Dynamic Suffix Array with Polylogarithmic Queries and Updates

The suffix array SA [1..n] of a text T of length n is a permutation of {1, ..., n} describing the lexicographical ordering of suffixes of T and is considered to be one of the most important data structures for string processing, with dozens of applications in data compression, bioinformatics, and information retrieval. One of the biggest drawbacks of the suffix array is that it is very difficult to maintain under text updates: even a single character substitution can completely change the contents of the suffix array. Thus, the suffix array of a dynamic text is modelled using suffix array queries, which return the value SA [i] given any i is an element of [1..n]. Prior to this work, the fastest dynamic suffix array implementations were by Amir and Boneh, who showed how to answer suffix array queries in (O) over tilde (k) time, where k is an element of [1..n] is a trade-off parameter, with (O) over tilde (n/k)-time text updates [ISAAC 2020]. In a very recent preprint, they also provided a solution with O (log(5)n)-time queries and (O) over tilde (n(2/3))-time updates [arXiv 2021]. We propose the first data structure that supports both suffix array queries and text updates in O(polylog n) time (achieving O(log(4) n) and O(log(3+o (1)) n) time, respectively). Our data structure is deterministic and the running times for all operations are worst-case. In addition to the standard single-character edits (character insertions, deletions, and substitutions), we support (also in O(log (3+o(1)) n) time) the "cut-paste" operation that moves any (arbitrarily long) substring of T to any place in T. To achieve our result, we develop a number of new techniques which are of independent interest. This includes a new flavor of dynamic locally consistent parsing, as well as a dynamic construction of string synchronizing sets with an extra local sparsity property; this significantly generalizes the sampling technique introduced at STOC 2019. We complement our structure by a hardness result: unless the Online Matrix-Vector Multiplication (OMv) Conjecture fails, no data structure with O (polylog n)-time suffix array queries can support the "copy-paste" operation in O (n(1-epsilon)) time for any epsilon > 0.