CF1537E2.Erase and Extend (Hard Version)

This is the hard version of the problem. The only difference is the constraints on $n$ and $k$ . You can make hacks only if all versions of the problem are solved.

You have a string $s$ , and you can do two types of operations on it:

• Delete the last character of the string.
• Duplicate the string: $s:=s+s$ , where $+$ denotes concatenation.

You can use each operation any number of times (possibly none).

Your task is to find the lexicographically smallest string of length exactly $k$ that can be obtained by doing these operations on string $s$ .

A string $a$ is lexicographically smaller than a string $b$ if and only if one of the following holds:

• $a$ is a prefix of $b$ , but $a\ne b$ ;
• In the first position where $a$ and $b$ differ, the string $a$ has a letter that appears earlier in the alphabet than the corresponding letter in $b$ .

The first line contains two integers $n$ , $k$ ( $1 \leq n, k \leq 5\cdot 10^5$ ) — the length of the original string $s$ and the length of the desired string.

The second line contains the string $s$ , consisting of $n$ lowercase English letters.

Print the lexicographically smallest string of length $k$ that can be obtained by doing the operations on string $s$ .

In the first test, it is optimal to make one duplication: "dbcadabc" $\to$ "dbcadabcdbcadabc".

In the second test it is optimal to delete the last $3$ characters, then duplicate the string $3$ times, then delete the last $3$ characters to make the string have length $k$ .

"abcd" $\to$ "abc" $\to$ "ab" $\to$ "a" $\to$ "aa" $\to$ "aaaa" $\to$ "aaaaaaaa" $\to$ "aaaaaaa" $\to$ "aaaaaa" $\to$ "aaaaa".