Question

Suppose there is given two String:

String s1= "MARTHA"
String s2= "MARHTA"

here we exchange positions of T and H. I am interested to write code which counts how many changes are necessary to transform from one String to another String.

Answer 1

Assuming that the distance counts only swaps, here is an idea based on permutations, that runs in linear time.

The first step of the algorithm is ensuring that the two strings are really equivalent in their character contents. This can be done in linear time using a hash table (or a fixed array that covers all the alphabet). If they are not, then s2 can’t be considered a permutation of s1, and the “swap count” is irrelevant.

The second step counts the minimum number of swaps required to transform s2 to s1. This can be done by inspecting the permutation p that corresponds to the transformation from s1 to s2. For example, if s1=”abcde” and s2=”badce”, then p=(2,1,4,3,5), meaning that position 1 contains element #2, position 2 contains element #1, etc. This permutation can be broke up into permutation cycles in linear time. The cycles in the example are (2,1) (4,3) and (5). The minimum swap count is the total count of the swaps required per cycle. A cycle of length k requires k-1 swaps in order to “fix it”. Therefore, The number of swaps is N-C, where N is the string length and C is the number of cycles. In our example, the result is 2 (swap 1,2 and then 3,4).

Now, there are two problems here, and I think I’m too tired to solve them right now 🙂

1) My solution assumes that no character is repeated, which is not always the case. Some adjustment is needed to calculate the swap count correctly.

2) My formula #MinSwaps=N-C needs a proof… I didn’t find it in the web.