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The Suffix Array $SA_S[1\ldots n]$ of an $n$-length string $S$ is alexicographically sorted array of the suffixes of $S$. The suffix array is oneof the most well known and widely used data structures in string algorithms. Wepresent a data structure for maintaining a representation of the suffix arrayof a dynamic string which undergoes symbol substitutions, deletions, andinsertions.  For every string manipulation, our data structure can be updated in$O(n^{\frac{2}{3}})$ time (ignoring multiplicative polylogarithmic factors)with $n$ being the current length of the string. For an input query $i\in[1\ldots n]$, our data structure reports $SA_S[i]$ in $O(\log^5(n))$ time.  We also present a faster data structure, with $O(\sqrt{n})$ update time(ignoring multiplicative polylogarithmic factors), for maintaining the InvertedSuffix Array of a dynamic string undergoing symbol substitutions updates. Foran input query $i\in [1\ldots n]$, our data structure reports the $i$'th entryin the inverted suffix array in $O(\log^4(n))$ time.  Our data structures can be used to obtain sub-linear dynamic algorithms forseveral classical string problems for which efficient dynamic solutions werenot previously known.

Dynamic Suffix Array with Sub-linear update time and Poly-logarithmic Lookup Time