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Rank/Select on Dynamic Compressed Sequences and Applications

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Rodrigo González and Gonzalo Navarro

Operations *rank* and *select* over a sequence of symbols have many
applications to the design of succinct and compressed data structures
managing text collections, structured text, binary relations, trees, graphs,
and so on. We are interested in the case where the collections can be updated
via insertions and deletions of symbols. Two current solutions stand out as the
best in the tradeoff of space versus time (when considering all the operations).
One solution, by Mäkinen and Navarro, achieves compressed space (i.e.,
*nH_0+o(n log s)* bits) and *O(log n log s)* worst-case time for all
the operations, where *n* is the sequence length, *s* is the alphabet
size, and *H_0* is the zero-order entropy of the sequence. The other
solution, by Lee and Park, achieves *O(log n (1+(log s)/(log log n)))*
amortized time and uncompressed space, i.e. *n log_2 s +O(n)+o(n log s)*
bits. In this paper we show that the best of both worlds can be achieved. We
combine the solutions to obtain *nH_0+o(n log s)* bits of space and
*O(log n (1+(log s)/(log log n)))* worst-case time for all the
operations. Apart from the best current solution to the problem, we obtain
several byproducts of independent interest applicable to partial sums, text
indexes, suffix arrays, the Burrows-Wheeler transform, and others.