De Bruijn Sequences for the Binary Strings with Maximum Specified Density

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1 De Bruij Sequeces for the Biary Strigs with Maximum Specified Desity Joe Sawada 1, Brett Steves 2, ad Aaro Williams 2 1 jsawada@uoguelph.ca School of Computer Sciece, Uiversity of Guelph, CANADA 2 brett@math.carleto.ca Departmet of Mathematics ad Statistics, Carleto Uiversity, CANADA 3 haro@uvic.ca Departmet of Mathematics ad Statistics, Carleto Uiversity, CANADA Abstract. A de Bruij sequece is a circular biary strig of legth 2 that cotais each biary strig of legth exactly oce as a substrig. A maximum-desity de Bruij sequece is a circular biary strig of legth 0) + 1) + 2) + + m) that cotais each biary strig of legth with desity umber of 1s) betwee 0 ad m, iclusively. I this paper we efficietly geerate maximum-desity de Bruij sequeces for all values of ad m. Whe = 2m + 1 our result gives a complemet-free de Bruij sequece which is a circular biary strig of legth 2 1 that cotais each biary strig of legth or its complemet exactly oce as a substrig. Keywords: de Bruij sequece, fixed-desity de Bruij sequece, Gray codes, ecklaces, Lydo words, cool-lex order 1 Itroductio Let B) be the set of biary strigs of legth. The desity of a biary strig is its umber of 1s. Let B d ) be the subset of B) whose strigs have desity d. Let B,m) = B 0 ) B 1 ) B m ) be subset of B) whose strigs have desity at most m. A de Bruij sequece or de Bruij cycle) is a circular biary strig of legth 2 that cotais each strig i B) exactly oce as a substrig [2]. De Bruij sequeces were itroduced by de Bruij [2] ad see earlier [3]) ad have may geeralizatios, variatios, ad applicatios. For example, oe ca refer to the recetly published proceedigs of the Geeralizatios o de Bruij Sequeces ad Gray Codes workshop [6]. I this paper we cosider a ew geeralizatio of de Bruij sequeces that specifies the maximum desity of the substrigs. A maximum-desity de Bruij sequece is a biary strig of legth ) 0 + ) 1 + ) ) m that cotais each strig i B,m) exactly oce as a substrig. For example, is a maximum-desity de Bruij sequece sice its 16 substrigs of legth 5 icludig those that wrap-aroud ) are precisely B5, 3). Our mai results are 1) a explicit costructio of maximumdesity de Bruij sequeces for all values of ad m, ad 2) a efficiet algorithm that geerates these de Bruij sequeces. We make four simple observatios ivolvig maximum-desity de Bruij sequeces for B, m): 1. A maximum-desity de Bruij sequece is simply a de Bruij sequece whe = m. 2. Complemetig each bit i a maximum-desity de Bruij sequece results i a miimum-desity de Bruij sequece for the biary strigs of legth whose desity is at least m. 3. A maximum-desity de Bruij sequece is acomplemet-free de Bruij sequece whe = 2m + 1. This is because each biary strigs of legth either has desity at most m or has desity at least m. Research supported i part by NSERC

2 4. Reversig the order of the bits i a maximum-desity de Bruij sequece simply gives aother maximum-desity de Bruij sequece for the same values of ad m. It is easier to describe our sequeces i oe order, ad the geerate them i the reverse order. Sectio 2 provides backgroud results. Sectio 3 describes our costructio ad proves its correctess. Sectio 4 provides a algorithm that geerates our costructio ad aalyzes its efficiecy. 2 Backgroud A ecklace is a biary strig i its lexicographically smallest rotatio. The ecklaces over B) ad B d ) are deoted N) ad N d ), respectively. The aperiodic prefix of a strig α = a 1 a 2 a is its shortest prefix ρα) = a 1 a 2 a k such that ρα) /k = α. As is customary, the previous expressio uses expoetiatio to refer to repeated cocateatio. Observe that if ρα) = k, the α has k distict rotatios. For example, ρ ) = ad is a ecklace sice it is lexicographically smaller tha its other four distict rotatios , , , ad A ecklace α is aperiodic if ρα) = α. Oe of the most importat results i the study of de Bruij sequeces is due to Fredrickse, Kessler ad Maioraa [4, 5] also see Kuth [7]). These authors proved that a de Bruij sequece for B) ca be costructed by cocateatig the aperiodic prefixes of the strigs i N) i lexicographic order. For example, the lexicographic order of N6) is , , , , , , , , , , , , , ) ad so the followig is a de Bruij sequece for B6), where visually separates the aperiodic prefixes from 1) i the cocateatio ) Although 2) is writte liearly, we treat it as a circular strig so that its substrigs ca wraparoud from the ed to the begiig. Iterestigly, 2) is also the lexicographically smallest de Bruij sequece for B6) whe writte liearly). Subsequet aalysis by Ruskey, Savage, ad Wag [8] proved that these lexicographically smallest de Bruij sequeces ca be geerated efficietly for all values of. Recetly, it was show that this ecklace-prefix algorithm ca be modified to create a restricted type of de Bruij sequece. Reverse cool-lex order is a variatio of co-lexicographic order that was first defied for B d ) by Ruskey ad Williams [11], ad has sice bee geeralized to subsets of B d ) icludig N d ) by Ruskey, Sawada, Williams [10]. I this paper we let cool d ) deote the reverse cool-lex order of N d ). The order for = 8 ad d = 4 appears below cool 4 8) = , , , , , , , , , ) Let db d ) deote the cocateatio of the aperiodic prefixes of cool d + 1). For example, the cocateatio of the aperiodic prefixes of 3) gives the followig db 4 7) = )

3 Observe that the circular strig i 4) cotais each strig i B 3 7) B 4 7) exactly oce as a substrig, ad has o other substrigs of legth 7. For this reaso we describe it as a dual-desity de Bruij sequece for B 3 7) B 4 7) i this paper 4. More geerally, Ruskey, Sawada, ad Williams [9] proved the followig result. Theorem 1. [9] The circular strig db d ) is a dual-desity de Bruij sequece for B d 1 ) B d ) whe 1 < d <. I this paper we do ot eed to completely uderstad the proof of Theorem 1, but we do eed a simple property for its dual-desity de Bruij sequeces. I other words, we eed to treat each db d ) as a gray box. The specific property we eed is stated i the followig simple lemma ivolvig reverse cool-lex order. This lemma follows immediately from equatio 5.1) i [9]. Lemma 1. [9] If N d ) cotais at least three ecklaces, the the first ecklace i cool d + 1) is 0 d+1 1 d, ad the secod ecklace i cool d + 1) is 0 d 1 d 1 01, ad the last ecklace i cool d + 1) is 0 x 10 y 1 d 1 where x = + 1 d)/2 ad y = + 1 d)/2. Moreover, each of the ecklaces give above are distict from oe aother ad are aperiodic. I Sectio 3 we will be takig apart the dual-desity de Bruij sequece db d ) aroud the locatio of the ecklace 0 d+1 1 d from cool d + 1). For this reaso we make two auxiliary defiitios. Let cool d+1) equal cool d +1) except that the first ecklace 0 d+1 1 d omitted. Similarly, let db d) be the cocateatio of the aperiodic prefixes of cool d + 1). For example, we will be splittig db 4 7) i 4) ito ad db 47) = Costructio I this sectio we defie a circular strig db,m) of legth 1 + 1) + 2) + + m). The we prove that db,m) is a maximum-desity de Bruij sequece for B,m) i Theorem 2. { m+1 1 m db m) db 4) db 2) if m is eve 5a) db,m) = m+1 1 m db m) db 5) db 3) if m is odd 5b) Table 1 provides examples of db,m) whe = 7. To uderstad 5), observe that db,m) is obtaied by splicig together the dual-desity de Bruij sequeces db d +1) = 0 d+1 1 d db d+1) for d = 0,2,4,...,m i 5a) or d = 1,3,5,...,m i 5b). I particular, db 0 + 1) = 0 is the aperiodic prefix of 0 +1 o the left side of 5a), ad the empty db 0) ad db 1) are omitted from the right sides of 5a) ad 5b), respectively. For the order of the splicig, observe that db m ) = 0 m+1 1 m db m) appears cosecutively i db,m). That is, db,m) = db m ). More specifically, db,m) is obtaied by isertig db m ) ito the portio of db,m 2) that cotais db m 2 ). To be precise, db m ) is iserted betwee the first ad secod ecklaces of cool m 2 + 1). That is, db,m) = 0 m+3 1 m 2 }{{} first i cool m 2 +1) db m ) 0 m+2 1 m 3 01 }{{} secod i cool m 2 +1).

4 Cool-lex orders Maximum-desity sequeces eve desities) db7, 2) db7, 4) db7, 6) cool 28) cool 28) cool 48) cool 48) Cool-lex orders Maximum-desity sequeces odd desities) db7, 1) db7, 3) db7, 5) db7, 7) cool 38) cool 28) cool 58) cool 48) cool 68) cool 68) cool 78) cool 68) Table 1. Maximum-desity de Bruij sequeces costructed from cool-lex order of ecklaces whe = 7 ad m > 0. For example, db7, 2) = Theorem 2. The circular strig db,m) is a maximum-desity de Bruij sequece for B,m). Proof. The claim ca be verified whe 4. The proof for 5 is by iductio o m. The result is true whe m {0,1} sice db,0) = 0 ad db,1) = 0 1 are maximum-desity de Bruij sequeces for B,0) ad B,1), respectively. The remaiig base case of m = 2 gives db,2) = db 2) = 0 db 2 ), which is a maximum-desity de Bruij sequece for B,2) sice db 2 ) is a dual-desity de Bruij sequece for B 1 ) B 2 ) by Theorem 1. First we cosider the special case where m =. I this case, N 1 +1) cotais at least three ecklaces. Therefore, Lemma 1 implies that db, 2) ad db, ) ca be expressed as follows db, 2) = db,) = }{{} db +1) Observe that every substrig of legth that appears i db, 2) also appears i db,). Furthermore, the substrigs of legth that appear i db,) ad ot db, 2) are precisely , ,...,01,1,1 1 0, which are a orderig of B 1 ) B ). Sice db, 2) is 4 The strig i 4) is described as a fixed-desity de Bruij sequece i [9] sice each substrig i B 37) B 47) ca be uiquely exteded to a strig i B 48) by appedig its missig bit.

5 a maximum-desity de Bruij sequece for B, 2) by iductio, we have prove that db,) is a maximum-desity de Bruij sequece for B,). Otherwise m <. I this case, N m 2 +1) ad N m +1) both cotai at least three ecklaces. Therefore, Lemma 1 implies that db,m 2) ad db,m) ca be expressed as follows db,m 2) = 0 m+3 1 m 2 0 m+2 1 m ) db,m) = 0 m+3 1 m 2 0 m+1 1 m 0 m 1 m x 10 y 1 m 1 }{{} db m) 0 m+2 1 m ) where x = + 1 m)/2, y = + 1 m)/2, ad the bouds m ad imply that 0 x 10 y 1 m 1 ad 0 m+2 1 m 3 01 are aperiodic. The substrigs of legth i db,m 2) are B,m 2) by iductio, ad the substrigs of legth i db m ) are B m 1 ) B m ) by Theorem 1. Therefore, we ca complete the iductio by provig that the substrigs of legth i db,m) iclude those i a) db,m 2), ad b) db m ). To prove a), observe that the substrigs of legth i 0 m+3 1 m 2 0 m+2 from 6) are i 0 m+3 1 m 2 0 m+1 from 7), except for 1 m 2 0 m+2. Similarly, the substrigs of legth i 1 m 2 0 m+2 1 m 3 from 6) are i 1 m 1 0 m+2 1 m 3 from 7), ad the latter also icludes 1 m 2 0 m+2. To prove b), cosider how the isertio of db m ) ito db,m) i 7) affects its substrigs that ca o loger wrap-aroud i db m ). The substrigs of legth i the wrap-aroud 1 m 1 0 m+1 i db m ) are all i 1 m 1 0 m+2 from 7). Therefore, db,m) is a maximum-desity de Bruij sequece for B,m). Corollary 1 follows from the first three simple observatios made i Sectio 1. Corollary 1. The costructio of the maximum-weight de Bruij sequeces db,m) icludes 1. db,) is a de Bruij sequece for B), 2. db,m) is a miimum-weight de Bruij sequece for B m ) B m+1 ) B ), 3. db2m + 1, m) is a complemet-free de Bruij sequeces for B2m + 1). 4 Algorithm As metioed i Sectio 1, the reversal of db,m), deoted db,m) R also yields a maximumdesity de Bruij sequece. To efficietly produce db,m) R we ca use the recursive cool-lex algorithm described i [12] to produce the reversal of db d ). I that paper, details are provided to trim each ecklace to its logest Lydo prefix ad to output the strigs i reverse order. A aalysis shows that o average, each bits ca be visited i costat time. There are two data structures used to maitai the curret ecklace: a strig represetatio a 1 a 2 a, ad a block represetatio B c B c 1 B 1 where a block is defied to be a maximal substrig of the form 0 1. A block of the form 0 s 1 t is represeted by s,t). Sice the umber of blocks c is maitaied as a global parameter, it is easy to test if the curret ecklace is of the form 0 1 : simply test if c = 1. By addig this test, it is a straightforward matter to produce the reversal of db d). To be cosistet with the descriptio i [12], the fuctio Ge d,d) ca be used to produce db d). Usig this fuctio, the followig pseudocode ca be used to produce db,m) R : if m is eve the start := 2 else start := 3 for i from start by 2 to m do Iitialize + 1, i) Ge + 1 i,i) for i from m by 2 dowto 0 do Prit 1 i 0 +i 1 ) if m is eve the Prit 0 )

6 The Iitialize+1,i) fuctio sets a 1 a 2 a +1 to 0 +1 i 1 i, sets c = 1 ad sets B 1 = +1 i,i). The first time it is called it requires O) time, ad for each subsequet call, the updates ca be performed i O1) time. Also ote that the strig visited by the Prit) fuctio ca also be updated i costat time after the first strig is visited. Sice the extra work outside the calls to Ge requires O) time, ad because the umber of bits i db,m) is Ω 2 ) where 1 < m <, we obtai the followig theorem. Theorem 3. The maximum-desity de Bruij sequece db,m) R ca be geerated i costat amortized time for each bits visited, where 1 < m <. 5 Ope Problems A atural ope problem is to efficietly costruct desity-rage de Bruij sequeces for the biary strigs of legth whose desity is betwee i ad j iclusively) for ay 0 i < j. Refereces 1. F. Chug, P. Diacois, ad R. Graham, Uiversal cycles for combiatorial structures, Discrete Mathematics, ) N.G. de Bruij, A combiatorial problem, Koikl. Nederl. Acad. Wetesch. Proc. Ser A, ) N.G. de Bruij, Ackowledgemet of priority to C. Flye Saite-Marie o the coutig of circular arragemets of 2 zeros ad oes that show each -letter word exactly oce, T.H. Report 75-WSK-06, Techological Uiversity Eidhove 1975) 13 pages. 4. H. Frederickse ad J. Maioraa, Necklaces of beads i k colors ad kary de Bruij sequeces, Discrete Mathematics, ) H. Frederickse ad I. J. Kessler, A algorithm for geeratig ecklaces of beads i two colors, Discrete Mathematics, ) G. Hurlbert, B. Jackso, B. Steves Editors), Geeralisatios of de Bruij sequeces ad Gray codes, Discrete Mathematics, ) D.E. Kuth, The Art of Computer Programmig, Volume 4, Geeratig all tuples ad permutatios, Fascicle 2, Addiso-Wesley, F. Ruskey, C. Savage, ad T.M.Y. Wag, Geeratig ecklaces, J. Algorithms, ) F. Ruskey, J. Sawada, ad A. Williams Fixed-desity de Bruij sequeces, submitted) F. Ruskey, J. Sawada, ad A. Williams Biary bubble laguages ad cool-lex order, submitted) F. Ruskey ad A. Williams, The coolest way to geerate combiatios, Discrete Mathematics, ) J. Sawada ad A. Williams A Gray Code for fixed-desity ecklace ad Lydo words i costat amortized time, submitted) 2010.