Problem 87: Computing Runs on Ordered Alphabets
\(
\def\sa#1{\tt{#1}}
\def\dd{\dot{}\dot{}}
\def\Lyn{\mathit{Lyn}}
\)
The aim of the problem is to show that all runs (maximal
periodicities) in a word drawn from a linearly sortable alphabet can be
computed in linear time.
The solution is based on the results of Problem 86,
where it is shown that a run possesses a special position from which starts
a longest Lyndon factor of the whole word.
Tracking the longest Lyndon factors has to be done
according to the alphabet ordering $\lt$ but also to its inverse $\lt^{-1}$.
When a longest Lyndon factor is located, simple extensions from two positions to
the right and to the left (like in Problem 74)
can confirm the starting position of the Lyndon factor is a special position
of a run whose period is the length of the Lyndon factor.
To do so we first define the Lyndon table $\Lyn$ of a
(non-empty) word $x$.
For a position $i$ on $x$, $i=0,\dots,|x|-1$, $\Lyn[i]$
is the length of the longest Lyndon factor starting at $i$:
\[\Lyn[i] = \max\{\ell \mid x[i\dd i+\ell-1] \mbox{ is a Lyndon word}\}.\]
| $i$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
| $x[i]$ | $\sa{a}$ | $\sa{b}$ | $\sa{b}$ | $\sa{a}$ | $\sa{b}$ | $\sa{a}$ | $\sa{b}$ | $\sa{a}$ | $\sa{a}$ | $\sa{b}$ | $\sa{a}$ | $\sa{b}$ | $\sa{b}$ | $\sa{a}$ | $\sa{b}$ | $\sa{a}$ |
| $\Lyn[i]$ | 3 | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 5 | 1 | 3 | 1 | 1 | 2 | 1 | 1 |
Show that Algorithm LongestLyndon correctly computes the table $\Lyn$.
LongestLyndon$(x \textrm{ non-empty word of length } n)$
\begin{algorithmic}
\FOR{$i\leftarrow n-1$ \DOWNTO $0$}
\STATE $(Lyn[i],j)\leftarrow (1,i+1)$
\WHILE{$j\lt n \mbox{ and } x[i..j-1] \lt x[j..j+Lyn[j]-1]$}
\STATE $(Lyn[i],j)\leftarrow (Lyn[i]+Lyn[j],j+Lyn[j])$
\ENDWHILE
\ENDFOR
\RETURN{Lyn}
\end{algorithmic}
Extend Algorithm LongestLyndon to compute all runs occurring in a word.
Use the longest common extensions like in
Problem 74.
What is the total running time of the algorithm if a comparison of two
factors is done with the help of the ranks of their associated suffixes in
the alphabetic order and if the longest common extension techniques are used?
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