Fully Functional Suffix Trees and Optimal Text Searching in BWT-Runs Bounded Space

Abstract

Indexing highly repetitive texts—such as genomic databases, software repositories and versioned text collections—has become an important problem since the turn of the millennium. A relevant compressibility measure for repetitive texts is r , the number of runs in their Burrows-Wheeler Transforms (BWTs). One of the earliest indexes for repetitive collections, the Run-Length FM-index, used O ( r ) space and was able to efficiently count the number of occurrences of a pattern of length m in a text of length n (in O ( m log log n ) time, with current techniques). However, it was unable to locate the positions of those occurrences efficiently within a space bounded in terms of r . In this article, we close this long-standing problem, showing how to extend the Run-Length FM-index so that it can locate the occ occurrences efficiently (in O ( occ log log n ) time) within O ( r ) space. By raising the space to O ( r log log n ), our index counts the occurrences in optimal time, O ( m ), and locates them in optimal time as well, O ( m + occ ). By further raising the space by an O ( w / log σ) factor, where σ is the alphabet size and w = Ω (log n ) is the RAM machine size in bits, we support count and locate in O (⌈ m log (σ)/ w ⌉) and O (⌈ m log (σ)/ w ⌉ + occ ) time, which is optimal in the packed setting and had not been obtained before in compressed space. We also describe a structure using O ( r log ( n / r )) space that replaces the text and extracts any text substring of length ℓ in the almost-optimal time O (log ( n / r )+ℓ log (σ)/ w ). Within that space, we similarly provide access to arbitrary suffix array, inverse suffix array, and longest common prefix array cells in time O (log ( n / r )), and extend these capabilities to full suffix tree functionality, typically in O (log ( n / r )) time per operation. Our experiments show that our O ( r )-space index outperforms the space-competitive alternatives by 1--2 orders of magnitude in time. Competitive implementations of the original FM-index are outperformed by 1--2 orders of magnitude in space and/or 2--3 in time.