Sparse Dynamic Programming II: Convex and Concave Cost Functions

We consider dynamic programming solutions to a number of different recurrences for sequence comparison and for RNA secondary structure prediction. These recurrences are defined over a number of points that is quadratic in the input size; however only a sparse set matters for the result. We give efficient algorithms for these problems, when the weight functions used in the recurrences are taken to be linear. Our algorithms reduce the best known bounds by a factor almost linear in the density of the problems: when the problems are sparse this results in a substantial speed-up