[Paper Review] Fewer runs than word length

Maxime Crochemore, Robert Mercaş|arXiv (Cornell University)|Dec 15, 2014

Algorithms and Data Compression15 references3 citations

TL;DR

This paper provides a novel proof that the number of runs in a word of length $ n $ is strictly less than $ n $, using lexicographic ordering to assign each run a unique border-free Lyndon root suffix. The key contribution is a combinatorial argument based on unique suffix assignment that establishes the upper bound, reinforcing the conjecture that runs are rarer than word length.

ABSTRACT

The work takes another look at the number of runs that a string might contain and provides an alternative proof for the bound. We also propose another stronger conjecture that states that, for a fixed order on the alphabet, within every factor of a word there are at most as many occurrences of Lyndon roots corresponding to runs in a word as the length of the factor (only first such occurrences for each run are considered).

Motivation & Objective

• To provide an alternative, combinatorial proof that the number of runs in a word of length $ n $ is less than $ n $, confirming a long-standing conjecture.
• To establish a one-to-one correspondence between runs and their greatest proper suffixes under lexicographic or reverse lexicographic ordering.
• To analyze the density of runs by linking them to border-free Lyndon roots and critical positions in squares.
• To investigate the maximum possible density of runs in factors of a word, particularly in binary words.
• To refine bounds on the number of $ \mathrm{Oroot} $s (Lyndon root conjugates) in intervals, especially under lexicographic and reverse orderings.

Proposed method

• Assign to each run its greatest proper suffix using lexicographic ordering $ < $ or reverse lexicographic ordering $ \widetilde{<} $, depending on whether the run can be extended to the right with a larger character.
• Prove that each position $ k > 0 $ in the word can be the starting point of at most one such assigned suffix, ensuring injectivity.
• Use the property that the assigned suffix corresponds to a border-free conjugate of the run’s period root, which is a Lyndon word.
• Leverage the fact that Lyndon words are lexicographically smallest among their rotations and border-free, to ensure uniqueness of the assigned suffix.
• Apply the result to binary words by distinguishing $ \mathrm{Oroot} $s based on their starting character and ordering, and use overlap properties to bound their number.
• Use Lemma 13 (overlap property of $ \mathrm{Oroot} $s) to show that overlapping $ \mathrm{Oroot} $s must be nested or disjoint, and that their lengths differ by at least 2 unless unary.

Experimental results

Research questions

• RQ1 Can a simpler or alternative proof be given for the conjecture that the number of runs in a word of length $ n $ is less than $ n $?
• RQ2 What is the structural relationship between runs and their Lyndon root conjugates, particularly in terms of suffix assignment?
• RQ3 How many $ \mathrm{Oroot} $s (Lyndon root conjugates) can occur in a factor of length $ \ell $, and what configurations maximize this number?
• RQ4 What constraints do the overlap and nesting properties of $ \mathrm{Oroot} $s impose on their maximum density in binary words?
• RQ5 Can tighter bounds be established on the number of $ \mathrm{Oroot} $s in intervals of a binary word, especially when restricted to a single ordering (lex or reverse-lex)?

Key findings

• The number of runs in any word of length $ n $ is strictly less than $ n $, confirmed via a new injective assignment of runs to their greatest suffixes.
• Each run is uniquely associated with a border-free Lyndon word that is a conjugate of its period root, via lexicographic or reverse lexicographic ordering.
• No two distinct runs can be assigned the same starting position for their greatest suffix, ensuring injectivity and proving the upper bound.
• In binary words, any interval of length $ \ell $ contains at most $ \frac{\ell - 1}{2} $ $ \mathrm{Oroot} $s obtained under the same ordering (lex or reverse-lex).
• The maximum number of $ \mathrm{Oroot} $s in a factor is achieved only when the factor is of the form $ a(ab)^{(\ell-2)/2}b $ with $ a < b $, under the conjecture.
• For non-uniform $ \mathrm{Oroot} $s, the number of such roots in a factor is bounded by $ \min\{|w|_{ab}, |w|_{ba}\} $, and for unary runs, at most one extra $ \mathrm{Oroot} $ per maximal unary block.