Optimal-Time Text Indexing in BWT-runs Bounded Space

Travis Gagie, Gonzalo Navarro and Nicola Prezza

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 Transform (BWT). 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 the text (in loglogarithmic time per pattern symbol, with current techniques). However, it was unable to locate the positions of those occurrences efficiently within a space bounded in terms of r. Since then, a number of other indexes with space bounded by other measures of repetitiveness --- the number of phrases in the Lempel-Ziv parse, the size of the smallest grammar generating the text, the size of the smallest automaton recognizing the text factors --- have been proposed for efficiently locating, but not directly counting, the occurrences of a pattern. In this paper we close this long-standing problem, showing how to extend the Run-Length FM-index so that it can locate the occ occurrences efficiently within O(r) space (in loglogarithmic time each), and reaching optimal time O(m+occ) within O(r log (n/r)) space, on a RAM machine with words of w = Omega(log n) bits. Raising the space to O(r w log_s(n/r)), we support locate in O(m log(s)/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 efficiently extracts any text substring, with an O(log (n/r)) additive time penalty over the optimum. Preliminary experiments show that our new structure outperforms the alternatives by orders of magnitude in the space/time tradeoff map.