UVa 612 - DNA Sorting

Summary
A straight forward problem where we are required to sort strings by how many inversions they contain.

Explanation
We have to sort a group of equal length strings by their inversion count. An inversion is an out of order pair, or more formally, the inversion set of an array is the set of pairs $$\{ (i, j) : i < j$$ and $$ A_i > A_j\}$$. Since the strings are relatively short, a simple check all pairs $$O(n^2)$$ solution will suffice to count the number of inversions in each string. Remember that we need to break ties by keeping things in order, so a stable sort is required.

Optimizations
Since the string alphabet is so small, consisting of only 4 symbols, A, C, G, and T, we can do the inversion count in $$4n = O(n)$$ time with a dynamic programming algorithm. We maintain a 4 by n array that keeps track of at index i how many occurances of each character there are after the ith character in the string.

Implementation
In C++, the STL's stable_sort can be used to considerably improve coding time.

Input
10 6 AACATGAAGG TTTTGGCCAA TTTGGCCAAA GATCAGATTT CCCGGGGGGA ATCGATGCAT

Output
CCCGGGGGGA AACATGAAGG GATCAGATTT ATCGATGCAT TTTTGGCCAA TTTGGCCAAA