The problem is: given a set of objects drawn from several groups, put all of them in a row in a “uniform” way. Whatever that means.
For example, suppose we have 21 gemstones: 9 red, 5 blue, 3 green, 2 cyan, 1 magenta and 1 yellow. How to place them on a string to make a reasonably looking necklace? The criterion is subjective, but we can probably agree that
Arrangement #1
looks better than
Arrangement #2
(referring to the uniformity of distributions, not the aesthetics of color.)
The approach I took was to repeatedly add the “most underrepresented” gem, defined by maximal difference between projected frequency of appearance (e.g., 5/21 for blue) and the frequency of appearance so far in the string. (Taking the convention 0/0=0 here.) Here is this algorithm in python ― not in Ruby, which would be more topical.
count = {'r': 9, 'b': 5, 'g': 3, 'c': 2, 'm': 1, 'y': 1} length = sum(count.values()) str = '' while len(str) < length: deficit = {} for char in count: deficit[char] = count[char]/length - (str.count(char)/len(str) if str else 0) str += max(deficit, key=deficit.get) print(str)The output, “rbgcryrbrmgrbrcbrgrbr”, is what the first image in this post represents. The second image is the result of an ill-fated attempt to replace difference by ratio when determining the under-representation.
I initially thought that two unique gems (yellow and magenta) would end up together, but this hasn’t happened: after one is added, the frequency of more common gems drops, allowing them to come back into play for a while. Still, the left half of the string is noticeably more diverse than the right half. It’d be better if two unique gems were in roughly symmetrical position, and generally there would be no preference between left-right and right-left directions.
Perhaps the new character should be added to the string either on the right or on the left, in alternating fashion. That should make things nice and symmetric, right?
Wrong.
Alternating concatenation
The search continues…