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

TL;DR

This paper introduces new dynamic suffix array data structures that support symbol updates with sub-linear update times and poly-logarithmic query times, enabling efficient handling of dynamic string problems.

Contribution

The authors present novel data structures for dynamic suffix arrays with sub-linear update times and fast query capabilities, advancing the efficiency of dynamic string algorithms.

Findings

Update time for suffix array: O(n^{2/3})

Query time for suffix array: O(log^5 n)

Inverted suffix array update time: O(√n)

Abstract

The Suffix Array $SA_S[1\ldots n]$ of an $n$-length string $S$ is a lexicographically sorted array of the suffixes of $S$. The suffix array is one of the most well known and widely used data structures in string algorithms. We present a data structure for maintaining a representation of the suffix array of a dynamic string which undergoes symbol substitutions, deletions, and insertions. 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 Inverted Suffix Array of a dynamic string undergoing symbol substitutions updates. For an input query $i\in [1\ldots n]$, our data structure reports the $i$'th entry in the inverted suffix array in $O(\log^4(n))$ time. Our data structures can be used to obtain sub-linear dynamic algorithms for several classical string problems for which efficient dynamic solutions were not previously known.