Efficient Lempel-Ziv Prediction with the Run-Length BWT

Abstract

Lempel-Ziv Plurality (LZP) is an algorithm of Kosoy that predicts the next symbol of a string by using its Lempel-Ziv factorization, and makes at most $O(z\log n)$ mistakes on every string of length $n$ with $z$ LZ77 phrases. We give efficient online implementations of LZP and of a new variant of it. Our first implementation maintains a run-length encoded Burrows-Wheeler transform (RLBWT) of the reversed history. It implements LZP, uses $O(z\log z\log^2 n)$ bits, and has worst-case update and prediction times $O(\log z+\log\log n)$ and $O(σ(\log z+\log\log n))$, respectively, where $σ$ is the size of the alphabet. This is an exponential improvement over Kosoy's implementation, whose update time is $O(z\log^7 n)$. We then introduce the Lempel-Ziv Majority (LZM) predictor, which has the same mistake bound and update time, and predicts in $O(1)$ time independently of $σ$. These two predictors accept new symbols at any time, without knowing the alphabet in advance. For an alphabet known in advance, a different data structure gives $O(σ(1+\log z/\log\log n))$ time per symbol, which is constant when $σ=O(1)$ and $z=\mathrm{polylog}(n)$.