Given a sequence like:
1,2,1,2,1,1,1,2,1,2,1,3,1,2,1,2,1,3
What is an efficient way to obtain the minimum ordering:
1,1,1,2,1,2,1,3,1,2,1,2,1,3,1,2,1,2
The brute force approach is obvious so please don’t recommend it — unless to provide a convincing proof that the brute force approach is the only way to do it!
More detail
I have an algorithm that generates a list of numbers. The output of the algorithm is a list / array, but logically the numbers represent a loop, where only the relative order of the elements matters. In order to store these loops for later comparison I want to put store them in such a way that the one-dimensional list of numbers stored represents the minimum ordering of the elements in the loop. A picture would be most helpful:
This loop describes the path around the T tetromino, where 1 is move forward, 2 is turn right, and 3 is turn left. It doesn’t matter where you start or even which direction you go, following this sequence of 18 moves will get you a T tetromino. The output of the algorithm which produces this loop will return the elements with an arbitrary starting point and direction. So the returned array could be:
Arbitrary initial ordering:
1,2,1,2,1,1,1,2,1,2,1,3,1,2,1,2,1,3
There is one minimum ordering, though. It can be obtained from two different circuits, reflecting the fact that the T tetromino is symmetrical:
Minimum ordering:
1,1,1,2,1,2,1,3,1,2,1,2,1,3,1,2,1,2
Brute force
The obvious brute force method is to construct all possible orderings and take the minimum. My question is whether there is a more clever and efficient way of doing this.
This is a well-studied problem called the lexicographically minimal string rotation.
The better algorithms all run in O(n) time as opposed to O(n*n) for the naive algorithm.