|Codeforces Round #615 (Div. 3)|
Recall that MEX of an array is a minimum non-negative integer that does not belong to the array. Examples:
You are given an empty array $$$a=$$$ (in other words, a zero-length array). You are also given a positive integer $$$x$$$.
You are also given $$$q$$$ queries. The $$$j$$$-th query consists of one integer $$$y_j$$$ and means that you have to append one element $$$y_j$$$ to the array. The array length increases by $$$1$$$ after a query.
In one move, you can choose any index $$$i$$$ and set $$$a_i := a_i + x$$$ or $$$a_i := a_i - x$$$ (i.e. increase or decrease any element of the array by $$$x$$$). The only restriction is that $$$a_i$$$ cannot become negative. Since initially the array is empty, you can perform moves only after the first query.
You have to maximize the MEX (minimum excluded) of the array if you can perform any number of such operations (you can even perform the operation multiple times with one element).
You have to find the answer after each of $$$q$$$ queries (i.e. the $$$j$$$-th answer corresponds to the array of length $$$j$$$).
Operations are discarded before each query. I.e. the array $$$a$$$ after the $$$j$$$-th query equals to $$$[y_1, y_2, \dots, y_j]$$$.
The first line of the input contains two integers $$$q, x$$$ ($$$1 \le q, x \le 4 \cdot 10^5$$$) — the number of queries and the value of $$$x$$$.
The next $$$q$$$ lines describe queries. The $$$j$$$-th query consists of one integer $$$y_j$$$ ($$$0 \le y_j \le 10^9$$$) and means that you have to append one element $$$y_j$$$ to the array.
Print the answer to the initial problem after each query — for the query $$$j$$$ print the maximum value of MEX after first $$$j$$$ queries. Note that queries are dependent (the array changes after each query) but operations are independent between queries.
7 3 0 1 2 2 0 0 10
1 2 3 3 4 4 7
4 3 1 2 1 2
0 0 0 0
In the first example: