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C - Max MEX

Score : $300$ points

Problem Statement

You are given a length-$N$ sequence of non-negative integers.
When $B$ is a sequence obtained by choosing $K$ elements from $A$ and concatenating them without changing the order, find the maximum possible value of $MEX(B)$.

Here, for a sequence $X$, we define $MEX(X)$ as the unique non-negative integer $m$ that satisfies the following conditions:

• Every integer $i$ such that $0 \le i < m$ is contained in $X$.
• $m$ is not contained in $X$.

Constraints

• All values in the input are integers.
• $1 \le K \le N \le 3 \times 10^5$
• $0 \le A_i \le 10^9$

Input

The input is given from Standard Input in the following format:

$N$ $K$

$A_1$ $A_2$ $\dots$ $A_N$

Output

Print the answer.

7 3
2 0 2 3 2 1 9

3

In this input, $A=(2,0,2,3,2,1,9)$, and you obtain the sequence $B$ by picking $K=3$ elements. For example,

• If the $1$-st, $2$-nd, and $3$-rd elements are chosen, $MEX(B)=MEX(2,0,2)=1$.
• If the $3$-rd, $4$-th, and $6$-th elements are chosen, $MEX(B)=MEX(2,3,1)=0$.
• If the $2$-nd, $6$-th, and $7$-th elements are chosen, $MEX(B)=MEX(0,1,9)=2$.
• If the $2$-nd, $3$-rd, and $6$-th elements are chosen, $MEX(B)=MEX(0,2,1)=3$.

The maximum possible $MEX$ is $3$.