A15262 | Permutation Graph

A permutation is an array consisting of $n$ distinct integers from $1$ to $n$ in arbitrary order. For example, $[2,3,1,5,4]$ is a permutation, but $[1,2,2]$ is not a permutation ( $2$ appears twice in the array) and $[1,3,4]$ is also not a permutation ( $n=3$ but there is $4$ in the array).

You are given a permutation of $1,2,\dots,n$ , $[a_1,a_2,\dots,a_n]$ . For integers $i$ , $j$ such that $1\le i<j\le n$ , define $\operatorname{mn}(i,j)$ as $\min\limits_{k=i}^j a_k$ , and define $\operatorname{mx}(i,j)$ as $\max\limits_{k=i}^j a_k$ .

Let us build an undirected graph of $n$ vertices, numbered $1$ to $n$ . For every pair of integers $1\le i<j\le n$ , if $\operatorname{mn}(i,j)=a_i$ and $\operatorname{mx}(i,j)=a_j$ both holds, or $\operatorname{mn}(i,j)=a_j$ and $\operatorname{mx}(i,j)=a_i$ both holds, add an undirected edge of length $1$ between vertices $i$ and $j$ .

In this graph, find the length of the shortest path from vertex $1$ to vertex $n$ . We can prove that $1$ and $n$ will always be connected via some path, so a shortest path always exists.

Each test contains multiple test cases. The first line contains the number of test cases $t$ ( $1 \le t \le 5\cdot 10^4$ ). Description of the test cases follows.

The first line of each test case contains one integer $n$ ( $1\le n\le 2.5\cdot 10^5$ ).

The second line of each test case contains $n$ integers $a_1$ , $a_2$ , $\ldots$ , $a_n$ ( $1\le a_i\le n$ ). It's guaranteed that $a$ is a permutation of $1$ , $2$ , $\dots$ , $n$ .

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

For each test case, print a single line containing one integer — the length of the shortest path from $1$ to $n$ .