Editorial

The analysis uses the notation $x\mathrm{mod}y$, which means the remainder of dividing the number $x$ by $y$. For example, $7\mathrm{mod}3=1$.

If we number the letters in the alphabet from $0$ to $25$ (i.e., '$a$'$=0$, '$b$'$=1$, $\dots$, '$z$'$=25$) and create a new array $b$, where $b_i$ corresponds to the number of the letter $s_i$, then the addition described in the condition is $(b_i+a_i)\mathrm{mod}26$. Now our task is to make the array $(b_i+a_i)\mathrm{mod}26$ lexicographically minimal for a certain permutation $a$.

It can be noticed that $(b_i+a_i)\mathrm{mod}26=(b_i+(a_i\mathrm{mod}26))\mathrm{mod}26$. This means that each $a_i$ can be replaced by $a_i\mathrm{mod}26$.

Let's create a count array of the array $a$, i.e., an array $c$ such that $c_x$ is the number of occurrences of the number $x$ in the array $a$. Since the sequence in which the elements of the array are placed does not matter, we will work with their quantity.

Let's choose such $x$ for $b_1$ that $(b_1+x)\mathrm{mod}26$ is minimal. After that, we will use this $x$ and remove it from the count array. We will do the same for $b_2,b_3,\dots ,b_n$.

Thus, we always try to minimize the prefix of the array, which guarantees a lexicographically minimal array.