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.