Compressed Inverse Suffix Arrays

The suffix array ($\SA$) and inverse suffix array ($\ISA$) are fundamental data structures in string algorithms. Given a text of length $n$ over an integer alphabet of size $\sigma$, explicitly storing these structures requires $\Theta(n\log n)$ bits, motivating compressed representations that approach the information-theoretic minimum of $n\log\sigma$ bits. Two landmark solutions are the FM-index, which achieves succinct space and supports $\SA$ and $\ISA$ queries in roughly logarithmic time, and the Compressed Suffix Array (CSA), which uses $O(n\log\sigma)$ bits and answers both queries in $O(\log_\sigma^\epsilon n)$ time. Nearly all subsequent compressed indexes, including entropy-compressed and repetitiveness-aware variants, support $\SA$ and $\ISA$ using essentially the same techniques, leading to similar space--time trade-offs. This raises a fundamental question: under the same asymptotic space bound, do $\SA$ and $\ISA$ have the same inherent query complexity? We provide evidence that this symmetry is not intrinsic. Under an $O(n\log\sigma)$-bit space bound, we show that $\SA$ queries are unlikely to admit $\log^{o(1)} n$ query time, suggesting that improving the classical CSA bound would require a major breakthrough. In contrast, we present a new encoding supporting $\ISA$ queries in $O(\tau+\log\log n/\log\log\sigma)$ time using $(1+1/\tau+o(1))n\log\sigma$ bits, for any $\tau\ge1$. Our main insight is that the $\LF$-mapping, traditionally implemented using the Burrows--Wheeler Transform, can be adapted to support $\ISA$ queries directly from the packed text, eliminating the need to store the BWT while achieving substantially faster query times.