New Lower Bounds for the Maximum Number of Runs in a String

We show a new lower bound for the maximum number of runs in a string. We prove that for any e > 0, (α − e)n is an asymptotic lower bound, where α = 174719/184973 ≈ 0.944565. It is superior to the previous bound 3/(1 + √ 5) ≈ 0.927 given by Franěk et al. [6,7]. Moreover, our construction of the strings and the proof is much simpler than theirs.

[1]  Gregory Kucherov,et al.  Finding maximal repetitions in a word in linear time , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[2]  Qian Yang,et al.  An asymptotic lower bound for the maximal-number-of-runs function , 2006, Stringology.

[3]  H. Wilf,et al.  Uniqueness theorems for periodic functions , 1965 .

[4]  William F. Smyth,et al.  How many runs can a string contain? , 2008, Theor. Comput. Sci..

[5]  Lucian Ilie,et al.  Maximal repetitions in strings , 2008, J. Comput. Syst. Sci..

[6]  Wojciech Rytter,et al.  The Number of Runs in Sturmian Words , 2008, CIAA.

[7]  Chris D. Godsil,et al.  ALGEBRAIC COMBINATORICS , 2013 .

[8]  M. Lothaire.  Algebraic Combinatorics on Words , 2002 .

[9]  Mathieu Giraud,et al.  Not So Many Runs in Strings , 2008, LATA.

[10]  Wojciech Rytter,et al.  The number of runs in a string , 2007, Inf. Comput..

[11]  Wojciech Rytter,et al.  Jewels of stringology , 2002 .

[12]  Wojciech Rytter,et al.  The Number of Runs in a String: Improved Analysis of the Linear Upper Bound , 2006, STACS.

[13]  William F. Smyth,et al.  The maximum number of of runs in a string , 2003, IWOCA 2007.

[14]  Lucian Ilie,et al.  Towards a Solution to the "Runs" Conjecture , 2008, CPM.