Manhattan Subarrays

2000ms

262144K

Description:

Suppose you have two points $$$p = (x_p, y_p)$$$ and $$$q = (x_q, y_q)$$$. Let's denote the Manhattan distance between them as $$$d(p, q) = |x_p - x_q| + |y_p - y_q|$$$.

Let's say that three points $$$p$$$, $$$q$$$, $$$r$$$ form a bad triple if $$$d(p, r) = d(p, q) + d(q, r)$$$.

Let's say that an array $$$b_1, b_2, \dots, b_m$$$ is good if it is impossible to choose three distinct indices $$$i$$$, $$$j$$$, $$$k$$$ such that the points $$$(b_i, i)$$$, $$$(b_j, j)$$$ and $$$(b_k, k)$$$ form a bad triple.

You are given an array $$$a_1, a_2, \dots, a_n$$$. Calculate the number of good subarrays of $$$a$$$. A subarray of the array $$$a$$$ is the array $$$a_l, a_{l + 1}, \dots, a_r$$$ for some $$$1 \le l \le r \le n$$$.

Note that, according to the definition, subarrays of length $$$1$$$ and $$$2$$$ are good.

Input:

The first line contains one integer $$$t$$$ ($$$1 \le t \le 5000$$$) — the number of test cases.

The first line of each test case contains one integer $$$n$$$ ($$$1 \le n \le 2 \cdot 10^5$$$) — the length of array $$$a$$$.

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

It's guaranteed that the sum of $$$n$$$ doesn't exceed $$$2 \cdot 10^5$$$.

Output:

For each test case, print the number of good subarrays of array $$$a$$$.

Sample Input:

Sample Output:

10
12
3

Note:

In the first test case, it can be proven that any subarray of $$$a$$$ is good. For example, subarray $$$[a_2, a_3, a_4]$$$ is good since it contains only three elements and:

$$$d((a_2, 2), (a_4, 4)) = |4 - 3| + |2 - 4| = 3$$$ $$$<$$$ $$$d((a_2, 2), (a_3, 3)) + d((a_3, 3), (a_4, 4)) = 3 + 1 + 2 + 1 = 7$$$;
$$$d((a_2, 2), (a_3, 3))$$$ $$$<$$$ $$$d((a_2, 2), (a_4, 4)) + d((a_4, 4), (a_3, 3))$$$;
$$$d((a_3, 3), (a_4, 4))$$$ $$$<$$$ $$$d((a_3, 3), (a_2, 2)) + d((a_2, 2), (a_4, 4))$$$;

In the second test case, for example, subarray $$$[a_1, a_2, a_3, a_4]$$$ is not good, since it contains a bad triple $$$(a_1, 1)$$$, $$$(a_2, 2)$$$, $$$(a_4, 4)$$$:

$$$d((a_1, 1), (a_4, 4)) = |6 - 9| + |1 - 4| = 6$$$;
$$$d((a_1, 1), (a_2, 2)) = |6 - 9| + |1 - 2| = 4$$$;
$$$d((a_2, 2), (a_4, 4)) = |9 - 9| + |2 - 4| = 2$$$;

So, $$$d((a_1, 1), (a_4, 4)) = d((a_1, 1), (a_2, 2)) + d((a_2, 2), (a_4, 4))$$$.

Informação

Provedor Codeforces

Código CF1550C