Fast Fourier Transform for Fitness Landscapes

✍ Scribed by Dan Rockmore; Peter Kostelec; Wim Hordijk; Peter F. Stadler

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 307 KB
• Volume: 12
• Category: Article
• ISSN: 1063-5203
• DOI: 10.1006/acha.2001.0346

✦ Synopsis

We cast some classes of fitness landscapes as problems of spectral analysis on various Cayley graphs. In particular, landscapes derived from RNA folding are realized on Hamming graphs and analyzed in terms of Walsh transforms; assignment problems are interpreted as functions on the symmetric group and analyzed in terms of the representation theory of S n . We show that explicit computation of the Walsh/Fourier transforms is feasible for landscapes with up to 10 8 configurations using fast Fourier transform techniques. We find that the cost function of a linear sum assignment problem involves only the defining representation of the symmetric group, while quadratic assignment problems are superpositions of the representations indexed by the partitions (n), (n -1, 1), (n -2, 2), and (n -2, 1, 1). These correspond to the four smallest eigenvalues of the Laplacian of the Cayley graph obtained by using transpositions as the generating set on S n .