Bounds for parametric sequence comparison

We consider the problem of computing a global alignment between two or more sequences subject to varying mismatch and indel penalties. We prove a tight 3(n/2/spl pi/)/sup 2/3/+O(n/sup 1/3/logn) bound on the worst-case number of distinct optimum alignments for two sequences of length n as the parameters are varied. This refines a O(n/sup 2/3/) upper bound by D. Gusfield et al. (1994). Our lower bound requires an unbounded alphabet. For strings over a binary alphabet, we prove a /spl Omega/(n/sup 1/2/) lower bound. For the parametric global alignment of k/spl ges/2 sequences under sum-of-pairs scoring, we prove a 3((k/2)n/2/spl pi/)/sup 2/3/+O(k/sup 2/3/n/sup 1/3/logn) upper bound on the number of distinct optimality regions and a /spl Omega/(n/sup 2/3/) lower bound. Based on experimental evidence, we conjecture that for two random sequences, the number of optimality regions is approximately /spl radic/n with high probability.