Efficient All Path Score Computations on Grid Graphs

We study the Integer-weighted Grid All Paths Scores (IGAPS) problem, which is given a grid graph, to compute the maximum weights of paths between every pair of a vertex on the first row of the graph and a vertex on the last row of the graph. We also consider a variant of this problem, periodic IGAPS, where the input grid graph is periodic and infinite. For these problems, we consider both the general (dense) and the sparse cases. For the sparse IGAPS problem with 0-1 weights, we give an O(r log 3 (n 2 /r)) time algorithm, where r is the number of (diagonal) edges of weight 1. Our result improves upon the previous O(n √ r) result by Krusche and Tiskin for this problem. For the periodic IGAPS problem we give an O(Cn2) time algorithm, where C is the maximum weight of an edge. This improves upon the previous O(C 2 n 2) algorithm of Tiskin. We also show a reduction from periodic IGAPS to IGAPS. This reduction yields o(n 2) algorithms for this problem.