AirlineDog / hirschberg-sequencing

A python script implementing Hirschberg's Algorithm to find optimal sequence alignments

Hirschberg Sequencing Algorithm

Let's say we got two DNA sequences GATTACA and GCATGCA and we want to get the optimal alignments of those two:

G-ATTACA | G-ATTACA | G-ATTACA
GCA-TGCG | GCAT-GCG | GCATG-CG

How are these alignments defined as optimal?

We calculate the alignment score as:

We begin we score 0

• If two letters are the same we add m (match) to the score
• If two letters are different we subtract d (differ) to the score
• If we add a gap we subtract g (gap) to the score

To find all possible alignments:

1. Split the first sequence in half
2. Calculate the last line of the Needleman-Wunsch matrix for the first half and the whole second sequence
3. Calculate the last line of the Needleman-Wunsch matrix for the reversed second half and the whole reversed second sequence
4. Add the elements of step 2 and reversed step 3 together
5. Find the indexes of the elements equal to the max element of step 4

Then recursively split the first sequence in half and the second one at the index found in step 5 until the length of a sequence is 0 or 1. The reqursion terminates when you end up with a sequence of length 0 or 1. In the first case add gaps to the allignment. In the second one you have to aplly the Needleman-Wunsch algorithm, but this time the matrix will either have 2 rows or 2 columns

This is an efficient way to reduce space complexity of the Needleman-Wunsch algorithm invented by Dan Hirschberg

Needleman-Wunsch algorithm

• Time complexity: O(mn)
• Space complexity: O(mn)

Hirschberg algorithm

• Time complexity: O(mn)
• Space complexity: O(min{m, n})

More inforamation:

• wikipedia
• Hirschberg, D. S. 1975. “A Linear Space Algorithm for Computing Maximal Common Subsequences.”
• Needleman, Saul B., and Christian D. Wunsch. 1970. “A General Method Applicable to the Search for Similarities in the Amino Acid Sequence of Two Proteins.”