On Morphisms Generating Run-Rich Strings
description
Transcript of On Morphisms Generating Run-Rich Strings
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
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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
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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
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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
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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
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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
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Basic Facts & Conjectures on
(“The Run Conjecture”)
The best upper-bound 1.029 [Crochemore+2011]
1.029
omitted deep history in this talk
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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
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0.500
0.600
0.700
0.800
0.900
0.9270509
History of Lower Bounds
1 10 1000.200
0.300
0.400
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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