# Problem 97: Lempel-Ziv Factorisation

The problem deals with a Lempel-Ziv factorisation of words. The factorisation considered here is the decomposition of a word $w$ into the product $w_0w_1\cdots w_k$ where each $w_i$ is the longest prefix of $w_iw_{i+1}\cdots w_k$ that occurs in $w$ before the present position $|w_0w_1\cdots w_{i-1}|$. If there is no previous occurrence, $w_i$ is the first letter of $w_iw_{i+1}\cdots w_k$.

The factorisation is stored in the array $\LZ$: $\LZ[0] = 0$ and, for $1\le i \le k$, $\LZ[i] = |w_0w_1\cdots w_{i-1}|$.

Show how to compute the array $\LZ$ of a word in linear time assuming a fixed-size alphabet.

The same running time can be achieved when the alphabet is linearly sortable, which is a weaker condition than the above one. This is done from the longest previous array ($\LPF$) array of the word that computes in linear time under this condition (see Problem 53).

The $\LPF$ array of a word $w$ is defined, for each position $i$ on $w$, by: $\LPF[i]$ is the length of the longest factor of $w$ that starts both at position $i$ and at a smaller position.

Design an algorithm that computes in linear time the Lempel-Ziv factorisation of a word given its $\LPF$ array.

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