Undiscretized Dynamic Programming and Ordinal Embeddings

OF THE DISSERTATION Undiscretized Dynamic Programming and Ordinal Embeddings by Rahul Shah Dissertation Director: Martin Farach-Colton Many optimization problems which are known to be NP-hard on graphs are polynomially solvable on trees using dynamic programming. Dynamic programming typically involves recursive functions stored as tables. Each entry of the table corresponds to the optimal subproblem solution. In many applications the complexity of brute-force dynamic programming can be improved especially when the functions involved are sparse, convex or concave. Algorithms for speeding up dynamic programming on path-like one dimensional structures are known. Here, we “undiscretize” the functions i.e. represent them as functions of continuous variable instead of storing the functions as the tables of discrete values. We use efficient data structures to store such functions and show how to quickly carry out operations involving these functions. We improve the complexity bounds for many tree dynamic programming problems, typically, from O(n) to O(n logn). These include problems like facility location, covering, economic lot sizing and multicast filtering. Given a set of pairwise distances on a set of n points, constructing an edge-weighted tree whose leaves are these n points such that the tree distances wouldmimic the original