In-Place Sparse Suffix Sorting

Suffix arrays encode the lexicographical order of all suffixes of a text and are often combined with the Longest Common Prefix array (LCP) to simulate navigational queries on the suffix tree in reduced space. In space-critical applications such as sparse and compressed text indexing, only information regarding the lexicographical order of a size-$b$ subset of all $n$ text suffixes is often needed. Such information can be stored space-efficiently (in $b$ words) in the sparse suffix array (SSA). The SSA and its relative sparse LCP array (SLCP) can be used as a space-efficient substitute of the sparse suffix tree. Very recently, Gawrychowski and Kociumaka [SODA 2017] showed that the sparse suffix tree (and therefore SSA and SLCP) can be built in asymptotically optimal $O(b)$ space with a Monte Carlo algorithm running in $O(n)$ time. The main reason for using the SSA and SLCP arrays in place of the sparse suffix tree is, however, their reduced space of $b$ words each. This leads naturally to the quest for in-place algorithms building these arrays. Franceschini and Muthukrishnan [ICALP 2007] showed that the full suffix array can be built in-place and in optimal running time. On the other hand, finding sub-quadratic in-place algorithms for building the SSA and SLCP for \emph{general} subsets of suffixes has been an elusive task for decades. In this paper, we give the first solution to this problem. We provide the first in-place algorithm building the full LCP array in $O(n\log n)$ expected time and the first Monte Carlo in-place algorithms building the SSA and SLCP in $O(n + b\log^2 n)$ expected time. We moreover describe the first in-place solution for the suffix selection problem: to compute the $i$-th smallest text suffix.