In this chapter we look at a number of important refinements that have been developed for certain core string edit and alignment problems. These refinements either speed up a dynamic programming solution, reduce its space requirements, or extend its utility.

Computing alignments in only linear space

One of the defects of dynamic programming for all the problems we have discussed is that the dynamic programming tables use Θ(nm) space when the input strings have length n and m. (When we talk about the space used by a method, we refer to the maximum space ever in use simultaneously. Reused space does not add to the count of space use.) It is quite common that the limiting resource in string alignment problems is not time but space. That limit makes it difficult to handle large strings, no matter how long we may be willing to wait for the computation to finish. Therefore, it is very valuable to have methods that reduce the use of space without dramatically increasing the time requirements.

Hirschberg [224] developed an elegant and practical space-reduction method that works for many dynamic programming problems. For several string alignment problems, this method reduces the required space from Θ(nm) to O(n) (for n < m) while only doubling the worst-case time bound. Miller and Myers expanded on the idea and brought it to the attention of the computational biology community [344].