Compressed Inverse Suffix Arrays

Abstract

The suffix array ($\SA$) and inverse suffix array ($\ISA$) are fundamental data structures in string algorithms. For a text $T[0 \dd n)$ over an alphabet $[0 \dd \sigma)$, explicitly storing either structure requires $\Theta(n\log n)$ bits, whereas the text itself requires only $n\log\sigma$ bits. Classical compressed indexes, including the FM-index of Ferragina and Manzini [FOCS 2000] and the compressed suffix array of Grossi and Vitter [STOC 2000], reduce the space to $O(n\log\sigma)$ bits while supporting both $\SA$ and $\ISA$ queries efficiently. Moreover, essentially all known approaches yield nearly identical space--time trade-offs for the two operations. This raises a natural question: \emph{under the same asymptotic space bound, do $\SA$ and $\ISA$ queries have the same inherent query-time complexity?} We provide strong evidence that they do not. Achieving $\log^{o(1)} n$ query time for $\SA$ using $O(n\log\sigma)$ bits would require a major breakthrough in computational geometry, by a straightforward consequence of a hardness result of Chien {\it et al.} [Algorithmica 2015]. In contrast, we give a near-succinct encoding using $n\log\sigma+o(n\log\sigma)+O(n)$ bits that supports $t_{\ISA}=O!\left(\tau+\frac{\log\log n}{\log\log\sigma}\right)$, where $\tau=\omega(1)$ can grow arbitrarily slowly, and $t_{\SA}=O(t_{\ISA}\tau\log_{\sigma} n)$. The leading $n\log\sigma$ bits are used solely to store the text in packed form, enabling optimal-time substring extraction and efficient packed pattern searching. Building on Sadakane [SODA 2002], we also give a suffix-tree encoding auxiliary to the text that uses only $O(n\log\log\log\sigma)$ bits and supports standard operations in polylogarithmic time. For growing alphabets, this auxiliary space is asymptotically smaller than the space required to store the text itself.