Array Eversion
Contest: Technocup 2022 - Elimination Round 3
Difficulty rating: 900
Tags: greedy

You are given an array $$$a$$$ of length $$$n$$$.

Let's define the eversion operation. Let $$$x = a_n$$$. Then array $$$a$$$ is partitioned into two parts: left and right. The left part contains the elements of $$$a$$$ that are not greater than $$$x$$$ ($$$\le x$$$). The right part contains the elements of $$$a$$$ that are strictly greater than $$$x$$$ ($$$> x$$$). The order of elements in each part is kept the same as before the operation, i. e. the partition is stable. Then the array is replaced with the concatenation of the left and the right parts.

For example, if the array $$$a$$$ is $$$[2, 4, 1, 5, 3]$$$, the eversion goes like this: $$$[2, 4, 1, 5, 3] \to [2, 1, 3], [4, 5] \to [2, 1, 3, 4, 5]$$$.

We start with the array $$$a$$$ and perform eversions on this array. We can prove that after several eversions the array $$$a$$$ stops changing. Output the minimum number $$$k$$$ such that the array stops changing after $$$k$$$ eversions.

Input format:
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 100$$$). Description of the test cases follows.

The first line contains a single integer $$$n$$$ ($$$1 \le n \le 2 \cdot 10^5$$$).

The second line contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le 10^9$$$).

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2 \cdot 10^5$$$.

Output format:
For each test case print a single integer $$$k$$$ — the number of eversions after which the array stops changing.

Constraints:
Time limit: 1.0 s
Memory limit: 256.0 MB

Note:
Consider the fist example.

- The first eversion: $$$a = [1, 4, 2, 5, 3]$$$, $$$x = 3$$$. $$$[2, 4, 1, 5, 3] \to [2, 1, 3], [4, 5] \to [2, 1, 3, 4, 5]$$$.
- The second and following eversions: $$$a = [2, 1, 3, 4, 5]$$$, $$$x = 5$$$. $$$[2, 1, 3, 4, 5] \to [2, 1, 3, 4, 5], [] \to [2, 1, 3, 4, 5]$$$. This eversion does not change the array, so the answer is $$$1$$$.

Consider the second example.

- The first eversion: $$$a = [5, 3, 2, 4, 1]$$$, $$$x = 1$$$. $$$[5, 3, 2, 4, 1] \to [1], [5, 3, 2, 4] \to [1, 5, 3, 2, 4]$$$.
- The second eversion: $$$a = [1, 5, 3, 2, 4]$$$, $$$x = 4$$$. $$$[1, 5, 3, 2, 4] \to [1, 3, 2, 4], [5] \to [1, 3, 2, 4, 5]$$$.
- The third and following eversions: $$$a = [1, 3, 2, 4, 5]$$$, $$$x = 5$$$. $$$[1, 3, 2, 4, 5] \to [1, 3, 2, 4, 5], [] \to [1, 3, 2, 4, 5]$$$. This eversion does not change the array, so the answer is $$$2$$$.

Example input:
3
5
2 4 1 5 3
5
5 3 2 4 1
4
1 1 1 1

Example output:
1
2
0

Editorial:
Lemma: If $$$x$$$ is max element of the array then eversion doesn't change the array.
Proof: In spite of the fact that division is stable, all elements will be passed to the left part. Their order won't be changed.
Lemma: The lastest element after eversion is the rightest element of the array which is greater than $$$x$$$ and lefter than $$$x$$$ in the array.
Proof: Look at all elements that are greater than $$$x$$$. This is the right part of the division. Due to stable division, the right element that is greater will be new $$$x$$$. There're no elements greater than $$$x$$$ and righter than $$$x$$$ because of eversion definition.
Let's build a sequence $$$x_a, x_{a-1}, \dots x_0$$$, where $$$x_0 = a_n$$$, $$$x_{i + 1}$$$ — the rightest element lefter than $$$x_i$$$ and greater than $$$x_i$$$. The answer is equals to $$$a$$$ because $$$\{x_i\}$$$ is a sequence of last elements overall eversions.
Example: $$$$$$6\ 10\ 4\ 17\ 9\ 2\ 8\ 1$$$$$$
Sequence $$$\{x_i\}$$$ — is $$$1, 8, 9, 17$$$. Answer is $$$3$$$.