Graphical representations of codes facilitate the design of computationally efficient decoding algorithms. This is an example of a general connection between dependency graphs, as arise in the representations of Markov random fields, and the dynamic programming principle. We concentrate on two computational tasks: finding the maximum-likelihood codeword and finding its posterior probability, given a signal received through a noisy channel. These two computations lend themselves to a particularly elegant version of dynamic programming, whereby the decoding complexity is particularly transparent. We explore some codes and some graphical representations designed specifically to facilitate computation. We further explore a coarse-to-fine version of dynamic programming that can produce an exact maximum-likelihood decoding many orders of magnitude faster than ordinary dynamic programming.