[i] denoting the range [l,r] for the ith special interval.
Next line: q denoting the number of non-special intervals
Next q lines: Two integers denoting lnonspecial[i] and rnonspecial[i] denoting the range [l,r] for the ith non-special interval.
Output format
print q space-seperated integers denoting the answer for each of the q non-special integers.
Constraints
1<=n<=10^5
-10^9<=lspecial[i]<=10^9
-10^9<=rspecial[i]<=10^9
1<=q<= 5 * 10^4
-10^9<=lnonspecial[i]<=10^9
-10^9<=rnonspecial[i]<=10^9
Sample Input
3
1 2
1 5
1 7
3
1 3
3 3
6 7
Sample Output
3 2 1
Time Limit 1 second
Hi, I guess I have found a solution, so: Let's divide special intervals that are in one non-special interval in two types: 1) starting before the left bound and ending at or after the left bound; 2) starting inside (including bounds) the non-special interval and ending somewhere (it doesn't matter, we have it inside query). Two groups do not intersect so we can count independently.
We will answer offline.
Let's count first type by using scan-line algorithm and adding the amount of opened special intervals to answer of opening now non-special intervals (so, when we meet the event like "left bound of non-special cut" we add to answer of it).
For second type we can use binsearch for sorted vector(or array) of special intervals (we have to sort by left bound). The answer is amount of intervals which have left bound in non-special interval of query (between left and right of non-special). It's easy to do using binsearch.
So, we have O((N + Q)log(N + Q)) for first types and O(Nlog(N) + Qlog(N)) for second types. The answer for interval is the sum of first type and second.
I guess it works fast enough to pass the time constraint.
There is no need to answer offline. We just need to count the intervals which overlap with [l, r]. The answer would be nothing but
Total special intervals — intervals ending before l — intervals starting after r.
The later 2 terms can be found in O(1) time with some obvious preprocessing
Ah, yeah, you're right, I made it a lot harder.