Abelian Repetitions in Sturmian Words

Transcription

1 Aelin Repetitions in Sturmin Words Griele Fici 1 Alessio Lngiu 2 Thierry Lecroq 3 Arnud Lefevre 3 Filippo Mignosi 4 Élise Prieur-Gston 3 1 Università di Plermo, Itly 2 King s College London, UK 3 Normndie Université, Université de Rouen, LITIS EA 4108, Frnce 4 Università dell Aquil, Itly SeqBio 2013 Novemer 25th-26th 2013 Montpellier, Frnce TL (LITIS) Aelin Periods SeqBio / 27

2 2nd Interntionl Conference on Algorithms for Big Dt Sl Gill (Yellow Room) of Plzzo dei Normnni Pizz del Prlmento Plermo, Itly, April 2014 TL (LITIS) Aelin Periods SeqBio / 27

3 Outline 1 Introduction 2 Sturmin words nd elin repetitions TL (LITIS) Aelin Periods SeqBio / 27

4 Outline 1 Introduction 2 Sturmin words nd elin repetitions TL (LITIS) Aelin Periods SeqBio / 27

5 Nottion nd definitions Given word w = w[1.. n] of length n over lphet Σ = { 1,..., σ } of crdinlity σ we denote y: w[i] its i-th symol w[i.. j] the fctor from the i-th to the j-th symols w the numer of occurrences of symol in w Pw = ( w 1,..., w σ ) its Prikh vector TL (LITIS) Aelin Periods SeqBio / 27

6 Nottion nd definitions Given word w = w[1.. n] of length n over lphet Σ = { 1,..., σ } of crdinlity σ we denote y: w[i] its i-th symol w[i.. j] the fctor from the i-th to the j-th symols w the numer of occurrences of symol in w Pw = ( w 1,..., w σ ) its Prikh vector TL (LITIS) Aelin Periods SeqBio / 27

7 Nottion nd definitions Given word w = w[1.. n] of length n over lphet Σ = { 1,..., σ } of crdinlity σ we denote y: w[i] its i-th symol w[i.. j] the fctor from the i-th to the j-th symols w the numer of occurrences of symol in w Pw = ( w 1,..., w σ ) its Prikh vector TL (LITIS) Aelin Periods SeqBio / 27

8 Nottion nd definitions Given word w = w[1.. n] of length n over lphet Σ = { 1,..., σ } of crdinlity σ we denote y: w[i] its i-th symol w[i.. j] the fctor from the i-th to the j-th symols w the numer of occurrences of symol in w Pw = ( w 1,..., w σ ) its Prikh vector TL (LITIS) Aelin Periods SeqBio / 27

9 Remrks on Prikh vectors Consider Pw = ( w 1,..., w σ ) then Pw[i] = w i Pw = σ i=1 P w[i] = w Pw Q iff Pw[i] Q[i] for every 1 i σ nd Pw < Q TL (LITIS) Aelin Periods SeqBio / 27

10 Remrks on Prikh vectors Consider Pw = ( w 1,..., w σ ) then Pw[i] = w i Pw = σ i=1 P w[i] = w Pw Q iff Pw[i] Q[i] for every 1 i σ nd Pw < Q TL (LITIS) Aelin Periods SeqBio / 27

11 Remrks on Prikh vectors Consider Pw = ( w 1,..., w σ ) then Pw[i] = w i Pw = σ i=1 P w[i] = w Pw Q iff Pw[i] Q[i] for every 1 i σ nd Pw < Q TL (LITIS) Aelin Periods SeqBio / 27

12 Remrks on Prikh vectors Consider Pw = ( w 1,..., w σ ) then Pw[i] = w i Pw = σ i=1 P w[i] = w Pw Q iff Pw[i] Q[i] for every 1 i σ nd Pw < Q Exmple P term TL (LITIS) Aelin Periods SeqBio / 27

13 Remrks on Prikh vectors Consider Pw = ( w 1,..., w σ ) then Pw[i] = w i Pw = σ i=1 P w[i] = w Pw Q iff Pw[i] Q[i] for every 1 i σ nd Pw < Q Exmple P term P remote TL (LITIS) Aelin Periods SeqBio / 27

14 Remrks on Prikh vectors Consider Pw = ( w 1,..., w σ ) then Pw[i] = w i Pw = σ i=1 P w[i] = w Pw Q iff Pw[i] Q[i] for every 1 i σ nd Pw < Q Exmple P term P remote P montpellier TL (LITIS) Aelin Periods SeqBio / 27

15 Aelin periods [Constntinescu nd Ilie, 2006] introduced the notion of Aelin period. Definition A word w hs Aelin period (h, p) iff w = u 0 u 1 u k 1 u k such tht: Pu 0 Pu 1 = = Pu k 1 Pu k Pu 0 = h, Pu 1 = p u 0 is clled the hed nd u k is clled the til. P w will denote the set of Aelin periods of w. TL (LITIS) Aelin Periods SeqBio / 27

16 Aelin periods w = TL (LITIS) Aelin Periods SeqBio / 27

17 Aelin periods w = P w = {(0, 6)} TL (LITIS) Aelin Periods SeqBio / 27

18 Aelin periods w = P w = {(0, 6), (0, 10)} TL (LITIS) Aelin Periods SeqBio / 27

19 Aelin periods w = P w = {(0, 6), (0, 10), (0, 12)} TL (LITIS) Aelin Periods SeqBio / 27

20 Aelin periods w = P w = {(0, 6), (0, 10), (0, 12), (0, 24)} TL (LITIS) Aelin Periods SeqBio / 27

21 Aelin periods w = P w = {(0, 6), (0, 10), (0, 12), (0, 24), (1, 9)} TL (LITIS) Aelin Periods SeqBio / 27

22 Aelin periods w = P w = {(0, 6), (0, 10), (0, 12), (0, 24), (1, 9), (1, 11)} TL (LITIS) Aelin Periods SeqBio / 27

23 Aelin periods w = P w = {(0, 6), (0, 10), (0, 12), (0, 24), (1, 9), (1, 11), (2, 8)} TL (LITIS) Aelin Periods SeqBio / 27

24 Aelin periods w = P w = {(0, 6), (0, 10), (0, 12), (0, 24), (1, 9), (1, 11), (2, 8), (3, 9)} TL (LITIS) Aelin Periods SeqBio / 27

25 Aelin periods w = P w = {(0, 6), (0, 10), (0, 12), (0, 24), (1, 9), (1, 11), (2, 8), (3, 9), (4, 7)} TL (LITIS) Aelin Periods SeqBio / 27

26 Aelin periods w = P w = {(0, 6), (0, 10), (0, 12), (0, 24), (1, 9), (1, 11), (2, 8), (3, 9), (4, 7), (5, 7)} TL (LITIS) Aelin Periods SeqBio / 27

27 Aelin periods w = P w = {(0, 6), (0, 10), (0, 12), (0, 24), (1, 9), (1, 11), (2, 8), (3, 9), (4, 7), (5, 7), (5, 9)} TL (LITIS) Aelin Periods SeqBio / 27

28 Aelin periods w = P w = {(0, 6), (0, 10), (0, 12), (0, 24), (1, 9), (1, 11), (2, 8), (3, 9), (4, 7), (5, 7), (5, 9)} Aelin powers (wek Ap) TL (LITIS) Aelin Periods SeqBio / 27

29 Aelin periods Remrk n hs n 2 Aelin periods. TL (LITIS) Aelin Periods SeqBio / 27

30 Motivtions Bioinformtics finding CpG islnds finding clusters of genes proteomics: mss spectrometry Other fields pproximte pttern mtching gmes (letters) TL (LITIS) Aelin Periods SeqBio / 27

31 Sturmin words Definition 1 Infinite words over inry lphet tht hve exctly n + 1 distinct fctors of length n for ech n 0 TL (LITIS) Aelin Periods SeqBio / 27

32 Fioncci words Fioncci numers F 0 = 0, F 1 = 1, F j = F j 1 + F j 2 for j 2 (0, 1, 1, 2, 3, 5, 8, 13, 21, 34,...) Fioncci words f 1 =, f 2 =, f j = f j 1 f j 2 for j 3 (,,,,,,,...) Fioncci words re Sturmin words TL (LITIS) Aelin Periods SeqBio / 27

33 Outline 1 Introduction 2 Sturmin words nd elin repetitions TL (LITIS) Aelin Periods SeqBio / 27

34 Our strting point G. Fici, T. L., A. Lefevre nd É. Prieur-Gston Computing Aelin periods in words In J. Holu nd J. Žďárek editors, Proceedings of the Prgue Stringology Conference 2011 (PSC 2011), Prgue, Tcheque Repulic, Pges , 2011 G. Fici, T. L., A. Lefevre, É. Prieur-Gston nd W. F. Smyth Qusi-Liner Time Computtion of the Aelin Periods of Word In J. Holu nd J. Žďárek editors, Proceedings of the Prgue Stringology Conference 2012 (PSC 2012), Prgue, Tcheque Repulic, Pges , 2012 G. Fici, T. L., A. Lefevre nd É. Prieur-Gston Algorithms for Computing Aelin Periods of Words. Discrete Applied Mthemtics, 2013, to pper TL (LITIS) Aelin Periods SeqBio / 27

35 Our strting point 2 i F i p i F i p i F i p TL (LITIS) Aelin Periods SeqBio / 27

36 Sturmin words Definition 2 Let α nd ρ, α (0, 1) irrtionl. The frctionl prt of r is defined y {r} = r r. Therefore, for α (0, 1), one hs tht { α} = 1 α. The sequence {nα + ρ}, n > 0, defines n infinite word s α,ρ = 1 (α, ρ) 2 (α, ρ) y the rule n (α, ρ) = { if {nα + ρ} [0, { α}), if {nα + ρ} [{ α}, 1). 0 { α} 1 ( n ) For α = φ 1 nd ρ = 0, φ = (1 + 5)/2, f = TL (LITIS) Aelin Periods SeqBio / 27

37 The Sturmin ijection 1 Proposition For ny n, i, with n > 0, if { (i + 1)α} < { iα} then n+i = {nα + ρ} [{ (i + 1)α}, { iα}), wheres if { iα} < { (i + 1)α}) then n+i = {nα + ρ} [0, { iα}) [{ (i + 1)α}, 1). 0 { α} { 2α} 1 ( n+1 ) When α = φ (thus { α} 0.382) for i = 1. If {nα + ρ} [0, { α}) [{ 2α}, 1), then n+1 = ; otherwise n+1 =. TL (LITIS) Aelin Periods SeqBio / 27

38 The Sturmin ijection 2 { 3α} { 6α}{ α} { 4α} { 2α} { 5α} c 0 (α, 6) c 1 (α, 6) c 2 (α, 6)c 3 (α, 6) c 4 (α, 6) α c 5 (α, 6) c 6 (α, 6)c 7 (α, 6) ( n ) ( n+1 ) ( n+2 ) ( n+3 ) ( n+4 ) ( n+5 ) The suintervls of the Sturmin ijection otined for α = φ 1 nd m = 6. Below ech intervl there is the fctor of s α of length 6 ssocited with tht intervl. For ρ = 0 nd n = 1, the prefix of length 6 of the Fioncci word is ssocited with [c 4 (α, 6), c 5 (α, 6)), which is the intervl contining α. TL (LITIS) Aelin Periods SeqBio / 27

39 The Sturmin ijection nd elin repetitions { 3α} { 6α}{ α} { 4α} { 2α} { 5α} c 0 (α, 6) c 1 (α, 6) c 2 (α, 6)c 3 (α, 6) c 4 (α, 6) α c 5 (α, 6) c 6 (α, 6)c 7 (α, 6) ( n ) ( n+1 ) ( n+2 ) ( n+3 ) ( n+4 ) ( n+5 ) All fctors of length m to the right of { mα} hve the sme Prikh vector. All fctors of length m to the left of { mα} hve the sme Prikh vector. The two Prikh vectors re different. TL (LITIS) Aelin Periods SeqBio / 27

40 The Sturmin ijection nd elin repetitions 2 Min Ide All the points in the sequence {nα}, {(n + m)α}, {(n + 2m)α},..., {(n + km)α} re one fter the other in the unitry thorus with step = { mα}, i.e. the distnce etween {(n + im)α} nd {(n + (i + 1)m)α} is { mα} in the unitry thorus. HENCE, if { mα} is smll nd {nα} is close to zero, there is ig numer k such tht ll previous points re ll to the left of { mα}, in the unitry intervl. In turn, y the Sturmin ijection, the fctors of length m strting t letters n, n+m, n+2m,..., n+km hve the sme Prikh vector. We hve n elin power of exponent k (nd conversely). TL (LITIS) Aelin Periods SeqBio / 27

41 The Sturmin ijection nd elin repetitions 3 Min result Theorem Let m e positive integer such tht {mα} < 0.5 (resp. {mα} > 0.5). Then: 1 In s α there is n elin power of period m nd exponent k 2 if nd only if {mα} < 1 k (resp. { mα} < 1 k ). 2 If in s α there is n elin power of period m nd exponent k 2 strting in position i with {iα} {mα} (resp. {iα} {mα}), then {mα} < k+1 (resp. { mα} < k+1 ). Conversely, if {mα} < k+1 (resp. { mα} < 1 k+1 ), then there is n elin power of period m nd exponent k 2 strting in position m. TL (LITIS) Aelin Periods SeqBio / 27

42 Consequences using Numer Theory Theorem Let s α e Sturmin word. For ny integer q > 1, let k q e the mximl exponent of n elin repetition of period q in s α. Then nd the equlity holds if α = φ 1. lim sup k q q 5, TL (LITIS) Aelin Periods SeqBio / 27

43 Other results on Fioncci words Theorem Let j > 1. The longest prefix of the Fioncci infinite word tht is n elin repetition of period F j hs length F j (F j+1 + F j 1 + 1) 2 if j is even or F j (F j+1 + F j 1 ) 2 if j is odd. Corollry Let j > 1 nd k j e the mximl exponent of prefix of the Fioncci word tht is n elin repetition of period F j. Then k j lim = 5. j F j TL (LITIS) Aelin Periods SeqBio / 27

44 Other results on Fioncci words 2 Theorem For j 3, the (smllest) elin period of the word f j is the n-th Fioncci numer F n, where n = j/2 if j = 0, 1, 2 mod 4, or n = 1 + j/2 if j = 3 mod 4. 2, 2, 2, 3, 5, 5, 5, 8, 13, 13, 13, 21, 34, 34, 34, 55, 89, 89, 89 2 is the elin period of, nd of. Not of tht hs elin period 3. Insted 5 is the elin period of nd of TL (LITIS) Aelin Periods SeqBio / 27

45 Open prolems 1 Is it possile to find the exct vlue of lim sup kq words s α with slope α different from φ 1? q for other Sturmin 2 Is it possile to give the exct vlue of this superior limit when α is n lgeric numer of degree 2? TL (LITIS) Aelin Periods SeqBio / 27

46 References G. Fici, A. Lngiu, T. L., A. Lefevre, F. Mignosi, nd É. Prieur-Gston Aelin repetitions in sturmin words In M.-P. Bél nd O. Crton, editors, Proceedings of the 17th Interntionl Conference on Developments in Lnguge Theory (DLT 2013), volume 7907 of Lecture Notes in Computer Science, pges , Mrne-l-Vllée, Frnce, Springer-Verlg, Berlin G. Fici, A. Lngiu, T. L., A. Lefevre, F. Mignosi, nd É. Prieur-Gston Aelin repetitions in sturmin words Report rxiv: v3 TL (LITIS) Aelin Periods SeqBio / 27

47 THANK YOU FOR YOUR ATTENTION! TL (LITIS) Aelin Periods SeqBio / 27