We show that the compressed suffix array and the compressed suffix tree for a string of length $n$ over an integer alphabet of size $σ\leq n$ can both be built in $O(n)$ (randomized) time using only $O(n\logσ)$ bits of working space. The previously fastest construction algorithms that used $O(n\logσ)$ bits of space took times $O(n\log\logσ)$ and $O(n\log^εn)$ respectively (where $ε$ is any positive constant smaller than $1$). In the passing, we show that the Burrows-Wheeler transform of a string of length $n$ over an alphabet of size $σ$ can be built in deterministic $O(n)$ time and space $O(n\logσ)$. We also show that within the same time and space, we can carry many sequence analysis tasks and construct some variants of the compressed suffix array and compressed suffix tree.

Expanded version of a paper appeared in proceedings of STOC 2014 conference