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LZ77 previous-match length and recurrence time

Definition

Consider a finite alphabet $A$ and a two-sided sequence $x \in A^{Z}$ .

For positive integers $n, W$ , define the backward recurrence time of the current length- $n$ block by $R_{n} (x) = \inf {j \geq 1 : (x_{- j}, \dots, x_{- j + n - 1}) = (x_{0}, \dots, x_{n - 1})},$ with $R_{n} = + \infty$ if the set is empty. The start offset is positive, so the current block is not counted as its own match.

Define its previous-match phrase length in a window of width $W$ by $L_{W} (x) = \max ({0} \cup {n \geq 1 : R_{n} (x) \leq W}),$ where the maximum is interpreted as $+ \infty$ when the set is unbounded. The empty match of length zero is always allowed. For a periodic sequence, $L_{W}$ can be infinite; the recurrence-time theorem will only be used to infer the logarithmic inverse relation in probability when $h > 0$ .

Facts & Assumptions

Given: A finite alphabet $A$ and a two-sided sequence $x \in A^{Z}$ .

[F1]

Finite-alphabet stationary source: the source coordinates are indexed over $Z$ and shifted by $T (x)_{j} = x_{j + 1}$ .

Proof

Given: The sequence and positive integers $n, W$ .

1.1F1givenalgebra

Define the backward recurrence time of the current length- $n$ block by $R_{n} (x) = \inf {j \geq 1 : (x_{- j}, \dots, x_{- j + n - 1}) = (x_{0}, \dots, x_{n - 1})},$ with $R_{n} = + \infty$ if the set is empty. The start offset is positive, so the current block is not counted as its own match.

2.1F1step 1.1algebra

Define its previous-match phrase length in a window of width $W$ by $L_{W} (x) = \max ({0} \cup {n \geq 1 : R_{n} (x) \leq W}),$ with maximum $+ \infty$ when the set is unbounded. The set is nonempty because it contains $0$ ; if a length- $m$ block matches at offset $j$ , every shorter positive prefix also matches at the same offset, so the matched lengths form an initial segment. The empty match of length zero is always allowed.

3.1step 2.1algebra∎

For a periodic sequence, $L_{W}$ can be infinite; the recurrence-time theorem will only be used to infer the logarithmic inverse relation in probability when $h > 0$ .

Depends on

Finite-alphabet stationary source

Used by

Stationary-ergodic LZ77 phrase-length theorem Theorem

Dependency tree · two levels

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Sources

Ioannis Kontoyiannis, Recurrence and Waiting Times in Stationary Processes and Their Applications in Data Compression, Stanford University dissertation (standard reference, not scraped)