Dynamic Programming for Set Data Types

We present an efficient generalization of algebraic dynamic programming (ADP) to unordered data types and a formalism for the automated derivation of outside grammars from their inside progenitors. These theoretical contributions are illustrated by ADP-style algorithms for shortest Hamiltonian path problems. These arise naturally when asking whether the evolutionary history of an ancient gene cluster can be explained by a series of local tandem duplications. Our framework makes it easy to compute Maximum accuracy solutions, which in turn require the computation of the probabilities of individual edges in the ensemble of Hamiltonian paths. The expansion of the Hox gene clusters is investigated as a show-case application. http://www.tbi.univie.ac.at/~choener/setgram/