Example

$ \def\sa#1{\tt{#1}} \def\dd{\dot{}\dot{}} $

Consider the word $\sa{abcacbabcbac}$. Let $L=\{\sa{a},\sa{c}\}$, $R=\{\sa{b}\}$. Then, substituting $\sa{d}$ for $\sa{ab}$ and $\sa{e}$ for $\sa{cb}$ produces the word $\sa{dcaedeac}$ of length $8\lt \frac{3}{4}12=9$.

On the contrary, setting $L=\{\sa{a}\}$, $R=\{\sa{b}\}$ and substituting $\sa{d}$ for $\sa{ab}$ in the word $\sa{aaabbb}$ containing two unary runs produces the word $\sa{aadbb}$ of length $5\gt \frac{3}{4}6=4.5$, which do not meet the desired bound.

Additional Problem 4: Shrinking a Text by Pairing Adjacent Symbols

$ \newcommand{\Comp}{\mathit{Compress}} $

One of the most powerfull compression techniques is recompression. In this technique there are two crucial operations: shrinking unary runs and pairing letters.

A unary run is a maximal occurrence of a factor of length at least 2 that is a repetition of the same letter. The first phase of the recompression technique consists in shrinking each unary run into a single letter.

The second phase it to apply the operation $\Comp(x,L,R)$, where $(L,R)$ is a partition of the alphabet $\Sigma$ of letters, $L\cup R=\Sigma$ and $L\cap R=\emptyset$. The compressed word $\Comp(x,L,R)$ results from $x$ by substituting a single letter (identifier of a pair of letters) for each of its 2-letter factor $ab$, whenever $a\in L$ and $b\in R$.

The Pairing problem consists in computing a partition $(L,R)$ of the alphabet of $x$ for which $|\Comp(x,L,R)|\le \frac{3}{4}|x|$.

Let $x$, $|x|\geq 2$, be a word over an integer alphabet containing no unary runs. Show how to compute in linear time a partition $(L,R)$ of the alphabet of $x$ for which $|\Comp(x,L,R)|\le \frac{3}{4}|x|$.

References

A. Jez, Recompression: A simple and powerful technique for word equations, J. ACM 63(1):4:1-4:51, 2016.

125 Problems in Text Algorithms

Example

Additional Problem 4: Shrinking a Text by Pairing Adjacent Symbols

References