Approximating LZ77 in Small Space

Given a positive $\epsilon \leq 1$ and read-only access to a string $S [1..n]$ whose LZ77 parse consists of $z$ phrases, with high probability we can build an LZ77-like parse of $S$ that consists of $\Oh{z / \epsilon}$ phrases using $\Oh{n^{1 + \epsilon}}$ time, $\Oh{n^{1 + \epsilon} / B}$ I/Os (where $B$ is the size of a disk block) and $\Oh{z / \epsilon}$ space.