Analytic combinatorics for bioinformatics I: seeding methods

Seeding heuristics are the most widely used strategies to speed up sequence alignment in bioinformatics. Such strategies are most successful if they are calibrated, so that the speed-versus-accuracy trade-off can be properly tuned. In the widely used case of read mapping, it has been so far impossible to predict the success rate of competing seeding strategies for lack of a theoretical framework. Here I present an approach to estimate such quantities based on the theory of analytic combinatorics. In a nutshell, the strategy is to specify a combinatorial construction of reads where the seeding heuristic fails, translate this specification into a generating function using formal rules, and finally extract the probabilities of interest from the singularities of the generating function. I use this approach to construct simple estimators of the success rate of the seeding heuristic under different types of sequencing errors. I also show how the analytic combinatorics strategy can be used to compute the associated type I and type II error rates (mapping the read to the wrong location, or being unable to map the read). Finally, I show how analytic combinatorics can be used to estimate average quantities such as the expected number of errors in reads where the seeding heuristic fails. Overall, this work introduces a theoretical and practical framework to find the success rate of seeding heuristics and related problems in bioinformatics.