# Algorithm details¶

Since the publication of the EMBnet journal application note about Cutadapt, the alignment algorithm used for finding adapters has changed significantly. An overview of this new algorithm is given in this section. An even more detailed description is available in Chapter 2 of my PhD thesis Algorithms and tools for the analysis of high-throughput DNA sequencing data.

The algorithm is based on semiglobal alignment, also called free-shift, ends-free or overlap alignment. In a regular (global) alignment, the two sequences are compared from end to end and all differences occuring over that length are counted. In semiglobal alignment, the sequences are allowed to freely shift relative to each other and differences are only penalized in the overlapping region between them:

```   FANTASTIC
ELEFANT
```

The prefix `ELE` and the suffix `ASTIC` do not have a counterpart in the respective other row, but this is not counted as an error. The overlap `FANT` has a length of four characters.

Traditionally, alignment scores are used to find an optimal overlap aligment: This means that the scoring function assigns a positive value to matches, while mismatches, insertions and deletions get negative values. The optimal alignment is then the one that has the maximal total score. Usage of scores has the disadvantage that they are not at all intuitive: What does a total score of x mean? Is that good or bad? How should a threshold be chosen in order to avoid finding alignments with too many errors?

For Cutadapt, the adapter alignment algorithm uses unit costs instead. This means that mismatches, insertions and deletions are counted as one error, which is easier to understand and allows to specify a single parameter for the algorithm (the maximum error rate) in order to describe how many errors are acceptable.

There is a problem with this: When using costs instead of scores, we would like to minimize the total costs in order to find an optimal alignment. But then the best alignment would always be the one in which the two sequences do not overlap at all! This would be correct, but meaningless for the purpose of finding an adapter sequence.

The optimization criteria are therefore a bit different. The basic idea is to consider the alignment optimal that maximizes the overlap between the two sequences, as long as the allowed error rate is not exceeded.

Conceptually, the procedure is as follows:

1. Consider all possible overlaps between the two sequences and compute an alignment for each, minimizing the total number of errors in each one.
2. Keep only those alignments that do not exceed the specified maximum error rate.
3. Then, keep only those alignments that have a maximal number of matches (that is, there is no alignment with more matches).
4. If there are multiple alignments with the same number of matches, then keep only those that have the smallest error rate.
5. If there are still multiple candidates left, choose the alignment that starts at the leftmost position within the read.

In Step 1, the different adapter types are taken into account: Only those overlaps that are actually allowed by the adapter type are actually considered.

## Quality trimming algorithm¶

The trimming algorithm implemented in Cutadapt is the same as the one used by BWA, but applied to both ends of the read in turn (if requested). That is: Subtract the given cutoff from all qualities; compute partial sums from all indices to the end of the sequence; cut the sequence at the index at which the sum is minimal. If both ends are to be trimmed, repeat this for the other end.

The basic idea is to remove all bases starting from the end of the read whose quality is smaller than the given threshold. This is refined a bit by allowing some good-quality bases among the bad-quality ones. In the following example, we assume that the 3’ end is to be quality-trimmed.

Assume you use a threshold of 10 and have these quality values:

42, 40, 26, 27, 8, 7, 11, 4, 2, 3

Subtracting the threshold gives:

32, 30, 16, 17, -2, -3, 1, -6, -8, -7

Then sum up the numbers, starting from the end (partial sums). Stop early if the sum is greater than zero:

(70), (38), 8, -8, -25, -23, -20, -21, -15, -7

The numbers in parentheses are not computed (because 8 is greater than zero), but shown here for completeness. The position of the minimum (-25) is used as the trimming position. Therefore, the read is trimmed to the first four bases, which have quality values 42, 40, 26, 27.