Linear-time computation of local Linear-time computation of local periods periods Gregory Kucherov INRIA/LORIA Nancy, France joint work with Roman Kolpakov (Moscow) and Jean-Pierre Duval, Thierry Lecroq, Arnaud Lefebvre (Rouen) Haifa Stringology Workshop, April 3-8 2005
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Linear-time computation of local periods Linear-time computation of local periods Gregory Kucherov INRIA/LORIA Nancy, France joint work with Roman Kolpakov.
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Linear-time computation of local Linear-time computation of local periodsperiods
Gregory KucherovINRIA/LORIA
Nancy, France
joint work with Roman Kolpakov (Moscow) and Jean-Pierre Duval, Thierry Lecroq, Arnaud
Lefebvre (Rouen)
Haifa Stringology Workshop, April 3-8 2005
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Periodicities (repetitions) in stringsPeriodicities (repetitions) in strings
period: the (global) period: minimal period periodicity = word of period Example: square, cube : fractional periodicity periodicities = “runs” of squares (cyclic) root, 8/3 exponent
Some work has been done ...Some work has been done ...
... see R.Kolpakov,G.Kucherov, Periodic structures in words, chapter of the 3rd Lothaire volume Applied Combinatorics on Words, Cambridge University Press, 2005
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Some work has been done ...Some work has been done ...
... see R.Kolpakov,G.Kucherov, Periodic structures in words, chapter of the 3rd Lothaire volume Applied Combinatorics on Words, Cambridge University Press, 2005
different results based on common simple techniques: extension functions and s-factorization
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Rest of this talkRest of this talk
Basics– extension functions– computing periodicities in time– s-factorisation (Lempel-Ziv factorization)– computing periodicities in time
focus on squares crossing the left border of focus on those of them centered inside general idea: compute squares and pick the minimal ones be careful, the number of squares can be super-linear!!
focus on squares crossing the left border of focus on those of them centered inside general idea: compute squares and pick the minimal ones be careful, the number of squares can be super-linear!! compute maximal periodicities in increasing order of periods
focus on squares crossing the left border of focus on those of them centered inside general idea: compute squares and pick the minimal ones be careful, the number of squares can be super-linear!! compute maximal periodicities in increasing order of periods only a linear number of squares need to be tested for
minimality!!
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Sketch of the proofSketch of the proof
assume we are looking at squares of period
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Sketch of the proofSketch of the proof
assume we are looking at squares of period consider largest period for which squares have
been found
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Sketch of the proofSketch of the proof
assume we are looking at squares of period consider largest period for which squares have
been found if , then test all squares of period (at most )
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Sketch of the proofSketch of the proof
assume we are looking at squares of period consider largest period for which squares have
been found if , then test all squares of period (at most ) if , then either , or
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Sketch of the proofSketch of the proof
assume we are looking at squares of period consider largest period for which squares have
been found if , then test all squares of period (at most ) if , then either , or
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Sketch of the proofSketch of the proof
assume we are looking at squares of period consider largest period for which squares have
been found if , then test all squares of period (at most ) if , then either , or
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Sketch of the proofSketch of the proof
assume we are looking at squares of period consider largest period for which squares have
been found if , then test all squares of period (at most ) if , then either , or
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Sketch of the proofSketch of the proof
assume we are looking at squares of period consider largest period for which squares have
been found if , then test all squares of period (at most ) if , then either , or at most squares need to be tested