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- W2736164374 abstract "The so-called one-bit catastrophe for the compression algorithm LZ'78 asks whether the compression ratio of an infinite word can change when a single bit is added in front of it. We answer positively this open question raised by Lutz and others: we show that there exists an infinite word $w$ such that $rho_{sup}(w)=0$ but $rho_{inf}(0w)>0$, where $rho_{sup}$ and $rho_{inf}$ are respectively the $limsup$ and the $liminf$ of the compression ratios $rho$ of the prefixes. To that purpose we explore the behaviour of LZ'78 on finite words and show the following results: - There is a constant $C>0$ such that, for any finite word $w$ and any letter $a$, $rho(aw)leq Csqrt{rho(w)log|w|}$. Thus, sufficiently compressible words ($rho(w)=o(1/log|w|)$) remain compressible with a letter in front; - The previous result is tight up to a multiplicative constant for any compression ratio $rho(w)=O(1/log|w|)$. In particular, there are infinitely many words $w$ satisfying $rho(w)=O(1/log|w|)$ but $rho(0w)=Omega(1)$." @default.
- W2736164374 created "2017-07-21" @default.
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- W2736164374 date "2017-07-13" @default.
- W2736164374 modified "2023-09-26" @default.
- W2736164374 title "Lempel-Ziv: a one-bit catastrophe but not a tragedy" @default.
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