CF1575H.Holiday Wall Ornaments

The Winter holiday will be here soon. Mr. Chanek wants to decorate his house's wall with ornaments. The wall can be represented as a binary string $a$ of length $n$ . His favorite nephew has another binary string $b$ of length $m$ ( $m \leq n$ ).

Mr. Chanek's nephew loves the non-negative integer $k$ . His nephew wants exactly $k$ occurrences of $b$ as substrings in $a$ .

However, Mr. Chanek does not know the value of $k$ . So, for each $k$ ( $0 \leq k \leq n - m + 1$ ), find the minimum number of elements in $a$ that have to be changed such that there are exactly $k$ occurrences of $b$ in $a$ .

A string $s$ occurs exactly $k$ times in $t$ if there are exactly $k$ different pairs $(p,q)$ such that we can obtain $s$ by deleting $p$ characters from the beginning and $q$ characters from the end of $t$ .

The first line contains two integers $n$ and $m$ ( $1 \leq m \leq n \leq 500$ ) — size of the binary string $a$ and $b$ respectively.

The second line contains a binary string $a$ of length $n$ .

The third line contains a binary string $b$ of length $m$ .

Output $n - m + 2$ integers — the $(k+1)$ -th integer denotes the minimal number of elements in $a$ that have to be changed so there are exactly $k$ occurrences of $b$ as a substring in $a$ .

For $k = 0$ , to make the string $a$ have no occurrence of 101, you can do one character change as follows.

100101011 $\rightarrow$ 100100011

For $k = 1$ , you can also change a single character.

100101011 $\rightarrow$ 100001011

For $k = 2$ , no changes are needed.