On Singularities Of Extremal Periodic Strings F . Franek (joint - PowerPoint PPT Presentation

Motivation and background Basic properties of σ d ( n ) Basic properties of ρ d ( n ) Computational Substantiation Thanks On Singularities Of Extremal Periodic Strings F . Franek (joint work with A. Deza) Advanced Optimization Laboratory Department of Computing and Software McMaster University, Hamilton, Ontario, Canada CanaDAM 2013 Memorial University, St. John’s Newfoundland, June 10-13, 2013 On Singularities Of Extremal Periodic Strings CanaDAM 2013

Motivation and background Basic properties of σ d ( n ) Basic properties of ρ d ( n ) Computational Substantiation Thanks Outline Motivation and background 1 Basic properties of σ d ( n ) 2 Basic properties of ρ d ( n ) 3 Computational Substantiation 4 On Singularities Of Extremal Periodic Strings CanaDAM 2013

Motivation and background Basic properties of σ d ( n ) Basic properties of ρ d ( n ) Computational Substantiation Thanks In1998 by Fraenkel and Simpson showed that the number of distinct squares in a string of length n is at most 2 n and hypothesized that the bound should be n . In 2005 Ilie provided a simpler proof and in 2007 presented an asymptotic upper bound of 2 n − Θ( log n ) . In 1999 Kolpakov and Kucherov proved that the maximum number of runs in a string is linear in the string’s length and conjectured that it is in fact bounded by the length. Many additional authors ( Rytter, Smyth, Simpson, Puglisi, Crochemore, Ilie, Kusano, Matsubara, Ishino, Bannai, Shinohara, FF ) contributed to improving the lower and upper bounds to the current asymptotic 0 . 944565 n ≤ ρ ( n ) ≤ 1 . 029 n On Singularities Of Extremal Periodic Strings CanaDAM 2013

Motivation and background Basic properties of σ d ( n ) Basic properties of ρ d ( n ) Computational Substantiation Thanks We consider the role played by the size of the alphabet of the string in both problems and investigate the functions σ d ( n ) and ρ d ( n ) , i.e. the maximum number of distinct primitively rooted squares, respectively runs, over all strings of length n containing exactly d distinct symbols. We revisit earlier results and conjectures and express them in terms of singularities of the two functions where a pair ( d , n ) is a singularity if σ d ( n ) − σ d − 1 ( n − 2 ) ≥ 2, or ρ d ( n ) − ρ d − 1 ( n − 2 ) ≥ 2 respectively. On Singularities Of Extremal Periodic Strings CanaDAM 2013

Motivation and background Basic properties of σ d ( n ) Basic properties of ρ d ( n ) Computational Substantiation Thanks On Singularities Of Extremal Periodic Strings CanaDAM 2013

Motivation and background Basic properties of σ d ( n ) Basic properties of ρ d ( n ) Computational Substantiation Thanks On Singularities Of Extremal Periodic Strings CanaDAM 2013

Motivation and background Basic properties of σ d ( n ) Basic properties of ρ d ( n ) Computational Substantiation Thanks Proposition ( s 1 ) 0 ≤ σ d ( n + 1 ) − σ d ( n ) ≤ 2 for n ≥ d ≥ 2 ( s 2 ) σ d ( n ) ≤ σ d + 1 ( n + 1 ) for n ≥ d ≥ 2 ( s 3 ) σ d ( n ) < σ d + 1 ( n + 2 ) for n ≥ d ≥ 2 ( s 4 ) σ d ( n ) = σ d + 1 ( n + 1 ) for 2 d ≥ n ≥ d ≥ 2 ( s 5 ) σ d ( n ) ≥ n − d, σ d ( 2 d + 1 ) ≥ d and σ d ( 2 d + 2 ) ≥ d + 1 for 2 d ≥ n ≥ d ≥ 2 ( s 6 ) σ d − 1 ( 2 d − 1 ) = σ d − 2 ( 2 d − 2 ) and 0 ≤ σ d ( 2 d ) − σ d − 1 ( 2 d − 1 ) ≤ 1 for d ≥ 4 ( s 7 ) 1 ≤ σ d + 1 ( 2 d + 2 ) − σ d ( 2 d ) ≤ 2 for d ≥ 2 . On Singularities Of Extremal Periodic Strings CanaDAM 2013

Motivation and background Basic properties of σ d ( n ) Basic properties of ρ d ( n ) Computational Substantiation Thanks Corollary ( c 1 ) σ 2 ( n ) ≤ 2 n − 51 for n ≥ 41 ( c 2 ) σ ( n ) ≤ 2 n − 19 for n ≥ 30 . Conjecture For any n ≥ d ≥ 2 , σ d ( n ) ≤ n − d Theorem Let ( d , 2 d ) be the first singularity on the main diagonal, i.e. the least d such that σ d ( 2 d ) − σ d − 1 ( d − 2 ) ≥ 2 . Then any square-maximal ( d , 2 d ) -string does not contain a pair but must contain at least ⌈ 2 d 3 ⌉ singletons. On Singularities Of Extremal Periodic Strings CanaDAM 2013

Motivation and background Basic properties of σ d ( n ) Basic properties of ρ d ( n ) Computational Substantiation Thanks Theorem ( e 1 ) no ( d , 2 d ) singularity ⇐ ⇒ { σ d ( n ) ≤ n − d for n ≥ d ≥ 2 } ( e 2 ) { σ d ( n ) ≤ n − d for n ≥ d ≥ 2 } ⇐ ⇒ { σ d ( 4 d ) ≤ 3 d for d ≥ 2 } ( e 3 ) { σ d ( n ) ≤ n − d for n ≥ d ≥ 2 } ⇐ ⇒ { σ d ( 2 d + 1 ) − σ d ( 2 d ) ≤ 1 for d ≥ 2 } ( e 4 ) no ( d , 2 d + 1 ) singularity = ⇒ { no ( d , 2 d ) singularity and σ d ( n ) ≤ n − d − 1 for n > 2 d ≥ 4 } ( e 5 ) { σ d ( 2 d ) = σ d ( 2 d + 1 ) for d ≥ 2 } = ⇒ { no ( d , 2 d ) singularity and σ d ( n ) ≤ n − d − 1 for n > 2 d ≥ 4 } ( e 6 ) { σ d ( 2 d ) = σ d ( 2 d + 1 ) for d ≥ 2 } = ⇒ { square-maximal ( d , 2 d ) -strings are, up to relabelling, unique and equal to a 1 a 1 a 2 a 2 a 2 . . . a d a d } On Singularities Of Extremal Periodic Strings CanaDAM 2013

Motivation and background Basic properties of σ d ( n ) Basic properties of ρ d ( n ) Computational Substantiation Thanks Proposition ( r 1 ) ρ d ( n ) ≤ ρ d + 1 ( n + 1 ) for n ≥ d ≥ 2 ( r 2 ) ρ d ( n ) ≤ ρ d ( n + 1 ) for n ≥ d ≥ 2 ( r 3 ) ρ d ( n ) < ρ d + 1 ( n + 2 ) for n ≥ d ≥ 2 ( r 4 ) ρ d ( n ) = ρ d + 1 ( n + 1 ) for 2 d ≥ n ≥ d ≥ 2 ( r 5 ) ρ d ( n ) ≥ n − d, ρ d ( 2 d + 1 ) ≥ d and ρ d ( 2 d + 2 ) ≥ d + 1 for 2 d ≥ n ≥ d ≥ 2 ( r 6 ) ρ d − 1 ( 2 d − 1 ) = ρ d − 2 ( 2 d − 2 ) = ρ d − 3 ( 2 d − 3 ) and 0 ≤ ρ d ( 2 d ) − ρ d − 1 ( 2 d − 1 ) ≤ 1 for d ≥ 5 On Singularities Of Extremal Periodic Strings CanaDAM 2013

Motivation and background Basic properties of σ d ( n ) Basic properties of ρ d ( n ) Computational Substantiation Thanks Proposition Let ( d , 2 d ) be the first singularity on the main diagonal, i.e. the least d such that ρ d ( 2 d ) − ρ d − 1 ( 2 d − 2 ) ≥ 2 . Then any run-maximal ( d , 2 d ) -string does not contain a symbol occurring exactly 2 , 3 , . . . , 7 or 8 times, and must contains at least ⌈ 7 d 8 ⌉ singletons. Conjecture For any n ≥ d ≥ 2 , ρ d ( n ) ≤ n − d On Singularities Of Extremal Periodic Strings CanaDAM 2013

Motivation and background Basic properties of σ d ( n ) Basic properties of ρ d ( n ) Computational Substantiation Thanks Theorem ( e 1 ) no ( d , 2 d ) singularity ⇐ ⇒ { ρ d ( n ) ≤ n − d for n ≥ d ≥ 2 } ( e 2 ) { ρ d ( n ) ≤ n − d for n ≥ d ≥ 2 } ⇐ ⇒ { ρ d ( 9 d ) ≤ 8 d for d ≥ 2 } ( e 3 ) { ρ d ( n ) ≤ n − d for n ≥ d ≥ 2 } ⇐ ⇒ { ρ d ( 2 d + 1 ) − ρ d ( 2 d ) ≤ 1 for d ≥ 2 } ( e 4 ) no ( d , 2 d + 1 ) singularity = ⇒ { no ( d , 2 d ) singularity and ρ d ( n ) ≤ n − d − 1 for n > 2 d ≥ 4 } ( e 5 ) { ρ d ( 2 d ) = ρ d ( 2 d + 1 ) for d ≥ 2 } = ⇒ { no ( d , 2 d ) singularity and ρ d ( n ) ≤ n − d − 1 for n > 2 d ≥ 4 } ( e 6 ) { ρ d ( 2 d ) = ρ d ( 2 d + 1 ) for d ≥ 2 } = ⇒ { square-maximal ( d , 2 d ) -strings are, up to relabelling, unique and equal to a 1 a 1 a 2 a 2 a 2 . . . a d a d } On Singularities Of Extremal Periodic Strings CanaDAM 2013

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Motivation and background Basic properties of σ d ( n ) Basic properties of ρ d ( n ) Computational Substantiation Thanks d = 2, n = 10 Covered: 154 not Covered: 357 ———————————————————————————– d = 2, n = 15 Covered: 4074 not Covered: 12,309 ———————————————————————————– d = 2, n = 20 Covered: 109,437 not Covered: 414,850 ———————————————————————————– d = 3, n = 10 Covered: 183 not Covered: 9,147 ———————————————————————————– d = 3, n = 15 Covered: 21,681 not Covered: 2,353,420 ———————————————————————————– d = 3, n = 20 Covered: 1,908,923 not Covered: 578,697,523 On Singularities Of Extremal Periodic Strings CanaDAM 2013