All highest scoring paths in weighted grid graphs and their application to finding all approximate repeats in strings

Weighted paths in directed grid graphs of dimension (m x n) can be used to model the string edit problem, which consists of obtaining optimal (weighted) alignments between substrings of A, \A\ = m, and substrings of B, \B\ = n. We build a data structure (in O(mn log m) time) that supports O(log m) time queries about the weight of any of the O(m(2)n) best paths from the vertices in column 0 of the graph to all other vertices. Using these techniques we present a simple O(n(2) log n) time and Theta(n(2)) space algorithm to find all (the locally optimal) approximate tandem (or nontandem) repeats xy within a string of size n. This improves (by a factor of log n) upon several previous algorithms for this problem and is the first algorithm to fond all locally optimal repeats. For edit graphs with weights in {0,-1,1}, a slight modification of our techniques yields an O(n(2)) algorithm for the cyclic string comparison problem, as compared to O(n log n) for the case of general weights.