Problem:
I have N (~100k-1m) strings each D (e.g. 2000) characters long and with a low alphabet (eg 3 possible characters). I would like to sort these strings such that there are as few possible changes between adjacent strings (eg hamming distance is low). Solution doesn't have to be the best possible but closer the better.
Example
N=4
D=5
//initial strings
1. aaacb
2. bacba
3. acacb
4. cbcba
//sorted so that hamming distance between adjacent strings is low
1. aaacb
3. acacb (Hamming distance 1->3 = 1)
4. cbcba (Hamming distance 3->4 = 4)
2. bacba (Hamming distance 4->2 = 2)
Thoughts about the problem
I have a bad feeling this is a non trivial problem. If we think of each string as a node and the distances to other strings as an edge, then we are looking at a travelling salesman problem. The large number of strings means that calculating all of the pairwise distances beforehand is potentially infeasible, I think turning the problem into some more like the Canadian Traveller Problem.
At the moment my solution has been to use a VP tree to find a greedy nearest neighbour type solution to the problem
curr_string = a randomly chosen string from full set
while(tree not empty)
found_string = find nearest string in tree
tree.remove(found_string)
sorted_list.add(curr_string)
curr_string = found_string
but initial results appear to be poor. Hashing strings so that more similar ones are closer may be another option but I know little about how good a solution this will provide or how well it will scale to data of this size.
Even if you consider this problem as similar to the travelling salesman problem (TSP), I believe that Hamming distances will follow the triangle inequality (Hamming(A,B) + Hamming(B,C) ≤ Hamming(A,C)), so you're only really dealing with ∆TSP (the metric travelling salesman problem), for which there are a number of algorithms which give good approximations at an ideal result. In particular, the Christofides algorithm will always give you a path of at most 1.5x the minimum possible length.