The similarity of two nucleotide sequences is often expressed in terms of evolutionary distance, a measure of the amount of change needed to transform one sequence into the other. Given two sequences with a small distance between them, can their similarity be explained by their base composition alone? The nucleotide order of these sequences contributes to their similarity if the distance is much smaller than their average permutation distance, which is obtained by calculating the distances for many random permutations of these sequences. To determine whether their similarity can be explained by their dinucleotide and codon usage, random sequences must be chosen from the set of permuted sequences that preserve dinucleotide and codon usage. The problem of choosing random dinucleotide and codon-preserving permutations can be expressed in the language of graph theory as the problem of generating random Eulerian walks on a directed multigraph. An efficient algorithm for generating such walks is described. This algorithm can be used to choose random sequence permutations that preserve (1) dinucleotide usage, (2) dinucleotide and trinucleotide usage, or (3) dinucleotide and codon usage. For example, the similarity of two 60-nucleotide DNA segments from the human beta-1 interferon gene (nucleotides 196-255 and 499-558) is not just the result of their nonrandom dinucleotide and codon usage.