Computing k-th Lyndon Word and Decoding Lexicographically Minimal de Bruijn Sequence

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1 Computing k-th Lyndon Word nd Decoding Lexicogrphiclly Miniml de Bruijn Sequence Tomsz Kociumk 1, Jkub Rdoszewski 1, nd Wojciech Rytter 1,2 1 Fculty of Mthemtics, Informtics nd Mechnics, University of Wrsw, Wrsw, Polnd [kociumk,jrd,rytter]@mimuw.edu.pl 2 Fculty of Mthemtics nd Computer Science, Copernicus University, Toruń, Polnd Abstrct. Let Σ be finite ordered lphbet. We present polynomiltime lgorithms for computing the k-th in the lexicogrphic order Lyndon word of given length n over Σ nd counting Lyndon words of length n tht re smller thn given word. We lso use the connections between Lyndon words nd miniml de Bruijn sequences (theorem of Fredricksen nd Miorn) to develop the first polynomil time lgorithm for decoding miniml de Bruijn sequence of ny rnk n (it determines the position of n rbitrry word of length n within the de Bruijn sequence). Our tools mostly rely on combintorics on words nd utomt theory. 1 Introduction We consider finite words over finite ordered lphbet Σ. A Lyndon word over Σ is word tht is strictly smller thn ll its nontrivil cyclic rottions. Lyndon words hve number of combintoril properties (see, e.g., [10]) including the fmous Lyndon fctoriztion theorem which sttes tht every word cn be uniquely written s conctention of lexicogrphiclly non incresing sequence of Lyndon words (due to this theorem Lyndon words re lso clled prime words, see [9]). They re lso relted to necklces of n beds in k colors, tht is, equivlence clsses of k-ry n-tuples under rottion [6, 7]. Lyndon words hve number of pplictions in the field of text lgorithms, see e.g. [1 3, 12]. A de Bruijn sequence of rnk n is cyclic sequence of length Σ n in which every possible word of length n ppers s subword exctly once. De Bruijn sequences re present in vriety of contexts, such s digitl fult testing, pseudorndom number genertion, nd modern public-key cryptogrphic schemes; there re numerous lgorithms for generting such sequences nd their generliztions to other combintoril structures hve been investigted, see [5, 9]. Fredricksen nd Miorn [7] hve shown surprising deep connection between de Bruijn sequences nd Lyndon words: the lexicogrphiclly miniml de Bruijn sequence Supported by Polish budget funds for science in s reserch project under the Dimond Grnt progrm.

2 over Σ is conctention, in lexicogrphic order, of ll Lyndon words over Σ whose length is divisor of n. All Lyndon words of length t most n cn be generted in lexicogrphic order by lgorithm of Fredricksen, Kessler nd Miorn (FKM) [6, 7] (nother lgorithm ws developed by Duvl in [4]). The nlysis from [14] shows tht the FKM lgorithm genertes the subsequent Lyndon words in constnt mortized time. We give the first polynomil time lgorithm for generting Lyndon words of rbitrry rnk in lexicogrphic order. We lso generlize the known simple formul for the number of Lyndon words of length n over Σ (see [9, 10]) by showing polynomil time lgorithm tht computes the number of Lyndon words of length n smller thn given word. For severl de Bruijn sequences decoding lgorithms exist which find the position of n rbitrry word of length n in given de Bruijn sequence in polynomil time [11, 15]. Such lgorithms prove useful in certin types of position sensing pplictions of de Bruijn sequences [15]. We obtin the first decoding lgorithm for the lexicogrphiclly miniml de Bruijn sequence by exploiting its connections with Lyndon words. We lso obtin polynomil-time lgorithm for rndom ccess of symbols in this sequence nd in relted sequence defined in [8]. Note tht the FKM lgorithm cn be used to compute the subsequent symbols of the lexicogrphiclly miniml de Bruijn sequence with O(n 2 ) time dely (or even with worst-cse O(1) time dely [13]), however it does it only in order. We denote by L nd L n the set of ll Lyndon words nd ll Lyndon words of length n, respectively, nd define Lynd(w) = {x L w : x w}. Exmple 1. For Σ = {, b} we hve Lynd(bbb) = 8 since we hve the following Lyndon words of length 6 smller thn bbb (note tht bbb itself is not Lyndon word): b, bb, bb, bbb, bbb, bbb, bbbb, bbbb. Let L (n) = d n L d. By db n we denote the lexicogrphiclly first de Bruijn sequence of rnk n over the given lphbet Σ. It is the conctention of ll Lyndon words in L (n) in lexicogrphic order. Exmple 2. For n = 6 nd binry lphbet we hve the following decomposition of db 6 into Lyndon words: Recently vrint of de Bruijn words ws introduced in [8]. Let db n be the conctention in lexicogrphic order of Lyndon words of length n over Σ. Then db n is cyclic sequence contining ll primitive words of length n. Exmple 3. For n = 6 nd binry lphbet we hve the following decomposition of db 6:

3 Our results. We ssume tht Σ fits in single mchine word. We present n O(n 3 )-time lgorithm for computing Lynd(w) for word w of length n. Using binry serch this lgorithm implies n O(n 4 log Σ )-time lgorithm for computing the k-th Lyndon word of length n (in the lexicogrphic order) for given k. Next we show n O(n 3 )-time decoding lgorithm tht finds the position of n rbitrry w Σ n in db n. We lso obtin O(n 4 log Σ )-time lgorithms computing the k-th symbol of db n nd db n for given k. 2 Preliminries Let Σ be finite ordered lphbet. By Σ nd Σ n we denote the set of ll words over Σ nd the set of ll such words of length n. If w is word then w denotes its length, w[i] its i-th letter (for 1 i w ), w[i, j] its fctor w[i]w[i+1]... w[j] nd w (i) its prefix w[1, i]. Additionlly w k is conctention of k copies of w nd w is n infinite word composed of n infinite number of copies of w. By rot(w, c) let us denote cyclic rottion of w obtined by moving (c mod n) first letters of w to its end (preserving the order of the letters). We sy tht the words w nd rot(w, c) re cycliclly equivlent (sometimes clled conjugtes). By w we denote the lexicogrphiclly miniml cyclic rottion of w. We sy tht w is primitive if w = u k for k Z + implies tht u = w, otherwise w is clled non-primitive. We sy tht λ Σ is Lyndon word if it is primitive nd λ = λ. All cyclic rottions of Lyndon word re different primitive words [10]. The technicl proofs of the following lemms re left for the full version. Lemm 4. Let x, y Σ. Assume x = x nd x y. Then x y. Lemm 5. For given word w Σ n we cn compute in O(n 2 ) time the lexicogrphiclly lrgest word w Σ n such tht w = w w. 3 Combintoril Tools Our bsic gol is to compute Lynd(w), tht is, the number of Lyndon words in Σ n not exceeding w (n = w ). It suffices to compute Lynd(w) for words w such tht w = w. We show how to reduce it to the computtion of the crdinlity of the following set: CS(v) = {x Σ v : x v} for some prefixes v of w. Computtion of CS(v) is lso of independent interest, we pply it in the decoding scheme for miniml de Bruijn sequences. Let us introduce the following uxiliry sets: CS l (v) = {x Σ l : x v } CS l(v) = {x Σ l : x is primitive, x v }. Note tht if x = v then x v is simply equivlent to x v. Thus CS(v) = CS v (v). 3

4 Observtion 6. Lynd(w) = 1 n CS n(w) Proof. Observe tht CS n(w) is the set of ll primitive words of length n tht hve cyclic rottion not exceeding w. Ech Lyndon word of length n not exceeding w corresponds to n such words: ll its cyclic rottions. Observtion 7. CS l (w) = d l CS d(w). Proof. For word x of length l there exists exctly one primitive word y such tht y k = x where k Z +. Thus: CS l (w) = { y Σ d : y is primitive, y l/d } w, d l nd the sum is disjoint. Now y l/d = y implies the requested formul. From Observtion 7, using Möbius inversion formul, we obtin: CS l(w) = µ( l d ) CS d(w). d l Observtion 8. Let w Σ n stisfy w = w. Then CS d (w) = CS(w (d) ). Proof. If d = n the equlity is trivil. Assume d < n. Let y Σ d be word. By Lemm 4, w(d) w, so y w(d) implies y w. On the other hnd, if y w, then y w (d), so y w(d). Applying for y = x we conclude tht x w(d) if nd only if x w, which proves the clim. We conclude with simple formul for Lynd(w) tht combines the results of this section. Lemm 9. Let w Σ n stisfy w = w. Then Lynd(w) = 1 n µ( n d ) CS(w(d) ). d n Exmple 10. Let w = bbbb. We hve w (1) =, w (2) = b, w (3) = b nd CS(w (1) ) = {}, CS(w (2) ) = {, b, b}, CS(w (3) ) = {, b, b, b}, CS(w) = 54, Lynd(w) = 1 6 (µ(1) CS(w) + µ(2) CS(w(3) ) + µ(3) CS(w(2) ) The set Lynd(w) contins the following words: + µ(6) CS(w(1) ) ) = 1 6 ( ) = 8. b, bb, bb, bbb, bbb, bbb, bbbb, bbbb. 4 Automt-Theoretic Tools: Computing CS In this section we design n lgorithm computing CS(w) for word w Σ n. Note tht we my ssume tht w = w, since CS(w) = CS(w ) where w Σ n is the lrgest word such tht w = w w. 4

5 Let Pref (w) = {w (i) s : i [0, n 1], s Σ, s < w[i + 1]} {w}. Consider lnguge L(w) contining words tht hve fctor y Pref (w). Equivlently, x L(w) if there exists fctor of x which is smller thn or equl to w, but is not proper prefix of w. For lnguge L Σ let L = {x : x 2 L}. Fct 11. CS(w) = L(w) Σ n Proof. Consider word x Σ n. If x CS(w) then x w. Tke y = x, which is fctor of x 2. Note tht y w, so some prefix of y belongs to Pref (w). This prefix is fctor of x 2, so x 2 L(w). Consequently, x L(w). On the other hnd, ssume x L(w), so x 2 contins fctor y Pref (w). Let us fix the first occurrence of y in x 2. Observe tht y cn be extended to cyclic rottion x of x. Note tht y Pref (w) implies tht x w, hence x x w nd x CS(w). We construct deterministic finite utomton A recognizing L(w). It hs n + 1 sttes: one for ech proper prefix of w, nd n uxiliry ccepting stte AC. The trnsitions re defined s follows: we set δ(ac, c) = AC for ny c Σ nd w (0) if c > w[i + 1], δ(w (i), c) = w (i+1) if c = w[i + 1] nd i n 1, AC otherwise. (A) strt (B) strt b b b b b b b b Fig. 1: Automt for word w = bbb: (A) ccepts ll words contining w s fctor, (B) ccepts ll words in L(w), i.e. contining fctor y Pref (w). Missing links led to the initil stte. The following fct implies tht L(A) = L(w). Fct 12. Let x Σ nd let q be the stte of A fter reding x. If x L(w) then q = AC. Otherwise q corresponds to the longest prefix of w which is suffix of x. 5

6 Proof. The proof goes by induction on x. If x = 0 the sttement is cler. Consider word x of length x 1. Let x = x c where c Σ. If x L(w) then clerly x L(w). By inductive ssumption fter reding x the utomton is in AC, nd A is constructed so tht it stys in AC once it gets there. Thus the conclusion holds in this cse. From now on we ssume tht x / L(w). Let w (i) be the stte of A fter reding x. If c < w[i + 1], clerly x L(w) (y = w (i) c Pref (w)), nd the utomton proceeds to AC s desired. Similrly, it behves correctly if i = n 1 nd c = w[i + 1]. Consequently we my ssume tht c w[i + 1] nd tht w is not suffix of x. Tke ny j such tht w (j) is suffix of x (possibly empty). Note tht then w (j) is border of w (i). Consequently w (j) w[i+1, n]w (i j) is cyclic rottion of w, so w (j) w[i+1, n]w (i j) w = w = w (j) w[j+1, n], hence c w[i+1] w[j+1]. This implies tht w (j) c could be prefix of w only if c = w[i + 1] = w[j + 1]. In prticulr, A indeed shifts to the longest prefix of w being suffix of x. Now we only need to prove tht x / L(w). For proof by contrdiction, choose fctor y of x such tht y Pref (w) nd y is miniml. Note tht y is suffix of x (since x / L(w)). We hve y = w (j) c for some j n 1 nd c < w[j + 1]. As we hve lredy noticed, such word cnnot be suffix of x. We sy tht n utomton with the set of sttes Q is sprse if the underlying directed grph hs O( Q ) edges counting prllel edges s one. Note tht the trnsitions from ny stte q of A led to t most 3 distinct sttes, so A is sprse. The following corollry summrizes the construction of A. Corollry 13. Let w Σ n stisfy w = w. One cn construct sprse utomton A with O(n) sttes recognizing L(w). For deterministic utomton A = (Q, q 0, F, δ) nd q, q Q let us define L A (q, q ) = {x Σ : δ(q, x) = q }. Then L A (q, q ) is recognized by (Q, q, {q }, δ). Note tht L(A) = q F L A(q 0, q) where the sum is disjoint. The proof of the following lemm is bsed on mtrix multipliction, we omit it in this version. Lemm 14. Let A = (Q, q 0, F, δ) be deterministic utomton with n sttes, nd let m Z 0. () In poly(n + m) time, one cn compute L A (q, q ) Σ k for ll q, q Q nd k m. (b) If A is sprse, it tkes O(m 2 n) time to compute ll vlues L A (q, q ) Σ k, k m, for fixed stte q or q. Observtion 15. Let A = (Q, q 0, F, δ) be deterministic utomton. Then L(A) = L A (q 0, q) L A (q, q ) nd the sum is disjoint. q Q,q F Proof. Consider word x Σ. Note tht x L A (q 0, q) L A (q, q ) for q = δ(q 0, x) nd q = δ(q, x) nd no other pir of sttes q, q. Clerly x L(A) if nd only if q F. 6

7 Corollry 16. If w Σ n stisfies w = w, then one cn compute CS(w) in poly(n) time. Proof. We construct the utomton A with L(A) = L(w) s in Corollry 13. By Fct 11, it suffices to compute L(A) Σ n. Observtion 15 reduces this to computing L A (w (0), q) L A (q, AC) Σ n for ll sttes q of A. One cn construct n utomton with O(n 2 ) sttes representing pirs of sttes of A tht recognizes L A (w (0), q) L A (q, AC). Now it is enough to pply Lemm 14() to determine L A (w (0), q) L A (q, AC) Σ n. Corollry 16 lredy gives polynomil-time lgorithm to compute CS(w), however, the pproch it tkes is rther inefficient. Below, we give n lgorithm working with more resonble O(n 3 ) bound for the running time, which exploits the structure of both the utomton A nd the lnguge L(w). Lemm 17. If w Σ n stisfies w = w, then one cn compute CS(w) in O(n 3 ) time. Proof. As before we pply Fct 11 with Corollry 13 nd ctully compute {x Σ n : x 2 L(A)}. If x L(A), then obviously x 2 L(A). Moreover, Lemm 14(b) lets us compute Σ n L(A) in O(n 3 ) time. Thus, it suffices to count x Σ n such tht x 2 L(A) but x / L(A). This is done bsed on the following clim, see Fig. 2. Clim. Assume x = n nd x 2 L(A) but x / L(A). Then there is unique decomposition x = x 1 x 2 x 3 such tht x 1, x 3 ε, x 3 x 1 Pref (w) nd x 1 x 2 L A (w (0), w (0) ). Proof (of the clim). Let v (for v Σ, Σ) be the shortest prefix of x 2 which belongs to L(A). Let w (i) = δ(w (0), v) be the stte of A fter reding v. Also, let u be the prefix of v of length v i. The structure of the utomton implies tht δ(w (0), u) = w (0), ctully u is the longest prefix of x 2 which belongs to L A (w (0), w (0) ). Note tht v = uw (i) nd w (i) Pref (w), so x / L(A) implies u < n v. We set the decomposition so tht x 1 x 2 = u nd x 3 x 1 = w (i). Uniqueness follows from deterministic behviour of the utomton. u v x x x 1 x 2 x 3 x 1 x 2 x 3 w (0) w (j) w (0) w (i) AC AC Fig. 2: Illustrtion of the clim. Both lines represent different fctoriztions of the sme word x 2. Blck circles represent sttes of the utomton. Only shded letters re not necessrily uniquely determined by x 3 nd x 1 for fixed w. The lgorithm considers ll n 2 choices of x 1 nd x 3, nd counts the number of x s conditioned on these vlues. Let x 1 = x 1 where x 1 Σ, Σ. Note tht 7

8 x 3 x 1 = x 3 x 1 Pref (w), so x 3 x 1 is prefix w (i) of w nd δ(w (i), ) = AC. Hence, i is uniquely determined by x 1 nd x 3. In prticulr x 3 nd x 1 re uniquely determined, the ltter lets us compute w (j) = δ(w (0), x 1). If we obtin δ(w (0), x 1) = AC, we would hve x 1 L(w), contrdiction, which implies tht no x s for our choice of x 1 nd x 3 exist. We need to count x 2 such tht x 2 = n i 1, x 2 L A (w (j), w (0) ) nd δ(w (i), ) = AC. Note tht δ(w (j), ) {w (0), w (j+1) }, since δ(w (j), ) = AC would imply tht x 2 L A (w (j), AC) rther thn x 2 L A (w (j), w (0) ). Thus the number of words x 2 is equl to L(q, w (0) ) Σ n i 1 summed for q {w (0), w (j+1) } tken with coefficients { Σ : δ(w (j), ) = q δ(w (i), ) = AC}. Lemm 14(b) lets us compute ll the vlues of the type L(q, w (0) ) Σ n i 1 tht we my need ltogether in O(n 3 ) time. The coefficient cn be computed for ny j, i nd q in O(n) time, nd in totl we need to sum O(n 2 ) integers fitting into O(n) mchine words ech. This concludes the O(n 3 )-time lgorithm. Lemm 18. For n rbitrry word w Σ n one cn compute CS(w) in O(n 3 ) time. Proof. Let w Σ n be the lrgest word such tht w = w w. Note tht CS(w ) = CS(w). By Lemm 5 we cn compute w in O(n 2 ) time. 5 Rnking Lyndon words nd De Bruijn Sequences Fct 19. Let α > 1 be rel number. Then d n dα = O(n α ). Proof. Recll tht for α > 1 we hve n=1 1 n = O(1). Consequently α d α = ( n ) α ( d n ) α d = n α 1 d = O(n α ). α d n d n d=1 d=1 Theorem 20. We cn compute Lynd(w) in O(n 3 ) time. Proof. We use the formul given by Lemm 9 nd the lgorithm of Lemm 18. The time complexity is O( d n d3 ) which, by Fct 19, reduces to O(n 3 ). Theorem 21. The k-th Lyndon word of length n cn be found in O(n 4 log Σ ) time. Proof. By definition we look for the smllest w Σ n such tht Lynd(w) k. We binry serch Σ n with respect to the lexicogrphic order, using the lgorithm of Theorem 20 to check whether Lynd(w) k. The size of the serch spce is Σ n, which gives n dditionl n log Σ -time fctor. The proof of the theorem of Fredricksen nd Miorn [7] is constructive, i.e. for ny word w of length n it shows the conctention of constnt number of consecutive Lyndon words of length dividing n tht contin w. This, together with the following lemm which reltes db n to CS, lets us compute the exct position where w occurs in db n. Recll tht L (n) is the set of Lyndon words whose length is divisor of n. 8

9 Lemm 22. Let w Σ n nd L(w) = {λ L (n) : λ w }. Then the conctention, in lexicogrphic order, of words λ L(w) forms prefix of db n nd its length, λ L(w) λ, is equl to CS(w). Moreover, if w = λd for some λ L nd d Z +, then λ is the lexicogrphiclly lrgest element of L(w). Proof. First, observe tht by Lemm 4 the lexicogrphic order on L, the set of ll Lyndon words, coincides with the lexicogrphic order of the infinite powers of these words. In prticulr, this remins true in L (n) nd shows tht conctention of elements of L(w) indeed forms prefix of db n, nd tht if w = λ d, then λ is the lexicogrphiclly lrgest element of L(w). It remins to show tht λ L(w) λ = CS(w). We shll build mpping φ : Σ n L (n) such tht φ 1 (λ) = λ nd x w if nd only if φ(x) L(w). Let x Σ n. There is unique primitive word y nd positive integer k such tht x = y k. We set φ(x) = y, note tht it indeed belongs to L (n). Moreover, to ech Lyndon word λ of length d n we hve ssigned v n d for ech cyclic rottion v of λ. Also, x = y n d, so x w if nd only if φ(x) n d w, i.e. φ(x) w, i.e. φ(x) L(w). Theorem 23. Given word w Σ n, its position in the de Bruijn sequence db n cn be found in O(n 3 ) time. Proof. Let λ 1 < λ 2 <... < λ p be ll Lyndon words in L (n) (we hve λ 1 λ 2... λ p = db n ). The proof of theorem of Fredricksen nd Miorn [7, 9] describes the loction of w in db n which cn be stted succinctly s follows. Clim ([7, 9]). Assume tht w = (αβ) d, where d Z + nd βα = λ k L (n). Denote = min Σ nd z = mx Σ. () If w = z i n i for i 1, then w occurs t position Σ n i. (b) If α z α then w is fctor of λ k λ k+1. (c) If α = z α nd d > 1 then w is fctor of λ k 1 λ k λ k+1. (d) If α = z α nd d = 1 then w is fctor of λ k 1λ k λ k +1, where λ k lrgest λ L (n) such tht λ < β. is the In cse () it is esy to locte w in db n, we omit it from further considertions. Observe tht λ k cn be retrieved s the primitive root of w. Also note tht, by Lemm 4, λ k is the primitive root of the lrgest w Σ n such tht w = w nd w < β α, nd thus it cn be computed in O(n 2 ) time using Lemm 5. Once we know λ k nd λ k, depending on the cse, we need to find the successor in L (n) nd possibly the predecessor in L (n) of one of them. For ny λ L (n) the successor in L (n) cn be generted by iterting single step of the FKM lgorithm t most (n 1)/2 times [6], i.e. in O(n 2 ) time. For the predecessor in L (n), version of the FKM lgorithm tht visits the Lyndon words in reverse lexicogrphic order cn be used [9], it lso tkes O(n 2 ) time to find the predecessor. In ll cses we obtin in O(n 2 ) time the Lyndon words whose conctention contins w. Then we perform pttern mtching for w in the conctention. This gives us reltive position of w in db n with respect to the position of the cnonicl 9

10 occurrence of λ k or λ k in db n. Lemm 22 proves tht such n occurrence of λ L (n) ends t position CS(λ n λ ), which cn be computed in O(n 3 ) time by Lemm 18. Applied to λ k or λ k this concludes the proof. To compute the k-th symbol of db n we hve to locte the Lyndon word from L (n) contining the k-th position of db n. We pply binry serch s in Theorem 21. The k-th symbol of db n is much esier to find due to simpler structure of the sequence. This gives rough ide of the proof of the following theorem. We omit the full proof in this version of the pper. Theorem 24. Given integers n nd k, the k-th symbol of db n nd db n cn be computed in O(n 4 log Σ ) time. References 1. S. Bonomo, S. Mntci, A. Restivo, G. Rosone, nd M. Sciortino. Suffixes, conjugtes nd Lyndon words. In M.-P. Bél nd O. Crton, editors, Developments in Lnguge Theory, volume 7907 of Lecture Notes in Computer Science, pges Springer, M. Crochemore, C. S. Iliopoulos, M. Kubic, J. Rdoszewski, W. Rytter, nd T. Wleń. Extrcting powers nd periods in word from its runs structure. Theor. Comput. Sci., doi: /j.tcs , M. Crochemore nd W. Rytter. Text Algorithms. Oxford University Press, J.-P. Duvl. Génértion d une section des clsses de conjugison et rbre des mots de Lyndon de longueur bornée. Theor. Comput. Sci., 60: , R. G. F. Chung, P. Diconis. Universl cycles for combintoril structures. Discrete Mthemtics, 110:43 59, H. Fredricksen nd I. J. Kessler. An lgorithm for generting necklces of beds in two colors. Discrete Mthemtics, 61(2-3): , H. Fredricksen nd J. Miorn. Necklces of beds in k colors nd k-ry de Bruijn sequences. Discrete Mthemtics, 23(3): , Y. Hin Au. Shortest sequences contining primitive words nd powers. ArXiv e-prints, Apr D. E. Knuth. The Art of Computer Progrmming, Volume 4, Fscicle 2. Addison- Wesley, M. Lothire. Combintorics on Words. Addison-Wesley, Reding, MA., U.S.A., C. J. Mitchell, T. Etzion, nd K. G. Pterson. A method for constructing decodble de Bruijn sequences. IEEE Trnsctions on Informtion Theory, 42(5): , M. Much. Lyndon words nd short superstrings. In S. Khnn, editor, SODA, pges SIAM, J. Rdoszewski. Genertion of lexicogrphiclly miniml de Bruijn sequences with prime words. Mster s thesis, University of Wrsw, 2008 (in Polish). 14. F. Ruskey, C. D. Svge, nd T. M. Y. Wng. Generting necklces. J. Algorithms, 13(3): , J. Tulini. De Bruijn sequences with efficient decoding lgorithms. Discrete Mthemtics, 226(1-3): ,