Appending Mex

1000ms

262144K

Description:

Initially Ildar has an empty array. He performs $$$n$$$ steps. On each step he takes a subset of integers already added to the array and appends the mex of this subset to the array.

The mex of an multiset of integers is the smallest non-negative integer not presented in the multiset. For example, the mex of the multiset $$$[0, 2, 3]$$$ is $$$1$$$, while the mex of the multiset $$$[1, 2, 1]$$$ is $$$0$$$.

More formally, on the step $$$m$$$, when Ildar already has an array $$$a_1, a_2, \ldots, a_{m-1}$$$, he chooses some subset of indices $$$1 \leq i_1 < i_2 < \ldots < i_k < m$$$ (possibly, empty), where $$$0 \leq k < m$$$, and appends the $$$mex(a_{i_1}, a_{i_2}, \ldots a_{i_k})$$$ to the end of the array.

After performing all the steps Ildar thinks that he might have made a mistake somewhere. He asks you to determine for a given array $$$a_1, a_2, \ldots, a_n$$$ the minimum step $$$t$$$ such that he has definitely made a mistake on at least one of the steps $$$1, 2, \ldots, t$$$, or determine that he could have obtained this array without mistakes.

Input:

The first line contains a single integer $$$n$$$ ($$$1 \leq n \leq 100\,000$$$) — the number of steps Ildar made.

The second line contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$0 \leq a_i \leq 10^9$$$) — the array Ildar obtained.

Output:

If Ildar could have chosen the subsets on each step in such a way that the resulting array is $$$a_1, a_2, \ldots, a_n$$$, print $$$-1$$$.

Otherwise print a single integer $$$t$$$ — the smallest index of a step such that a mistake was made on at least one step among steps $$$1, 2, \ldots, t$$$.

Sample Input:

4
0 1 2 1

Sample Output:

-1

Sample Input:

3
1 0 1

Sample Output:

Sample Input:

4
0 1 2 239

Sample Output:

Note:

In the first example it is possible that Ildar made no mistakes. Here is the process he could have followed.

$$$1$$$-st step. The initial array is empty. He can choose an empty subset and obtain $$$0$$$, because the mex of an empty set is $$$0$$$. Appending this value to the end he gets the array $$$[0]$$$.
$$$2$$$-nd step. The current array is $$$[0]$$$. He can choose a subset $$$[0]$$$ and obtain an integer $$$1$$$, because $$$mex(0) = 1$$$. Appending this value to the end he gets the array $$$[0,1]$$$.
$$$3$$$-rd step. The current array is $$$[0,1]$$$. He can choose a subset $$$[0,1]$$$ and obtain an integer $$$2$$$, because $$$mex(0,1) = 2$$$. Appending this value to the end he gets the array $$$[0,1,2]$$$.
$$$4$$$-th step. The current array is $$$[0,1,2]$$$. He can choose a subset $$$[0]$$$ and obtain an integer $$$1$$$, because $$$mex(0) = 1$$$. Appending this value to the end he gets the array $$$[0,1,2,1]$$$.

Thus, he can get the array without mistakes, so the answer is $$$-1$$$.

In the second example he has definitely made a mistake on the very first step, because he could not have obtained anything different from $$$0$$$.

In the third example he could have obtained $$$[0, 1, 2]$$$ without mistakes, but $$$239$$$ is definitely wrong.

Informação

Provedor Codeforces

Código CF1054B