Finding Founder Sequences from a Set of Recombinants

Inspired by the current interest in the so–called haplotype blocks we consider a related, complementary problem abstracted from the following scenario. We are given the DNA or SNP sequences of a sample of individuals from a (human) population. The population is assumed to have evolved as an isolate, founded some generations ago by a relatively small number of settlers. Then the sequences in our given sample should be a result of recombinations of the corresponding sequences of the founders, possibly corrupted by (rare) point mutations. We are interested in finding plausible reconstructions of the sequences of the founders. Formulated as a combinatorial string problem, one has to find a set of founder sequences such that given sequences can be composed from fragments taken from the corresponding locations of the founder sequences. The solution can be optimized for example with respect to the number of founders or the number of crossovers. We give polynomial– time algorithms for some special cases as well as a general solution by dynamic programming.