We describe two methods for constructing an optimal global alignment of, and an optimal local alignment between, a DNA sequence and a protein sequence. The alignment model of the methods addresses the problems of frameshifts and introns in the DNA sequence. The methods require computer memory proportional to the sequence lengths, so they can rigorously process very huge sequences. The simplified versions of the methods were implemented as computer programs named NAP and LAP. The experimental results demonstrate that the programs are sensitive and powerful tools for finding genes by DNA-protein sequence homology.