Dynamic Programming Algorithms in Semiring and Hypergraph Frameworks

Dynamic Programming (DP) is an important class of algorithms widely used in many areas of speech and language processing. Recently there have been a series of work trying to formalize many instances of DP under algebraic or graph-theoretic frameworks. This report surveys two such frameworks, namely semirings and directed hypergraphs, and draws connections between them. We formalize two particular types of DP algorithms under these frameworks: the Viterbi-style fixed-order algorithms and the Dijkstra-style best-first algorithms. Wherever relevant, we also discuss typical applications of these algorithms in Natural Language Processing.