Erase and Extend (Hard Version)

2000ms

262144K

Description:

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 $$$b$$$.

Input:

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.

Output:

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

Sample Input:

8 16
dbcadabc

Sample Output:

dbcadabcdbcadabc

Sample Input:

4 5
abcd

Sample Output:

aaaaa

Note:

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".

Informação

Provedor Codeforces

Código CF1537E2