On Morphisms Generating

Run-Rich Strings

Kazuhiko Kusano, Kazuyuki Narisawa and

Ayumi ShinoharaGSIS, Tohoku University, Japan

PSC2013

On Morphisms Generating

Run-Rich Strings

Kazuhiko Kusano, Kazuyuki Narisawa and

Ayumi ShinoharaGSIS, Tohoku University, Japan

PSC2013

April 2013~

We found good morphismsthat generate Run-Rich Strings

𝜙𝑟 (a )=abac𝜙𝑟 ( b )=aac𝜙𝑟 (c )=a

𝑢𝑖=h (𝜙𝑟𝑖 (a ))

𝑣 𝑖=𝜓𝑒(𝜙𝑟𝑖 (a ))

𝜎 (𝑣12 )|𝑣12|

=2.036982

: Number of Runs in : Sum of Exponents of Runs in

Know Best L.B. [Simpson2010]

Know Best L.B. [Crochemore+2011]

In Short,

We found good morphismsthat generate Run-Rich Strings

𝜙𝑟 (a )=abac𝜙𝑟 ( b )=aac𝜙𝑟 (c )=a

𝑢𝑖=h (𝜙𝑟𝑖 (a ))

𝑣 𝑖=𝜓𝑒(𝜙𝑟𝑖 (a ))

𝜎 (𝑣12 )|𝑣12|

=2.036982

: Number of Runs in : Sum of Exponents of Runs in

Know Best L.B. [Simpson2010]

Know Best L.B. [Crochemore+2011]

In Short,

𝜙𝑟 : Γ → Γ∗for Γ={a , b , c }

We found good morphismsthat generate Run-Rich Strings

𝜙𝑟 (a )=abac𝜙𝑟 ( b )=aac𝜙𝑟 (c )=a

𝑢𝑖=h (𝜙𝑟𝑖 (a ))

𝑣 𝑖=𝜓𝑒(𝜙𝑟𝑖 (a ))

𝜎 (𝑣12 )|𝑣12|

=2.036982

: Number of Runs in : Sum of Exponents of Runs in

Know Best L.B. [Simpson2010]

Know Best L.B. [Crochemore+2011]

In Short,

𝜙𝑟 (a )=abac

⋮

𝜙𝑟 : Γ → Γ∗for Γ={a , b , c }

We found good morphismsthat generate Run-Rich Strings

𝜙𝑟 (a )=abac𝜙𝑟 ( b )=aac𝜙𝑟 (c )=a

𝑢𝑖=h (𝜙𝑟𝑖 (a ))

𝑣 𝑖=𝜓𝑒(𝜙𝑟𝑖 (a ))

𝜎 (𝑣12 )|𝑣12|

=2.036982

: Number of Runs in : Sum of Exponents of Runs in

Know Best L.B. [Simpson2010]

Know Best L.B. [Crochemore+2011]

In Short,

We found good morphismsthat generate Run-Rich Strings

𝜙𝑟 (a )=abac𝜙𝑟 ( b )=aac𝜙𝑟 (c )=a

𝑢𝑖=h (𝜙𝑟𝑖 (a ))

𝑣 𝑖=𝜓𝑒(𝜙𝑟𝑖 (a ))

𝜎 (𝑣12 )|𝑣12|

=2.036982

: Number of Runs in : Sum of Exponents of Runs in

Know Best L.B. [Simpson2010]

Know Best L.B. [Crochemore+2011]

In Short,

𝜓𝑒 : Γ → {0 ,1 }∗

Run (maximal repetition) of Periodic substring of which is extendableneither to the left nor to the rightwith the same period.

• : the number of runs in string • : sum of exponents of runs in string

abaababaaab

period : 3exponent : 2

period : 2exponent : 2.5

period : 1exponent : 3

period : 1exponent : 2

𝜌 (𝑤 )=4𝜎 (𝑤 )=9.5

for any string

Maximum Number of Runs(Maximum Sum of Exponents of Runs) in a String of Length

for any integer

abaababaaab

aabaabbaabb

aaaaaaaaaaa

𝜎 (𝑤2 )=2+2+2+2+2+2+2.25𝜎 (𝑤3 )=11

𝜎 (𝑤1 )=4

𝜎 (𝑤2 )=7

𝜎 (𝑤3 )=1

aababaababb 𝜎 (𝑤4 )=7 𝜎 (𝑤2 )=2+2+2+2 .5+2+2+2

𝑛=11

run-maximal string

run-maximal string

SoE-maximal string

Run-Rich StringsAll run-maximal and SOE-maximal strings ≤ 27

𝜌 (𝑤)𝜎 (𝑤)

𝜌 (𝑤)𝜎 (𝑤)

𝜌 (𝑤)𝝈(𝒘 )

run-maximal

SOE-maximal

ab

Run-Rich StringsAll run-maximal and SOE-maximal strings ≤ 27

𝜌 (𝑤)𝜎 (𝑤)

𝜌 (𝑤)𝜎 (𝑤)

𝜌 (𝑤)𝝈(𝒘 )

run-maximal

SOE-maximal

ab

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

1 1 2 2 3 4 5 5 6 7 8 8 10 10 11 12 13 14 15 15 16 17 18 19 20 21

2.0 3.0 4.0 5.0 6.0 8.0 10.0 11.0 12.5 14.5 16.0 17.4 20.3 21.3 23.0 25.0 27.0 29.0 31.0 32.0 33.7 35.7 37.6 39.7 42.2 44.3

Exact Values of for

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

1 1 2 2 3 4 5 5 6 7 8 8 10 10 11 12 13 14 15 15 16 17 18 19 20 21

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 650.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0

10

20

30

40

50

60

𝜌(𝑛)𝑛

𝜌 (𝑛)ρ(11) = 7= run(aabaabbaabb )

ρ(11) / 11 = 7 / 11 = 0.63636...

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

1 1 2 2 3 4 5 5 6 7 8 8 10 10 11 12 13 14 15 15 16 17 18 19 20 21

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 650.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0

10

20

30

40

50

60

𝜌(𝑛)𝑛

𝜌 (𝑛)<𝜌 (𝑛+2)𝜌 (𝑛) ≤𝜌 (𝑛+1)

(“The Run Conjecture”)𝜌 (𝑛+1 ) ≤ 𝜌 (𝑛 )+2

: Max #runs in binary strings

Basic Facts & Conjectures on

ρ(14) = ρ(13) + 2

ρ(42) = ρ(41) + 2

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

1 1 2 2 3 4 5 5 6 7 8 8 10 10 11 12 13 14 15 15 16 17 18 19 20 21

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 650.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0

10

20

30

40

50

60

𝜌(𝑛)𝑛

𝜌 (𝑛)<𝜌 (𝑛+2)𝜌 (𝑛) ≤𝜌 (𝑛+1)

ρ(14) = ρ(13) + 2

ρ(42) = ρ(41) + 2

(“The Run Conjecture”) (“The Run Conjecture”)𝜌 (𝑛+1 ) ≤ 𝜌 (𝑛 )+2

: Max #runs in binary strings

Basic Facts & Conjectures on

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 650.00

0.10

0.20

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0.40

0.50

0.60

0.70

0.80

0.90

1.00

0

10

20

30

40

50

60

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

1 1 2 2 3 4 5 5 6 7 8 8 10 10 11 12 13 14 15 15 16 17 18 19 20 21

𝜌(𝑛)𝑛

(“The Run Conjecture”) (“The Run Conjecture”)

Basic Facts & Conjectures on

(“The Run Conjecture”)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 650.00

0.10

0.20

0.30

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0.50

0.60

0.70

0.80

0.90

1.00

0

10

20

30

40

50

60

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 650.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0

10

20

30

40

50

60

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

1 1 2 2 3 4 5 5 6 7 8 8 10 10 11 12 13 14 15 15 16 17 18 19 20 21

𝜌(𝑛)𝑛

(“The Run Conjecture”)

Basic Facts & Conjectures on

(“The Run Conjecture”)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 650.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0

10

20

30

40

50

60

𝜌(𝑛)𝑛

Basic Facts & Conjectures on

(“The Run Conjecture”)

The best upper-bound 1.029 [Crochemore+2011]

1.029

omitted deep history in this talk

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 650.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0

10

20

30

40

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𝜌(𝑛)𝑛

Basic Facts & Conjectures on

(“The Run Conjecture”)

The best upper-bound 1.029 [Crochemore+2011]

1.029

omitted deep history in this talk

The known lower-bounds for

• 0.9445757 [Simpson 2010, (Matsubara+2009)]• 0.9445756 [Matsubara+2009]• 0.9445648 [Matsubara+2008]• 0.9270509 [Franek+2003]

lim ¿𝑛→ ∞ 𝜌 (𝑛 )𝑛

History of Lower Bounds

1 10 1000.200

0.300

0.400

0.500

0.600

0.700

0.800

0.900

0.9270509

History of Lower Bounds

1 10 1000.200

0.300

0.400

0.500

0.600

0.700

0.800

0.900

Found a string with

0.9270509

History of Lower Bounds

1 10 100 0.200

0.300

0.400

0.500

0.600

0.700

0.800

0.900

t0 = 1001010010110100101t1 = 1001010010110t2 = 100101001011010010100101tk = tk-1 tk-2 (k mod 3 = 0, k>2)tk = tk-1 tk-4 (k mod 3 ≠ 0, k>2)

0.94457570.9445757Found a string with

0.9270509

NTT SoftwareApril 2012~

0.9270509

History of Lower Bounds

1 10 1000.200

0.300

0.400

0.500

0.600

0.700

0.800

0.900

Simpson2010Matsubara+2009Matsubara+2008Franek+2003ρ(n)/n

t0 = 1001010010110100101t1 = 1001010010110t2 = 100101001011010010100101tk = tk-1 tk-2 (k mod 3 = 0, k>2)tk = tk-1 tk-4 (k mod 3 ≠ 0, k>2)

The Modified Padovan Wordsf (a) = aacab f (b) = acabf (c) = ach(a) = 101001011001010010110100h(b) = 1010010110100h(c) = 10100101

pk=R(f(pk-5)) for k>5

0.9445757Found a string with

History of Lower Bounds

1 10 1000.200

0.300

0.400

0.500

0.600

0.700

0.800

0.900

Simpson2010Matsubara+2009Matsubara+2008Franek+2003ρ(n)/n

t0 = 1001010010110100101t1 = 1001010010110t2 = 100101001011010010100101tk = tk-1 tk-2 (k mod 3 = 0, k>2)tk = tk-1 tk-4 (k mod 3 ≠ 0, k>2)

The Modified Padovan Wordsf (a) = aacab f (b) = acabf (c) = ach(a) = 101001011001010010110100h(b) = 1010010110100h(c) = 10100101

pk=R(f(pk-5)) for k>5

0.9445757

Exactly Identical !!!

Found a string with

History of Lower Bounds

1 10 1000.200

0.300

0.400

0.500

0.600

0.700

0.800

0.900

Simpson2010Matsubara+2009Matsubara+2008Franek+2003ρ(n)/n

t0 = 1001010010110100101t1 = 1001010010110t2 = 100101001011010010100101tk = tk-1 tk-2 (k mod 3 = 0, k>2)tk = tk-1 tk-4 (k mod 3 ≠ 0, k>2)

The Modified Padovan Wordsf (a) = aacab f (b) = acabf (c) = ach(a) = 101001011001010010110100h(b) = 1010010110100h(c) = 10100101

pk=R(f(pk-5)) for k>5

0.9445757Found a string with

We found yet another good morphisms

1 10 1000.200

0.300

0.400

0.500

0.600

0.700

0.800

0.900

Simpson2010Matsubara+2009Matsubara+2008Franek+2003ρ(n)/n

t0 = 1001010010110100101t1 = 1001010010110t2 = 100101001011010010100101tk = tk-1 tk-2 (k mod 3 = 0, k>2)tk = tk-1 tk-4 (k mod 3 ≠ 0, k>2)

The Modified Padovan Wordsf (a) = aacab f (b) = acabf (c) = ach(a) = 101001011001010010110100h(b) = 1010010110100h(c) = 10100101 h()

pk=R(f(pk-5)) for k>5

0.9445757h() = 101001011001010010110100h() = 1010010110100h() = 10100101

h() for

New

A New Lower Bound for Maximum Sum of Exponents of Runs in a String

Run-Rich StringsAll run-maximal and SOE-maximal strings ≤ 27

𝜌 (𝑤)𝜎 (𝑤)

𝜌 (𝑤)𝜎 (𝑤)

𝜌 (𝑤)𝝈(𝒘 )

run-maximal

SOE-maximal

ab

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

1 1 2 2 3 4 5 5 6 7 8 8 10 10 11 12 13 14 15 15 16 17 18 19 20 21

2.0 3.0 4.0 5.0 6.0 8.0 10.0 11.0 12.5 14.5 16.0 17.4 20.3 21.3 23.0 25.0 27.0 29.0 31.0 32.0 33.7 35.7 37.6 39.7 42.2 44.3

Maximum Sum of Exponents of Runs in a string• The best

upper bound for is 4.087 [Crochemore+2011]lower bound is 2.035267 [Crochemore+2011]

Maximum Sum of Exponents of Runs in a string• The best

upper bound for is 4.087 [Crochemore+2011]lower bound is 2.035267 [Crochemore+2011]lower bound is 2.036992 [This paper]New

identical for

New Lower Bounds for • 2.035267 : current best [Crochemore+2011]• 2.036982 • 2.036992

New

Note: would be give a slightly better bound, but we have failed to evaluate it.

New

We found good morphismsthat generate Run-Rich Strings

𝜙𝑟 (a )=abac𝜙𝑟 ( b )=aac𝜙𝑟 (c )=a

𝑢𝑖=h (𝜙𝑟𝑖 (a ))

𝑣 𝑖=𝜓𝑒(𝜙𝑟𝑖 (a ))

𝜎 (𝑣12 )|𝑣12|

=2.036982

: Number of Runs in : Sum of Exponents of Runs in

Know Best L.B. [Simpson2010]

Know Best L.B. [Crochemore+2011]

Summary

Future Work• Can we get better lower bounds

by considering more general morphisms ?

e.g.

• Can we get general formulae for and from the definitions of and ?

(cf. for Strumian words [Franek+2000, Baturo+2008, Piątkowski2013] )

Thank you.

We found good morphismsthat generate Run-Rich Strings

𝜙𝑟 (a )=abac𝜙𝑟 ( b )=aac𝜙𝑟 (c )=a

𝑢𝑖=h (𝜙𝑟𝑖 (a ))

𝑣 𝑖=𝜓𝑒(𝜙𝑟𝑖 (a ))

𝜎 (𝑣12 )|𝑣12|

=2.036982

: Number of Runs in : Sum of Exponents of Runs in

Know Best L.B. [Simpson2010]

Know Best L.B. [Crochemore+2011]

In Short,

We found good morphismsthat generate Run-Rich Strings

𝜙𝑟 (a )=abac𝜙𝑟 ( b )=aac𝜙𝑟 (c )=a

𝑢𝑖=h (𝜙𝑟𝑖 (a ))

𝑣 𝑖=𝜓𝑒(𝜙𝑟𝑖 (a ))

𝜎 (𝑣12 )|𝑣12|

=2.036982

: Number of Runs in : Sum of Exponents of Runs in

Know Best L.B. [Simpson2010]

Know Best L.B. [Crochemore+2011]

In Short,

𝜙𝑟 (a )=abac

⋮

We found good morphismsthat generate Run-Rich Strings

𝜙𝑟 (a )=abac𝜙𝑟 ( b )=aac𝜙𝑟 (c )=a

𝑢𝑖=h (𝜙𝑟𝑖 (a ))

𝑣 𝑖=𝜓𝑒(𝜙𝑟𝑖 (a ))

𝜎 (𝑣12 )|𝑣12|

=2.036982

: Number of Runs in : Sum of Exponents of Runs in

Know Best L.B. [Simpson2010]

Know Best L.B. [Crochemore+2011]

In Short,

𝜙𝑟 (a )=abac

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

1 1 2 2 3 4 5 5 6 7 8 8 10 10 11 12 13 14 15 15 16 17 18 19 20 21

2.0 3.0 4.0 5.0 6.0 8.0 10.0 11.0 12.5 14.5 16.0 17.4 20.3 21.3 23.0 25.0 27.0 29.0 31.0 32.0 33.7 35.7 37.6 39.7 42.2 44.3