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On Morphisms Generating Run-Rich Strings

On Morphisms Generating Run-Rich Strings. Kazuhiko Kusano, Kazuyuki Narisawa and Ayumi Shinohara GSIS, Tohoku University, Japan. PSC2013. On Morphisms Generating Run-Rich Strings. Kazuhiko Kusano, Kazuyuki Narisawa and Ayumi Shinohara GSIS, Tohoku University, Japan. April 2013~.

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On Morphisms Generating Run-Rich Strings

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  1. On Morphisms GeneratingRun-Rich Strings Kazuhiko Kusano, Kazuyuki Narisawa and Ayumi Shinohara GSIS, Tohoku University, Japan PSC2013

  2. On Morphisms GeneratingRun-Rich Strings Kazuhiko Kusano, Kazuyuki Narisawa and Ayumi Shinohara GSIS, Tohoku University, Japan April 2013~ PSC2013

  3. In Short, We found good morphismsthat generate Run-Rich Strings Know Best L.B. [Simpson2010] Know Best L.B. [Crochemore+2011] : Sum of Exponents of Runs in : Number of Runs in

  4. In Short, We found good morphismsthat generate Run-Rich Strings Know Best L.B. [Simpson2010] Know Best L.B. [Crochemore+2011] : Sum of Exponents of Runs in : Number of Runs in

  5. In Short, We found good morphismsthat generate Run-Rich Strings Know Best L.B. [Simpson2010] Know Best L.B. [Crochemore+2011] : Sum of Exponents of Runs in : Number of Runs in

  6. In Short, We found good morphismsthat generate Run-Rich Strings Know Best L.B. [Simpson2010] Know Best L.B. [Crochemore+2011] : Sum of Exponents of Runs in : Number of Runs in

  7. In Short, We found good morphismsthat generate Run-Rich Strings Know Best L.B. [Simpson2010] Know Best L.B. [Crochemore+2011] : Sum of Exponents of Runs in : Number of Runs in

  8. 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 : 1exponent : 2 period : 1exponent : 3 period : 3exponent : 2 period : 2exponent : 2.5 for any string

  9. Maximum Number of Runs(Maximum Sum of Exponents of Runs) in a String of Length abaababaaab aabaabbaabb run-maximal string aaaaaaaaaaa run-maximal string SoE-maximal string aababaababb for any integer

  10. Run-Rich Strings a All run-maximal and SOE-maximal strings ≤ 27 b run-maximal SOE-maximal

  11. Run-Rich Strings a All run-maximal and SOE-maximal strings ≤ 27 b run-maximal SOE-maximal

  12. Exact Values of for ρ(11) = 7 = run(aabaabbaabb) ρ(11) / 11 = 7 / 11 = 0.63636...

  13. Basic Facts & Conjectures on ρ(42) = ρ(41) + 2 ρ(14) = ρ(13) + 2 (“The Run Conjecture”) : Max #runs in binary strings

  14. Basic Facts & Conjectures on ρ(42) = ρ(41) + 2 ρ(14) = ρ(13) + 2 (“The Run Conjecture”) (“The Run Conjecture”) : Max #runs in binary strings

  15. Basic Facts & Conjectures on (“The Run Conjecture”) (“The Run Conjecture”) (“The Run Conjecture”)

  16. Basic Facts & Conjectures on (“The Run Conjecture”) (“The Run Conjecture”)

  17. 1.029 Basic Facts & Conjectures on omitted deep history in this talk The best upper-bound 1.029 [Crochemore+2011] (“The Run Conjecture”)

  18. 1.029 Basic Facts & Conjectures on omitted deep history in this talk The best upper-bound 1.029 [Crochemore+2011] (“The Run Conjecture”) • The known lower-bounds for • 0.9445757 [Simpson 2010, (Matsubara+2009)] • 0.9445756 [Matsubara+2009] • 0.9445648 [Matsubara+2008] • 0.9270509 [Franek+2003]

  19. History of Lower Bounds 0.9270509

  20. History of Lower Bounds • Found a string with 0.9270509

  21. History of Lower Bounds • Found a string with 0.9445757 0.9445757 0.9270509 NTT Software April 2012~ • t0=1001010010110100101 • t1=1001010010110 • t2= 100101001011010010100101 tk = tk-1tk-2(k mod 3 = 0, k>2) tk = tk-1tk-4 (k mod 3 ≠ 0, k>2)

  22. History of Lower Bounds • Found a string with 0.9445757 0.9270509 • The Modified Padovan Words • f (a) = aacab • f (b) = acab • f (c) = ac • h(a) = 101001011001010010110100 • h(b) = 1010010110100 • h(c) = 10100101 • t0=1001010010110100101 • t1=1001010010110 • t2= 100101001011010010100101 tk = tk-1tk-2(k mod 3 = 0, k>2) tk = tk-1tk-4 (k mod 3 ≠ 0, k>2) pk=R(f(pk-5)) for k>5

  23. History of Lower Bounds • Found a string with 0.9445757 Exactly Identical !!! • The Modified Padovan Words • f (a) = aacab • f (b) = acab • f (c) = ac • h(a) = 101001011001010010110100 • h(b) = 1010010110100 • h(c) = 10100101 • t0=1001010010110100101 • t1=1001010010110 • t2= 100101001011010010100101 tk = tk-1tk-2(k mod 3 = 0, k>2) tk = tk-1tk-4 (k mod 3 ≠ 0, k>2) pk=R(f(pk-5)) for k>5

  24. History of Lower Bounds • Found a string with 0.9445757 • The Modified Padovan Words • f (a) = aacab • f (b) = acab • f (c) = ac • h(a) = 101001011001010010110100 • h(b) = 1010010110100 • h(c) = 10100101 • t0=1001010010110100101 • t1=1001010010110 • t2= 100101001011010010100101 tk = tk-1tk-2(k mod 3 = 0, k>2) tk = tk-1tk-4 (k mod 3 ≠ 0, k>2) pk=R(f(pk-5)) for k>5

  25. We found yet another good morphisms • h() = 101001011001010010110100 • h() = 1010010110100 • h() = 10100101h() for New 0.9445757 • The Modified Padovan Words • f (a) = aacab • f (b) = acab • f (c) = ac • h(a) = 101001011001010010110100 • h(b) = 1010010110100 • h(c) = 10100101 h() • t0=1001010010110100101 • t1=1001010010110 • t2= 100101001011010010100101 tk = tk-1tk-2(k mod 3 = 0, k>2) tk = tk-1tk-4 (k mod 3 ≠ 0, k>2) pk=R(f(pk-5)) for k>5

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

  27. Run-Rich Strings a All run-maximal and SOE-maximal strings ≤ 27 b run-maximal SOE-maximal

  28. 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]

  29. 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

  30. New Lower Bounds for New • 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.

  31. Summary We found good morphismsthat generate Run-Rich Strings Know Best L.B. [Simpson2010] Know Best L.B. [Crochemore+2011] : Sum of Exponents of Runs in : Number of Runs in

  32. Future Work • Can we get better lower boundsby 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] )

  33. Thank you.

  34. In Short, We found good morphismsthat generate Run-Rich Strings Know Best L.B. [Simpson2010] Know Best L.B. [Crochemore+2011] : Sum of Exponents of Runs in : Number of Runs in

  35. In Short, We found good morphismsthat generate Run-Rich Strings Know Best L.B. [Simpson2010] Know Best L.B. [Crochemore+2011] : Sum of Exponents of Runs in : Number of Runs in

  36. In Short, We found good morphismsthat generate Run-Rich Strings Know Best L.B. [Simpson2010] Know Best L.B. [Crochemore+2011] : Sum of Exponents of Runs in : Number of Runs in

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