Optimal Construction of Compressed Indexes for Highly Repetitive Texts

Abstract: We propose algorithms that, given the input string of length $n$ over integer alphabet of size $\sigma$, construct the Burrows-Wheeler transform (BWT), the permuted longest-common-prefix (PLCP) array, and the LZ77 parsing in $O(n/\log_{\sigma}n+r\,{\rm polylog}\,n)$ time and working space, where $r$ is the number of runs in the BWT of the input. These are the essential components of many compressed indexes such as compressed suffix tree, FM-index, and grammar and LZ77-based indexes, but also find numerous applications in sequence analysis and data compression. The value of $r$ is a common measure of repetitiveness that is significantly smaller than $n$ if the string is highly repetitive. Since just accessing every symbol of the string requires $\Omega(n/\log_{\sigma}n)$ time, the presented algorithms are time and space optimal for inputs satisfying the assumption $n/r\in\Omega({\rm polylog}\,n)$ on the repetitiveness. For such inputs our result improves upon the currently fastest general algorithms of Belazzougui (STOC 2014) and Munro et al. (SODA 2017) which run in $O(n)$ time and use $O(n/\log_{\sigma} n)$ working space. We also show how to use our techniques to obtain optimal solutions on highly repetitive data for other fundamental string processing problems such as: Lyndon factorization, construction of run-length compressed suffix arrays, and some classical "textbook" problems such as computing the longest substring occurring at least some fixed number of times.