This algorithm can solve some difficult data structure problems. Like this: Give an array a[] of size N, we query Q times. Each time we want to get the mode number(the value that appears most often in a set of data) of subsequence a[l], a[l + 1] .. a[r].

Maybe you want to solve it by segment tree or some other data structures. However, if you know the answer of [l, middle], [middle + 1, r], it's difficult to get the answer of [l, r].

Here is an offline algorithm which can solve this problem in O(N * sqrt(N) * log(N) + Q log Q).

First we need a data structure that can support: insert a number. delete a number. get the mode number. It can be solved easily by binary heap / balanced binary search tree. We call this data structure DS.

Secondly, if we have all the numbers in [l, r] organized in DS, we can transform to [l', r'] in O((|l-l'| + |r-r'|) * log N). We need a good tranform order.

S = the max integer number which is less than sqrt(N);
bool cmp(Query A, Query B)
{
  if (A.l / S ！= B.l / S) return A.l / S < B.l / S;
  return A.r > B.r
}

We sort all the queries by this comparator, and it can be proofed easily that the total transform costs is N sqrt(N) log N.