Consecutive interval query and dynamic programming on intervals

Given a set of n points (nodes) on a line and a set of m weighted intervals defined on the nodes, we consider a particular dynamic programming (DP) problem on these intervals. If the weight function of the DP has convex or concave property, we can solve this DP problem efficiently by using matrix searching in Monge matrices, together with a new query data structure, which we call the consecutive query structure. We invoke our algorithm to obtain fast algorithms for sequential partition of a graph and for maximum K-clique of an interval graph.