A new periodicity lemma

  • Authors:

  • Kangmin Fan;William F. Smyth;R. J. Simpson

  • Affiliations:

  • Algorithms Research Group, Department of Computing & Software, McMaster University, Hamilton, Ontario, Canada;Algorithms Research Group, Department of Computing & Software, McMaster University, Hamilton, Ontario, Canada;Department of Mathematics & Statistics, Curtin University, Perth, WA, Australia

  • Venue:
  • CPM'05 Proceedings of the 16th annual conference on Combinatorial Pattern Matching
  • Year:
  • 2005

Quantified Score

Hi-index 0.00

Visualization

Abstract

Given a string x=x[1..n], a repetition of period p in x is a substring ur=x[i..i+rp−1], p = |u|, r ≥ 2, where neither u=x[i..i+p−1] nor x[i..i+(r+1)p−1] is a repetition. The maximum number of repetitions in any string x is well known to be Θ(nlog n). A run or maximal periodicity of period p in x is a substring urt=x[i..i+rp+|t|−1] of x, where ur is a repetition, t a proper prefix of x, and no repetition of period p begins at position i – 1 of x or ends at position i + rp + |t|. In 2000 Kolpakov and Kucherov showed that the maximum number ρ(n) of runs in any string x is O(n), but their proof was nonconstructive and provided no specific constant of proportionality. At the same time, they presented experimental data strongly suggesting that ρ(n) n. In this paper, as a first step toward proving this conjecture, we present a periodicity lemma that establishes limitations on the number of squares, and their periods, that can occur over a specified range of positions in x. We then apply this result to specify corresponding limitations on the occurrence of runs.