Approximate Suffix-Prefix Dictionary Queries

In the all-pairs suffix-prefix (APSP) problem [Gusfield et al., Inf. Process. Lett. 1992], we are given a dictionary R of r strings, S1, . . ., Sr, of total length n, and we are asked to find the length SPLi,j of the longest string that is both a suffix of Si and a prefix of Sj, for all i, j ∈ [1 . . r]. APSP is a classic problem in string algorithms with applications in bioinformatics, especially in sequence assembly. Since r = |R| is typically very large in real-world applications, considering all r² pairs of strings explicitly is prohibitive. This is when the data structure variant of APSP makes sense; in the same spirit as distance oracles computing shortest paths between any two vertices given online. We show how to quickly locate k-approximate matches (under the Hamming or the edit distance) in R using a version of the k-errata tree [Cole et al., STOC 2004] that we introduce. Let SPL^k_i,j be the length of the longest suffix of Si that is at distance at most k from a prefix of Sj. In particular, for any k = O(1), we show an O(nlog^k n)-sized data structure to support the following queries: One-to-One^k(i, j): output SPL^k_i,j in O(log^k nlog log n) time. Report^k(i, d): output all j ∈ [1 . . r], such that SPL^k_i,j ≥ d, in O(log^k n(log n/log log n + output)) time, where output denotes the size of the output. In fact, our algorithms work for any value of k not just for k = O(1), but the formulas bounding the complexities get much more complicated for larger values of k.