Breaking the $O(n)$-Barrier in the Construction of Compressed Suffix Arrays and Suffix Trees

Abstract

The suffix array and the suffix tree are the two most fundamental data structures for string processing. For a length-$n$ text, however, they use $\Theta(n \log n)$ bits of space, which is often too costly. To address this, Grossi and Vitter [STOC 2000] and, independently, Ferragina and Manzini [FOCS 2000] introduced space-efficient versions of the suffix array, known as the compressed suffix array (CSA) and the FM-index. Sadakane [SODA 2002] then showed how to augment them to obtain the compressed suffix tree (CST). For a length-$n$ text over an alphabet of size $\sigma$, these structures use only $O(n\log\sigma)$ bits. The biggest remaining open question is how efficiently they can be constructed. After two decades, the fastest algorithms still run in $O(n)$ time [Hon et al., FOCS 2003], which is $\Theta(\log_{\sigma} n)$ factor away from the lower bound of $\Omega(n/\log_{\sigma}n)$. In this paper, we make the first in 20 years improvement in $n$ for this problem by proposing a new compressed suffix array and a new compressed suffix tree which admit $o(n)$-time construction algorithms while matching the space bounds and the query times of the original CSA/CST and the FM-index. More precisely, our structures take $O(n\log\sigma)$ bits, support SA queries and full suffix tree functionality in $O(\log^{\epsilon}n)$ time per operation, and can be constructed in $O(n \min(1,\log\sigma/\sqrt{\log n}))$ time using $O(n\log\sigma)$ bits of working space. We derive this result as a corollary from a much more general reduction: We prove that all parameters of a compressed suffix array/tree (query time, space, construction time, and construction working space) can essentially be reduced to those of a data structure answering new query types that we call prefix rank and prefix selection. Using the novel techniques, we also develop a new index for pattern matching.