Background: Multiple sequence alignment is fundamental. Exponential growth in computation time appears to be inevitable when an optimal alignment is required for many sequences. Exact costs of optimum alignments are therefore rarely computed. Consequently much effort has been invested in algorithms for alignment that are heuristic, or explore a restricted class of solutions. These give an upper bound on the alignment cost, but it is equally important to determine the quality of the solution obtained. In the absence of an optimal alignment with which to compare, lower bounds may be calculated to assess the quality of the alignment. As more effort is invested in improving upper bounds (alignment algorithms), it is therefore important to improve lower bounds as well. Although numerous cost metrics can be used to determine the quality of an alignment, many are based on sum-of-pairs (SP) measures and their generalizations.
Results: Two standard and two new methods are considered for using exact 2-way and 3-way alignments to compute lower bounds on total SP alignment cost; one new method fares well with respect to accuracy, while the other reduces the computation time. The first employs exhaustive computation of exact 3-way alignments, while the second employs an efficient heuristic to compute a much smaller number of exact 3-way alignments. Calculating all 3-way alignments exactly and computing their average improves lower bounds on sum of SP cost in v-way alignments. However judicious selection of a subset of all 3-way alignments can yield a further improvement with minimal additional effort. On the other hand, a simple heuristic to select a random subset of 3-way alignments (a random packing) yields accuracy comparable to averaging all 3-way alignments with substantially less computational effort.
Conclusion: Calculation of lower bounds on SP cost (and thus the quality of an alignment) can be improved by employing a mixture of 3-way and 2-way alignments.