The maximal number of runs in standard Sturmian words

Transcription

1 The miml numer of runs in stndrd Sturmin words Pwe l Bturo Mrcin Piątkowski Wojciech Rytter Fculty of Mthemtics nd Computer Science Nicolus Copernicus University Institute of Informtics Wrsw University {pturo, mrcin.pitkowski, rytter}@mt.umk.pl Sumitted: Jun 25, 2012; Accepted: Jn 7, 2013; Pulished: Jn 21, 2013 Mthemtics Suject Clssifiction: 05C88 Astrct We investigte some repetition prolems for very specil clss S of strings clled the stndrd Sturmin words, which hve very compct representtions in terms of sequences of integers. Usully the size of this word is eponentil with respect to the size of its integer sequence, hence we re deling with repetition prolems in compressed strings. An eplicit formul is given for the numer ρw) of runs in stndrd word w. We show tht ρw) w 4 5 for ech w S, nd there is n infinite sequence of strictly growing words w k S such tht lim k ρw k ) w k = 4 5. Moreover, we show how to compute the numer of runs in stndrd Sturmin word in liner time with respect to the size of its compressed representtion. 1 Introduction A run miml repetition) is non-etendle with the sme period) periodic segment in string, in which the period repets t lest twice. Runs re importnt in comintorics on words nd mny prcticl pplictions: dt compression, computtionl iology, pttern-mtching nd so on. The structure of repetitions is lmost completely understood The study is cofounded y the Europen Union from resources of the Europen Socil Fund. Project PO KL Informtion technologies: Reserch nd their interdisciplinry pplictions, Agreement UDA- POKL / Supported y the grnt N of the Ntionl Science Center the electronic journl of comintorics 201) 2013), #P13 1

2 for the clss of Fioncci words, see [11], [17], [9]. In this pper we investigte the structure of runs in clss S of stndrd Sturmin words nd give n ect formul nd tight symptotic ound for the numer of miml repetitions. We continue here the work of [6], where it ws shown how to compute the numer of runs for lock-complete Sturmin words not ll stndrd Sturmin words hve this property) in liner time with respect to the size of the whole word. We show lso the lgorithm, which computes the numer of runs in ny stndrd word in liner time with respect to the size of its compressed representtion the directive sequence), hence in logrithmic time with respect to the length of the word. Throughout the pper we use the stndrd notions of comintorics on words. In prticulr, words re finite sequences over finite set Σ of letters, clled the lphet. For word w = w 1 w 2... w n, y w[i] we denote its i-th letter nmely w i ), y w[i..j] the suword w i w i+1... w j, y w its length nd y w the numer of letters occurring in w. The numer i is period of the word w if w[j] = w[i + j] for ll i with i + j w. The miniml period of w will e denoted y periodw). We sy tht word w is periodic if periodw) w. A word w is sid to e primitive if w is not of the form 2 zk, where z is nonempty word nd k 2 is nturl numer. A miml repetition run, in short) in word w is n intervl α = [i..j], such tht w[i..j] = u k v k 2) is nonempty periodic suword of w, where u is of the miniml length nd v is proper prefi possily empty) of u, tht cn not e etended neither w[i 1..j] nor w[i..j + 1] is run with the period u ). A run α cn e properly included s n intervl in nother run β, ut in this cse periodα) < periodβ). The vlue of the run α = [i...j] is the fctor vlα) = w[i...j]. When it mkes no miguity we identify sometimes run with its vlue nd the period of the run α = [i...j] with the suword w[i..periodw)], clled lso the genertor of the repetition. The mening will lwys e cler from the contet. Oserve tht two different runs could correspond to the identicl suwords, if we disregrd their positions. Hence runs re lso clled the miml positioned repetitions. Emple 1. Let w =. There re 5 runs with the period : w[5..6] = 2, w[12..13] = 2, w[19..20] = 2, w[26..27] = 2, w[31..32] = 2, 5 runs with the period : w[1..5] = ) 2, w[6..12] = ) 3, w[13..19] = ) 3, w[20..26] = ) 3, w[27..31] = ) 2, 4 runs with the period : w[3..8] = ) 2, w[10..15] = ) 2, w[17..22] = ) 2, w[24..29] = ) 2, the electronic journl of comintorics 201) 2013), #P13 2

3 4 runs with the period : w[1..10] = ) 2, w[8..17] = ) 2, w[15..24] = ) 2, w[22..33] = ) 2, nd 1 run with the period : w[1..31] = ) 4. Alltogether we hve 19 runs, see Figure 1 for comprison. Figure 1: The structure of miml repetitions for the emple inry word. Denote y ρw) the numer of runs in the word w nd y ρn) the miml numer of runs in the words of length n. The most interesting relted open conjecture is: ρn) < n. In 1999 Kolpkov nd Kucherov see [10]) showed tht the numer ρw) of runs in string w is O w ), ut the ect multiplictive constnt coefficient is still unknown. The est known results relted to the vlue of ρn) re n ρn) n. The upper ound is y [3], [4] nd the lower ound is y [7], [8], [12]. Tle 1 t the end of this pper) shows the miml numer of runs nd the repetition rtio in ll inry words compred to stndrd words to e defined in the net section) for smll n. 2 Stndrd words A directive sequence is n integer sequence: γ = γ 0, γ 1,..., γ n ), where γ 0 0 nd γ i > 0 for i = 1, 2,..., n. The stndrd word corresponding to γ, denoted y n+1 = Swγ), is descried y the recurrences of the form: 1 =, 0 =,..., n = n 1 ) γ n 1 n 2, n+1 = n ) γn n 1. 1) the electronic journl of comintorics 201) 2013), #P13 3

4 The sequence of words { i } n+1 i=0 is clled stndrd sequence. Every word occurring in stndrd sequence is stndrd word, nd every stndrd word occurs in some stndrd sequence. We ssume tht the stndrd word given y the empty directive sequence is the single letter nd Sw0) =. The clss of ll stndrd words is denoted y S. Emple 2. Consider the directive sequence γ = 1, 2, 1, 3, 1). Then Sw1, 2, 1, 3, 1) = 5, where: 1 = ; 0 = ; 1 = 0 ) 1 1 = ; 2 = 1 ) 2 0 = ; 3 = 2 ) 1 1 = ; 4 = 3 ) 3 2 = ; 5 = 4 ) 1 3 =. For γ 0 > 0 we hve stndrd words strting with the letter nd for γ 0 = 0 we hve stndrd words strting with the letter. In fct the word Sw0, γ 1,..., γ n ) cn e otined from Swγ 1,..., γ n ) y switching the letters nd. Oserve tht for even n > 0 the stndrd word n hs the suffi, nd for odd n > 0 it hs the suffi. Moreover, every stndrd word consists either of repeted occurrences of the letter seprted y single occurrences of the letter or repeted occurrences of the letter seprted y single occurrences of the letter. Those letters re clled respectively the repeting letter nd the single letter. If the repeting letter is letter respectively), the word is clled the Sturmin word of the type type respectively), see Definition in [16] for comprison. Without loss of generlity we consider in this pper the stndrd Sturmin words of the type, therefore we ssume tht γ 0 > 0. The words of the type cn e considered similrly nd ll the results hold. For more interesting fcts relted to comintoril structure of stndrd words see for instnce in [1], [2], [13], [14] nd [15]. Remrk 3. Stndrd words re generliztion of Fioncci words, the well known fmily of strings. By definition, Fioncci words re stndrd words given y directive sequences of the form γ = 1, 1,..., 1). 3 Morphic reduction of stndrd words The recurrent definition of stndrd words from the previous section leds to simple chrcteriztion y composition of morphisms. Let γ = γ 0, γ 1,..., γ n ) e directive sequence. We ssocite with γ sequence of morphisms {h i } n i=0, defined s: h i : { γ i for 0 i n. 2) the electronic journl of comintorics 201) 2013), #P13 4

5 Lemm 4. For 0 i n the morphism h i trnsforms stndrd word into nother stndrd word, nd we hve: Swγ n ) = h n ), Swγ i, γ i+1,..., γ n ) = h i Swγi+1, γ i+2,..., γ n ) ). Proof. We prove the lemm y the induction on the length of the directive sequence. Recll tht the stndrd word given y the empty directive sequence is. For γ = 1 we hve, y definition of stndrd words nd the morphism h n, Swγ n ) = γn = h n ). Assume now tht γ = k 2 nd for directive sequences shorter thn k the sttement holds. Then we hve: Swγ i,..., γ n ) = [ Swγ i,..., γ n 1 ) ] γ n Swγi,..., γ n 2 ) ind. = nd this concludes the proof. )] γn ) [h i Swγ i+1,..., γ n 1 ) hi Swγ i+1,..., γ n 2 ) [Swγi+1 = h i,..., γ n 1 ) ] ) γ n Swγi+1,..., γ n 2 ) ) = h i Swγ i+1,..., γ n ), Remrk 5. As direct corollry to Lemm 4 we hve tht for the directive sequence γ = γ 0, γ 1,..., γ n ): Swγ 0, γ 1,..., γ n ) = h 0 h 1... h n ). Emple 6. Consider the directive sequence γ = 1, 2, 1, 3, 1). We hve compre with Emple 2): Sw1) = h 4 ) ) Sw3, 1) = h 3 Sw1) Sw1, 3, 1) = h 2 Sw3, 1) ) Sw2, 1, 3, 1) = h 1 Sw1, 3, 1) ) = = = = Sw1, 2, 1, 3, 1) = h 0 Sw2, 1, 3, 1) ) =. the electronic journl of comintorics 201) 2013), #P13 5

6 Reduction sequence Oserve tht the inverse morphism h 1 i cn e seen s reduction of the word w i = Swγ i,..., γ n ) to w i+1 = Swγ i+1,..., γ n ). We utilise this pproch to reduce the computtion of runs in w i to the sme computtion in w i+1. Our concept is similr to the one shown in [6], ut is more closely relted to the comintoril structure of stndrd words. Recll tht w denotes the numer of occurrences of the letters in the word w. In the rest of this pper we use the following nottion: N γ k) = Swγk, γ k+1,..., γ n ), M γ k) = Swγk, γ k+1,..., γ n ), 3) which enles us to simplify the formuls for the numer of runs. Remrk 7. The ove definition, together with equtions 1) nd 2), implies: N γ k) = γ k N γ k + 1) + N γ k + 2), M γ k) = N γ k + 1). 4) Remrk 8. Oserve tht for Fioncci word F n the numer of the letters in F n equls the length of the word F n 1, nd therefore N γ k) = F n k 1 nd M γ k) = F n k 2. Emple 9. Let γ = 1, 2, 1, 3, 1) e directive sequence. Then we hve: γ Swγ) Swγ) Swγ) 1) N γ 4) = 1 M γ 4) = 1 3, 1) N γ 3) = 4 M γ 3) = 1 1, 3, 1) N γ 2) = 5 M γ 2) = 4 2, 1, 3, 1) N γ 1) = 14 M γ 1) = 5 1, 2, 1, 3, 1) N γ 0) = 19 M γ 0) = 14 The following lemm enles to epress the length of ny stndrd word in terms of the numers N γ k) nd M γ k). Lemm 10. Let w = Swγ 0, γ 1,..., γ n ), A = N γ 2) nd B = N γ 3). Then w = γ 0 + 1) γ ) A + γ 0 + 1) B. the electronic journl of comintorics 201) 2013), #P13 6

7 Proof. By the definition of N γ k) nd M γ k) we hve w = N γ 0) + M γ 0) nd M γ 0) = N γ 1). By repeted ppliction of the formuls from Eqution 3) we otin: w = N γ 0) + N γ 1) = γ 0 + 1) N γ 1) + N γ 2) = γ 0 + 1) γ 1 + 1) N γ 2) + γ 0 + 1) N γ 3) = γ 0 + 1) γ 1 + 1) A + γ 0 + 1) B nd the proof is complete. 4 The formul nd the lgorithm In this section we present the formul for the numer of runs. The proof of its correctness is presented lter in Section 6. We egin with the definition of some useful zero-one functions for testing the prity of nonnegtive integer i: eveni) = { 1 for even i 0 for odd i nd oddi) = { 1 for odd i 0 for even i, nd for testing if positive integer i equls 1: unryi) = { 1 for i = 1 0 for i > 1. Theorem 11 Formul for the numer of runs in stndrd words). Let γ = γ 0,..., γ n ) e directive sequence nd n 3. The numer of runs in stndrd word w = Swγ 0,..., γ n ) is given y the following formul: ρw) = 2A + 2B + γ) 1 for γ 0 = γ 1 = 1 γ 1 + 2)A + B + γ) oddn) for γ 0 = 1; γ 1 > 1 2A + 3B + γ) evenn) for γ 0 > 1; γ 1 = 1 2γ 1 + 1)A + 2B + γ) for γ 0 > 1; γ 1 > 1, 5) where: A = N γ 2), B = N γ 3) nd γ) = n 1 γ γ n ) unryγ n ). the electronic journl of comintorics 201) 2013), #P13 7

8 The detiled proof of the ove theorem is shown in Section 6. The formul presented in Theorem 11 leds to simple nd fst lgorithm for computtion of the numer of runs in stndrd words. Theorem 12. We cn count the numer of runs in ny stndrd word Swγ 0,..., γ n ) in liner time with respect to the length of the directive sequence γ. Proof. The formul for the numer of runs in stndrd words from Theorem 11 depends directly on the components of the directive sequence γ = γ 0,..., γ n ) nd the numers N γ 2) nd N γ 3). It is sufficient to prove tht we cn compute the numers N γ k) for k = 1, 2, 3 in time On). For this purpose we cn iterte Eqution 1): Algorithm 1: Compute N γ k) Input: γ = γ 0,..., γ n ) Output: N γ k) num := 1 num := 0 for i := n downto k do tmp := num num := γ i num + num num := tmp 7 return num The correctness of the lgorithm follows from the following invrint which holds fter ech itertion of the min for-loop: num equls N γ i) nmely the numer of letters in Swγ i,..., γ n )); num equls M γ i) = N γ i + 1) nmely the numer of letters in Swγ i,..., γ n )). We strt the lgorithm with the empty directive sequence descriing the word Sw ) =, hence the initil vlues of the vriles re: num = 1 nd num = 0. In ech itertion of the for-loop vlues of N γ i) nd M γ i) re computed from N γ i + 1) nd M γ i + 1) y direct ppliction of Eqution 4). Therefore, the loop computes consecutively vlues: Nγ n), M γ n) ), N γ n 1), M γ n 1) ),..., N γ k), M γ k) ). The lgorithm performs n k + 1 itertions of the for loop to compute N γ k), hence its time compleity is On), where n denotes the length of the directive sequence γ. Using Algorithm 1 nd pplying the formul from Eqution 5) we cn count the numer of runs in ny stndrd word in liner time with respect to the size of the directive sequence logrithmic with respect to the length of the whole word). the electronic journl of comintorics 201) 2013), #P13 8

9 Now we cn use the formul from Eqution 5) to compute the numer of runs in some emple stndrd words. Emple 13. Let γ = 1, 2, 1, 3, 1) e directive sequence. We hve n = 4 nd In this cse Swγ) =. A = N γ 2) = 5, B = N γ 3) = 4, = 4 1) 7 1 = 5, odd4) = 0. Theorem 11 implies: ρw) = γ 1 + 2) A + B + odd4) = 4 A + B 5 = 19, see Figure 1 nd Emple 1 for comprison. It is known tht the numer of runs in the n-th Fioncci word F n is given y the formul ρf n ) = 2 F n 2 + 3, see [11] for the proof. As the net emple we derive this formul using Theorem 11. Emple 14. Recll tht F n = Sw1, 1,... 1) n ones) nd in this cse N γ k) = F n k 1. According to the formul from Theorem 11 we hve: ρf n ) = 2 N γ 2) + 2 N γ 3) + n 1 n 1 1 = 2 F n F n 4 3 = 2 F n Asympthotic ehviour of the numer of runs The following lemm gives the ound for the numer of runs in stndrd words descried y the directive sequences of the length t most 2. Lemm 15 Estimtion for short γ). Let γ = γ 0,..., γ n ) e directive sequence, w = Swγ) nd n 2. Then ρw) < 4 5 w. Proof. Recll tht the stndrd word given y the empty directive sequence is nd does not include ny repetition. Therefore, we hve to consider two cses: γ = 1 nd γ = 2. the electronic journl of comintorics 201) 2013), #P13 9

10 Cse 1: First ssume tht γ = 1. Then, γ = γ 0 ) nd w = Swγ 0 ) = γ 0 nd w = γ There is one run for γ 0 > 1, no run for γ 0 = 1 nd oviously ρw) < 4 w. 5 Cse 2: Assume now tht γ = 2. We hve γ = γ 0, γ 1 ) nd w = Swγ 0, γ 1 ) = γ 0 ) γ 1 nd w = γ 0 + 1)γ The numer of runs in w depends on the vlues of γ 0 nd γ 1 s follows: 0 for γ 0 = 1, γ 1 = 1 1 for γ 0 > 1, γ 1 = 1 ρw) =. 1 for γ 0 = 1, γ 1 > 1 γ for γ 0 > 1, γ 1 > 1 In ech cse we hve ρw) < 4 5 ) γ 0 + 1)γ = 4 5 w nd the proof is complete. Now we re redy to estimte the symptotic ound for the numer of runs in ll stndrd Sturmin words. Theorem 16 Upper ound). For ech stndrd word w we hve ρw) 4 5 w. Proof. Let γ = γ 0,..., γ n ) e directive sequence nd w = Swγ 0,..., γ n ) e stndrd word. Recll the formul 5) from Theorem 11 nd oserve tht γ) 0. The cse of n 2 follows from Lemm 15. It is sufficient to prove the sttement for n 3. We consider four cses depending on the vlues of γ 0 nd γ 1. Cse 1: γ 0 = γ 1 = 1. We hve, due to Lemm 10 nd eqution 5): Hence w = 3A + 2B nd ρw) 2 A + 2 B. ρw) w 2A + 2B 3A + 2B 4 5, due to inequlities A B 1. This completes the proof of this cse. Cse 2: γ 0 = 1; γ 1 > 1. We hve, due to Lemm 10 nd eqution 5): w = 2 γ 1 + 1) A + 2B nd ρw) γ 1 + 2) A + B. the electronic journl of comintorics 201) 2013), #P13 10

11 Consequently: ρw) w ecuse γ 1 2 nd γ γ Cse 3: γ 0 > 1; γ 1 = 1. Due to eqution 5) nd Lemm 10, we hve: γ 1 + 2) A + B 2 γ 1 + 1) A + 2B 4 5, w = ρw) 2A + 3B, ) γ 0 + 2) A + γ 0 + 1) B 4A + 3B, nd consequently: ρw) w 2A + 3B 4A + 3B 3A + 2B 4A + 3B 3 4 < 4 5. Cse 4: γ 0 > 1; γ 1 > 1. In this cse, due to eqution 5) nd Lemm 10, we hve: ρw) 2 γ 1 + 1) A + 2 B, nd consequently w = γ 0 + 1) γ ) A + γ 0 + 1) B, ρw) w 2 γ 1 + 1) A + 2 B γ0 + 1) γ ) A + γ 0 + 1) B 2 γ 1 + 1) A + 2 B 3 γ 1 + 1) A + 3 B 4 5, ecuse This completes the proof of the theorem. 2 γ γ The ove results give n symptotic ound for the numer of runs in stndrd words. Below we construct strictly growing sequence of stndrd words to show tht this estimtion is tight. Theorem 17. For the clss S of stndrd words we hve: { ρw) sup : w S w } = 0.8. the electronic journl of comintorics 201) 2013), #P13 11

12 Proof. Let γ = 1, 2, k, k) nd w k w k = = Sw1, 2, k, k). By definition we hve ) k ) k nd wk = 5k 2 + 2k + 5, nd due to Eqution 5): ρwk ) = 4k 2 k + 3, see Figure 2 for the cse k = 3. Consequently lim k ρw k ) w k = lim k 4k 2 k + 3 5k 2 + 2k + 5 = 0.8, nd this completes the proof. Figure 2: The structure of runs in the stndrd word Sw1, 2, k, k) for k = 3. There re 4k 2 k + 3 = 36 runs. 6 The proof of Theorem 11 the formul) This section is devoted to the proof of Theorem 11. We egin with chrcteriztion of the structure of possile periods of miml repetitions in stndrd words. Recll the recurrent definition given y eqution 1) nd the words i defined there. The following lemm is version of Theorem 1 in [5] using slightly different nottion. Lemm 18 Structurl Lemm). The period of ech miml repetition in the stndrd word Swγ 0, γ 1,..., γ n ) is of the form i ) j i 1, where 0 j < γ i. the electronic journl of comintorics 201) 2013), #P13 12

13 To prove the ove lemm it is sufficient to show tht no fctor of the word Swγ 0,..., γ n ) tht does not stisfy the condition given there could e the genertor of some repetition, see the proof of Theorem 1 in [5] for more detils. The min ide of the proof of Theorem 11 is prtition of the set of ll miml repetitions in the word Swγ 0,..., γ n ) into three seprte ctegories depending on the length of their periods. We sy tht run is: short if the length of its period does not eceed 1, lrge if the length of its period eceeds 2, medium otherwise. Denote y ρ S w), ρ M w) nd ρ L w) the numer of short, medium nd lrge runs in the word w. respectively. Emple 19. Recll the word w = Sw1, 2, 1, 3, 1) from Emple 1 nd the set of its miml repetitions. In this cse we hve: 10 short runs the periods re nd ), 8 medium runs the periods re nd ), 1 lrge run the period is ), see Figure 1 for comprison. Counting short runs We strt with the computtion of the numer of short runs. These re runs with the periods of the form or +. Their numer depends on the vlue of γ 0 nd γ 1. Lemm 20 Short Runs). Let w = Swγ 0,..., γ n ) e stndrd word. The numer of short runs in w is given y the formul: N γ 2) + N γ 3) 1 for γ 0 = 1, γ 1 = 1 2 N γ 2) oddn) for γ 0 = 1, γ 1 > 1 ρ S w) =. N γ 1) + N γ 3) evenn) for γ 0 > 1, γ 1 = 1 N γ 1) + N γ 2) for γ 0 > 1, γ 1 > 1 Proof. Short runs re miml repetitions with periods of the form or k. We estimte the numer of runs with periods of ech type seprtely. the electronic journl of comintorics 201) 2013), #P13 13

14 Cse 1: runs with periods of the form. First ssume tht γ 0 > 0. Every run with the period in Swγ 0,..., γ n ) equls γ 0 or γ 0+1 nd is followed y the single letter. Due to Lemm 4, every such run in Swγ 0,..., γ n ) corresponds to the letter in Swγ 1,..., γ n ). Hence in this cse we hve N γ 1) runs with the period. Assume now tht γ 0 = 1. In this cse the word Swγ 0,..., γ n ) consists of the locks of two types: or nd only the locks of the second type include the runs with the period. Due to Lemm 4 every such run in Swγ 0,..., γ n ) corresponds to the letter followed y the letter in Swγ 1,..., γ n ), hence the numer of such runs equls the numer of locks in Swγ 1,..., γ n ). Recll tht for n even length of the directive sequence γ 1,..., γ n ) n is even) the word Swγ 1,..., γ n ) ends with nd in this cse the numer of runs with the period in in Swγ 1,..., γ n ) equls the numer of the letters in Swγ 1,..., γ n ), which is N γ 2). For n odd length of the directive sequence γ 1,..., γ n ) n is odd) the word Swγ 1,..., γ n ) ends with nd the lst letter does not correspond to run in Swγ 0,..., γ n ). In this cse, the numer of runs with the period in Swγ 0,..., γ n ) is one less thn the numer of the letters in Swγ 1,..., γ n ), which is N γ 2) 1. Finlly, the ove resoning cn e summrized s: { Nγ 2) oddn) for γ 0 = 1 N γ 1) for γ 0 > 1. Cse 2: runs with periods of the form k. Notice tht, due to Eqution 2) nd Lemm 4, the runs with the periods γ 0 nd γ 0+1 in the word Swγ 0,..., γ n ) correspond to the runs with the periods in the word Swγ 1,..., γ n ). Similr rgumenttion s ove shows tht the numer of such runs in the word Swγ 0,..., γ n ) equls: { Nγ 3) evenn) for γ 1 = 1 N γ 2) for γ 1 > 1. Comining the results from the two ove cses we conclude the proof of the lemm. Counting medium runs Recll tht medium runs re miml repetitions with periods 1 ) k 0 for 0 < k < γ 1 nd 2, where i re s in Eqution 1). Oserve tht medium runs pper in stndrd words generted y directive sequences of the length t lest 3. We hve to consider two cses: the directive sequences of the length 3 nd the longer ones. The vlue of γ 0 does not ffect the numer of medium runs, hence to simplify clcultions we ssume in further proofs tht γ 0 = 1. We strt with counting medium runs in stndrd words generted y directive sequences of the length greter thn 3. the electronic journl of comintorics 201) 2013), #P13 14

15 Lemm 21 Medium runs, n 3). Let w = Swγ 0,..., γ n ) e stndrd word nd n 3. The numer of medium runs in w is given y the formul: ρ M w) = N γ 1) N γ 2) γ The sttement of Lemm 21 is corollry to the following stronger clim: Clim 22. Let w = Swγ 0,..., γ n ) e stndrd word. There re: 1) N γ 2) 1 runs with the period 1 ) i 0 for ech 0 < i < γ 1. 2) N γ 3) runs with the period 2. Proof. Point 1) The word Swγ 0,..., γ n ) hs the form: ) α 1 ) α 2... ) αs or ) α 1 ) α 2... ) αs, where 0 < α i γ 1 + 1) nd s = N γ 2), ecuse due to Lemm 4) every fctor ) α i in Swγ 0,..., γ n ) corresponds to the letter in Swγ 2,..., γ n ). For emple, see Figure 3, the word Sw1, 4, 2, 2) hs the form Sw1, 4, 2, 2) = ) 4 ) 4 ) 5 ) 4 ) 5. Ech pir of neighoring fctors: ) α i ) α i+1 produces γ 1 1 runs with the period ) k for ech 0 < k < γ 1. In the word Swγ 0,..., γ n ) we hve N γ 2) 1 such pirs nd therefore N γ 2) 1)γ 1 1) medium runs with the periods 1 ) k 0 for 0 < k < γ Figure 3: The structure of runs with the periods 1 < p < 2 for the word Sw1, 4, 2, 2) nd its decomposition into words 1, 2 nd 0, 1. the electronic journl of comintorics 201) 2013), #P13 15

16 Point 2) The word Swγ 0,..., γ n ) cn e represented s sequence of conctented words 1 nd 2 nd hs the form: ) : α α αs or ) : β β βs 2 1. For emple the word Sw1, 4, 2, 2) hs the decomposition , see Figure 3. First ssume the cse ). Ech run with the period 2 hs the form 2 ) k 1. By the definition of stndrd words, the fctor 1 2 hs 2 s prefi. Therefore, the numer of such runs in Swγ 0,..., γ n ) equls the numer of fctors 1 in the decomposition mentioned ove, which due to Lemm 4) corresponds to the numer of the letters in Swγ 2,..., γ n ), nmely N γ 3). Assume now the cse ). The word Swγ 0,..., γ n ) hs the suffi 1 ut in this cse we hve β s 2. Hence the numer of runs with the period 2 is the sme s in the previous cse. Proof of Lemm 21. Summing up the formuls from the points 1) nd 2) of Clim 22 we otin: ρ M = N γ 2) 1 ) γ 1 1) + N γ 3) = γ 1 N γ 2) + N γ 3) ) N γ 2) γ = N γ 1) N γ 2) γ nd this completes the proof of the lemm. The structure of the medium runs in stndrd words defined y directive sequences of the length 3 is slightly different. Lemm 23 Medium runs, n=2). Let w = Swγ 0, γ 1, γ 2 ) e stndrd word. The numer of medium runs in w is given y the formul: ρ M w) = N γ 1) N γ 2) γ unryγ 2 ) Proof. The proof of the cse γ 2 Lemm 21. > 1 uses the sme rgumenttion s the proof of In the cse γ 2 = 1, the word Swγ 0, γ 1, γ 2 ) hs the decomposition Swγ 0, γ 1, γ 2 ) = γ 0 ) γ 0 = 2 1. There is no run with the period 2, nd we hve to sutrct 1 from the numer of the fctors 1 in this cse. the electronic journl of comintorics 201) 2013), #P13 16

17 The recurrence for lrge runs Recll tht the run is clled lrge if it hs the period of the length greter thn 2, where 2 is s in Eqution 1). We reduce the prolem of counting lrge runs to counting medium runs, using the morphic representtion of stndrd words. Let h e morphism nd let v = k e the word of the length k. The morphism h defines the prtition of the word w = hv) into segments h 1 ), h 2 ),..., h t ). These segments re clled the h-locks. We sy tht fctor of the word w is synchronized with the morphism h in w if nd only if ech occurrence of in w strts t the eginning of some h-lock nd ends t the end of some h-lock. Oserve tht every fctor in w tht is synchronized with h corresponds to some fctor in v, hence the morphism h preserves the structure of the fctors tht re synchronized with it. h 0 Figure 4: The periods of the medium runs 1 0 = nd 2 = do not synchronize with the morphism h 0 in the word Sw1, 2, 1, 3, 1), while the period of the lrge run 3 = is synchronized with h 0 nd corresponds to the medium run with the period 1 0 = in the word Sw2, 1, 3, 1). Emple 24. Let w = Sw1, 2, 1, 3, 1) nd v = Sw2, 1, 3, 1) e stndrd words nd h 0 e the morphism defined s: { h 0 :. Recll tht Sw1, 2, 1, 3, 1) = h 0 Sw2, 1, 3, 1) ), Sw1, 2, 1, 3, 1) =, Sw2, 1, 3, 1) =. The fctors w[6..8] = nd w[13..17] = re not synchronized with the morphism h 0, ecuse oth of them ends in the middle some h 0 -lock. The fctor w[22..28] strts t the eginning of some h 0 -lock nd ends t the end of some h 0 -lock, hence is synchronized with the morphism h 0. Moreover it corresponds to the fctor v[13..16] =, see Figure 4 for comprison. the electronic journl of comintorics 201) 2013), #P13 17

18 Lemm 25 Synchroniztion Lemm). The periods of lrge runs in the word Swγ 0,..., γ n ) re synchronized with the morphism h 0. Proof. Let h 0 e the morphism defined s h 0 : { γ 0. Due to Lemm 4 we hve Swγ 0,..., γ n ) = h 0 Swγ1,..., γ n ) ). Moreover, h 0 determines the prtition of Swγ 0,..., γ n ) into h 0 -locks of the form γ 0 nd, see Figure 4 for the prtition of Sw1, 2, 1, 3, 1). Recll tht the period of ech lrge run in the word Swγ 0,..., γ n ) is of the form i ) k i 1, where 0 k < γ i nd i 2. By the definition of stndrd words the fctor i ) k i 1 strts with γ 0, hence t the eginning of some h 0 -lock. For even i 2, the suword i ) k i 1 ends with 1 = γ 0, hence t the end of some h 0 -lock, nd is oviously synchronized with h 0. For odd i 2 the fctor i ) k i 1 ends with 3 2 = γ γ = γ 2 2 γ 0 ) γ 1+1. First ssume tht i ) k i 1 is followed y the lock γ 0. The single letter t the end of i ) k i 1 is then the whole h 0 -lock nd i ) k i 1 is synchronized with the morphism h 0. Assume now tht i ) k i 1 ends with γ 0 ) γ1+1 nd is followed y γ0 1 ), nmely it ends in the middle of some h 0 -lock. In this cse we hve the occurrence of the fctor γ 0 ) γ1+2 in Swγ 0,..., γ n ), which is reduced y the morphism h 1 0 to the fctor γ1+2 in Swγ 1,..., γ n ). By the definition stndrd words, Swγ 1,..., γ n ) cn include only locks of two types: the short lock γ 1 nd the long lock γ1+1, hence we hve the contrdiction nd the proof is complete. The following lemm, which is direct corollry to Synchroniztion lemm, llows to reduce the prolem of counting the lrge runs in the word Swγ 0,..., γ n ) to those in Swγ 1,..., γ n ). Lemm 26 Recurrence Lemm). Let w = Swγ 0,..., γ n ) nd v = Swγ 1,..., γ n ) e stndrd words. The numer of lrge runs in w is given y the recurrence ρ L w) = ρ L v) + ρ M v). the electronic journl of comintorics 201) 2013), #P13 18

19 Proof. The synchroniztion lemm implies tht the morphism defined s in Eqution 2) preserve the structure of long runs in stndrd words. Recll tht Swγ 0,..., γ n ) is reduced y h 1 0 to Swγ 1,..., γ n ). Therefore, every lrge run α in Swγ 0,..., γ n ) corresponds to some run β in Swγ 1,..., γ n ). Due to Lemm 18, the period of the run α is of the form i ) k 1, where 0 < k γ i nd i 2. The corresponding run β is either lrge for i = 2) or medium for i = 2). Hence, to compute ll lrge runs in Swγ 0,..., γ n ) it is sufficient to compute ll lrge nd medium runs in Swγ 1,..., γ n ). The sttement of the net lemm gives the compct formul for the numer of the medium nd the lrge runs in stndrd words. Lemm 27 Lrge Runs). Let w = Swγ 0,..., γ n ) e stndrd word. We hve ρ L w) + ρ M w) = N γ 1) + n 1 γ γ n ) unryγ n ). Proof. Due to the formuls from Lemm 21 nd Lemm 23 nd the recurrence from Lemm 26 we hve n 2 ρ L w) + ρ M w) = ρ M Swγi,..., γ n ) ) = i=0 ) N γ 1) N γ 2) γ ) N γ n 2) N γ n 1) γ n ) N γ n 1) N γ n) γ n unryγ n ). Tking into ccount tht N γ n) = γ n the ove formul cn e written s ρ L w) + ρ M w) = N γ 1) + n 1) γ γ n ) unryγ n ), nd this concludes the proof. Now we re redy to prove the formul for the numer of runs in stndrd words given y Eqution 5). Proof of Theorem 11. Let w = Swγ 0,..., γ n ) e stndrd word. Recll tht we divide the set of ll runs in w into three disjoint susets of short, medium nd lrge runs, depending on the length of their periods. The numer ρw) of runs in w equls: ρw) = ρ S w) + ρ M w) + ρ L w), where ρ S w), ρ M w) nd ρ L w) denote the numer of short, medium nd lrge runs in w. + the electronic journl of comintorics 201) 2013), #P13 19

20 Due to Lemm 20, the numer of short runs in w is given y the formul: N γ 2) + N γ 3) 1 for γ 0 = 1, γ 1 = 1 2 N γ 2) oddn) for γ 0 = 1, γ 1 > 1 ρ S w) = N γ 1) + N γ 3) evenn) for γ 0 > 1, γ 1 = 1 N γ 1) + N γ 2) for γ 0 > 1, γ 1 > 1. Moreover, due to Lemm 27, the numer of lrge nd medium runs in w is given s: ρ L w) + ρ M w) = N γ 1) + n 1 γ γ n ) unryγ n ). Finlly, comining the ove formuls, we hve: 2A + 2B + γ) 1 for γ 0 = γ 1 = 1 γ 1 + 2)A + B + γ) oddn) for γ 0 = 1; γ 1 > 1 ρw) = 2A + 3B + γ) evenn) for γ 0 > 1; γ 1 = 1 2γ 1 + 1)A + 2B + γ) for γ 0 > 1; γ 1 > 1, where: A = N γ 2) = Swγ 2, γ 3,..., γ n ), B = N γ 3) = Swγ 3, γ 4,..., γ n ), γ) = n 1 γ γ n ) unryγ n ). This completes the proof of the theorem. 7 Finl remrks The im of this pper ws to study prolems relted to miml repetitions for one of the most thoroughly investigted clss of strings in comintorics on words the stndrd Sturmin words. We hve presented formuls for the numers of runs long with the detiled nlysis of their symptotic ehviour. The complete understnding of their comintoril structure for lrge clss of complicted words is step towrds etter understnding of this prolem in generl. The miml repetition rtio 0.8 for stndrd words hs een first discovered y us during computer eperiments with very long strings. Similrly, we were tuning mny intermedite formuls with the ssistnce of the computer. Our lgorithm for computing the numer of runs in stndrd words is n emple of very fst computtion on highly compressed tets in liner time with respect to the size of their compressed representtion. the electronic journl of comintorics 201) 2013), #P13 20

21 n All inry words Stndrd words ρn) ρn)/n Emple word ρn) ρn)/n Emple word , Tle 1: The comprison of the miml numer of runs nd the repetition rtio for the clss of ll inry words nd the clss of stndrd words of given length. References [1] P. Bturo, M. Piątkowski, nd W. Rytter. Usefulness of directed cyclic suword grphs in prolems relted to stndrd Sturmin words. Interntionl Journl of Foundtions of Computer Science, 206): , [2] P. Bturo nd W. Rytter. Compressed string-mtching in stndrd Sturmin words. Theoreticl Computer Science, ): , [3] M. Crochemore nd L. Ilie. Anlysis of miml repetitions in strings. In Proceedings of the 32nd Interntionl Conference on Mthemticl Foundtions of Computer Science, volume 4708 of Lecture Notes in Computer Science, pges Springer, [4] M. Crochemore, L. Ilie, nd L. Tint. Towrds solution of the runs conjecture. In Proceedings of the 19th nnul symposium on Comintoril Pttern Mtching, volume 5029 of Lecture Notes in Computer Science, pges Springer, [5] D. Dmnik nd D. Lenz. Powers in Sturmin sequences. Europen Journl of Comintorics, 244): , [6] F. Frnek, A. Krmn, nd W. F. Smyth. Repetitions in Sturmin strings. Theoreticl Computer Science, 2492): , the electronic journl of comintorics 201) 2013), #P13 21

22 [7] F. Frnek, R. J. Simpson, nd W. F. Smyth. The mimum numer of runs in string. In Proceedings of 14th Austrlin Workshop on Comintioril Algorithms, pges 26 35, [8] F. Frnek nd Q. Yng. An symptotic lower ound for the miml numer of runs in string. Jnterntionl Journl of Foundtions of Computer Science, 191): , [9] C. S. Iliopoulos, D. Moore, nd W. F. Smyth. A chrcteriztion of the squres in Fioncci string. Theoreticl Computer Science, ): , [10] R. Kolpkov nd G. Kucherov. Finding miml repetitions in word in liner time. In Proceedings of the 40th Annul Symposium on Foundtions of Computer Science, pges IEEE Computer Society, [11] R. M. Kolpkov nd G. Kucherov. On miml repetitions in words. In Proceedings of 12th Interntionl Symposium on Fundmentls of Computtion Theory, volume 1684 of Lecture Notes in Computer Science, pges Springer, [12] K. Kusno, W. Mtsur, A. Ishino, H. Bnni, nd A. Shinor. New lower ound for the mimum numer of runs in string. Computing Reserch Repository, s/ , [13] M. Piątkowski nd W. Rytter. Computing the numer of cuic runs in stndrd Sturmin words. In Proceedings of the 16-th Prgue Stringology Conference, pges Czech Technicl University, [14] M. Piątkowski nd W. Rytter. Asymptotic ehviour of the miml numer of squres in stndrd Sturmin words. Interntionl Journl of Foundtions of Computer Science, 232): , [15] M. Piątkowski nd W. Rytter. The numer of cues in Sturmin words. In Proceedings of the 17-th Prgue Stringology Conference, pges Czech Technicl University, [16] N. Pythes Fogg. Sustitutions in Dynmics, Arithmetics nd Comintorics, volume 1794 of Lecture Notes in Mthemtics. Springer, [17] W. Rytter. The structure of suword grphs nd suffi trees of Fioncci words. Theoreticl Computer Science, 3632): , the electronic journl of comintorics 201) 2013), #P13 22