Accessing the Suffix Array via $\phi ^{-1}$-Forest

Abstract

Kärkkainen et al. (CPM, 2009) defined the concept of $\phi$ that later became key to the construction of the r-index. Given a string S[1..n], its suffix array $\textrm{SA}$ and its inverse suffix array $\textrm{ISA}$, we define $\phi$ as the permutation of $\{1,\ldots ,n\}$ such that $\phi (i) = \textrm{SA}[\textrm{ISA}[i]-1]$ if $\textrm{ISA}[i] > 1$, and $\phi (i) = \textrm{SA}[n]$ otherwise. Gagie et al. (JACM, 2020) showed that it is possible to store $\mathcal {O}(r)$ words such that the permutations $\phi$ and $\phi ^{-1}$ are evaluated in $\mathcal {O}(\log \log _w(n/r))$-time, which was improved to $\mathcal {O}(1)$-time by Nishimoto and Tabei (ICALP, 2021). In this paper, we introduce the concept of $\phi ^{-1}$-forest, which is a data structure using sampled $\textrm{SA}$ values to speed up random access to $\textrm{SA}$. We implemented our approach and compared its performance with respect to the r-index.