We can implement a solution of time complexity ~\mathcal O(N^2)~ which for every substring calculates the number of different letters in it. If we use the so-called sliding window technique, we need to keep track of how many times each letter appears and note each time when a new letter starts or stops appearing.

Note that the numerator of the expression we want to minimize, i.e. the number of different characters in a substring, can either be ~1, 2, \dots, 26~. Therefore, it is enough to fix its value in every possible way and determine the largest possible substring which has exactly that many different characters. Now we have ~26~ values to compare and pick the smallest one.

A simple implementation calculates for each starting character the longest substring which contains exactly ~K~ different characters. If we calculate for each character and each position the first next appearance of that character, for each starting character we can quickly check ~\le 26~ strings which go to the next new character.