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Universal Compressed Text Indexing

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Gonzalo Navarro and Nicola Prezza

The rise of repetitive datasets has lately generated a lot of interest
in compressed self-indexes based on dictionary compression, a rich and
heterogeneous family of techniques that exploits text repetitions in different
ways. For each such compression scheme, several different indexing solutions
have been proposed in the last two decades. To date, the fastest indexes for
repetitive texts are based on the run-length compressed Burrows-Wheeler
transform (BWT) and on the Compact Directed Acyclic Word Graph (CDAWG). The
most space-efficient indexes, on the other hand, are based on the Lempel-Ziv
parsing and on grammar compression. Indexes for more universal schemes such as
collage systems and macro schemes have not yet been proposed. Very recently,
Kempa and Prezza [STOC 2018] showed that all dictionary compressors can be
interpreted as approximation algorithms for the smallest *string
attractor*, that is, a set of text positions capturing all distinct
substrings. Starting from this observation, in this paper we develop the first
*universal* compressed self-index, that is, the first indexing data
structure based on string attractors, which can therefore be built on top of
any dictionary-compressed text representation. Let *g* be the size of a
string attractor for a text of length *n*. From known reductions,
*g* can
be chosen to be asymptotically equal to any repetitiveness measure: number of
runs in the BWT, size of the CDAWG, number of Lempel-Ziv phrases, number of
rules in a grammar or collage system, size of a macro scheme. Our index takes
*O(g log(n/g))* words of space and supports locating the *occ*
occurrences of any pattern of length *m* in *O(m log n +
occ log^e n)* time, for any constant *e > 0*. This is, in
particular, the first index for general macro schemes and collage systems. Our
result shows that the relation between indexing and compression is much deeper
than what was previously thought: the simple property standing at the core of
all dictionary compressors is sufficient to support fast indexed queries.