Statement: We have an array A. Lets take every index i and find L[i] and R[i] — closest left and right index j such that A[j] >= A[i] respectively. Then I state that if for every i in 0..n-1 we sum min(i — L[i], R[i] — i) it will be O(N log N). It`s useful if we fix A[i] as the maximum number of segment and if the length of segments is fixed, we can naively iterate over all such segments.
Proof: Iterating over such i we also have the second end of this segment. Lets for every i calculate the maximum number of times that i can be the second end of such segment. Let index j be the first end of such segment. I will prove for j > i, because for j < i its symmetrical. Lets take maximum j > i such that j can be first end of this segment. Then A[j] > A[j — 1] and A[j] > A[j — 2] ... A[j] > A[i]. So now lets take second maximum j such than j can be first end of this segment. Lets call the maximum j j1, second maximum j2. Notice that j2 — i <= j1 — j2. Because otherwise we wouldnt take j2 with i, consequently 2 * j2 <= j1 + i. Consequently the number of such js is maximum log n.
Usage: The most beautiful usage of this I have recently seen in task https://codeforces.me/problemset/problem/1175/F. I was thinking about hashes, etc. , but there is a solution just fixing the maximum of permutation, let it be A[i], find L[i], R[i] and iterate all segments with length A[i] that include i, but neither L[i] nor R[i]. Notice that it`s even less that described above so we can explicitly get our candidates to be the permutations and their number is <= N log N.
