201263: [AtCoder]ARC126 D - Pure Straight

Problem Statement

Given is a sequence of $N$ positive integers $A = (A_1, A_2, \ldots, A_N)$, where each $A_i$ is $1$, $2$, $\ldots$, or $K$.

You can do the following operation on this sequence any number of times.

• Swap two adjacent elements, that is, choose $i$ and $j$ such that $|i-j|=1$ and swap $A_i$ and $A_j$.

Find the minimum number of operations needed to make $A$ satisfy the following condition.

• $A$ contains $(1, 2, \ldots, K)$ as a contiguous subsequence, that is, there is a positive integer $n$ at most $N-K+1$ such that $A_n = 1$, $A_{n+1} = 2$, $\ldots$, and $A_{n+K-1} = K$.

Constraints

• $2\leq K\leq 16$
• $K \leq N\leq 200$
• $A_i$ is $1$, $2$, $\ldots$, or $K$.
• $A$ contains at least one occurrence of each of $1, 2, \ldots, K$.

Input

Input is given from Standard Input in the following format:

$N$ $K$
$A_1$ $A_2$ $\ldots$ $A_N$

Output

Print the minimum number of operations needed to make $A$ satisfy the condition.

Sample Input 1

4 3
3 1 2 1

Sample Output 1

2

One optimal sequence of operations is as follows.

• Swap $A_1$ and $A_2$, changing $A$ to $(1,3,2,1)$.
• Swap $A_2$ and $A_3$, changing $A$ to $(1,2,3,1)$.
• Now we have $A_1 = 1$, $A_2 = 2$, $A_3 = 3$, satisfying the condition.

Sample Input 2

5 5
4 1 5 2 3

Sample Output 2

5

Sample Input 3

8 4
4 2 3 2 4 2 1 4

Sample Output 3

5