PROBLEM LINKS:
Author:sikander_nsit
Tester:sikander_nsit
Editorialist:sikander_nsit
DIFFICULTY:
MEDIUM
PROBLEM:
In the problem we had to find the sum of the smallest k elements in the given array range.
EXPLANATION:
Brute force solution would not work because of the constraints. Segment tree can be used for this.
Each node of the segment tree will consist of two arrays : first is the list of elements in the range in sorted order and the second stores the cumulative sum.
For every query we can find a list of S sorted arrays such that their union is the given range in query [L,R].
1. Go through all the arrays and if any have length > k, truncate to length k .
2. Identify the largest remaining array A. If more than one, pick one.
3. Pick the middle element M of the largest array A.
4. Use a binary search on the remaining arrays to find the same element (or the smallest element > M).
5. Based on the indexes of the various elements, calculate the total number of elements <= M. This should give you B, the count of numbers <= M.
6. If k < B, truncate all the arrays at the split points you've found and iterate on the smaller arrays (use the bottom halves).
7. If k > B, truncate all the arrays at the split points you've found and iterate on the smaller arrays (use the top halves, and search for element (k-B)). Add the cumulative sums till the split points to the answer.
When you get to the point where you only have one element per array (or 0), make a new array of size n with those data, sort, and calculate the sum of first k elements. Because you're always guaranteed to remove at least half of one array, in S iterations, you'll get rid of half the elements. That means there are S log k iterations. Each iteration is of order S log k (due to the binary searches), so the whole thing is S^2 (log k)^2 in worst case. But the actual performance would be much better than the worst case.
AUTHOR'S SOLUTION:
Author's solution can be found here.