Generating Balanced Parentheses and Binary Trees by Prefix Shifts

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1 Gnrtin Bln Prnthss n Binr Trs Pri Shits Frnk usk 1 Aron Willims 2 Dprtmnt o Computr Sin Univrsit o Vitori, Vitori BC, V8W 3P6, Cn, 1 UL: 2 Emil: hron@uvi. Astrt W show tht th st B n o ln prnthsis strins with n lt n n riht prnthss n nrt pri shits. I 1, 2,..., 2n is mmr o B n, thn th k-th pri shit is th strin 1, k, 2,..., k 1, k+1,..., 2n. Pri shit lorithms r lso known or omintions, n prmuttions o multist; th omintion lorithm pprs in sils o Knuth vol 4. W show tht th lorithm is losl rlt to th omintion lorithm, n lik it, hs looplss implmnttion, n rnkin lorithm tht uss O(n) rithmti oprtions. Aitionll, th lorithm n irtl trnslt to nrt ll inr trs looplss implmnttion tht mks onstnt numr o pointr hns or h sussivl nrt tr. Kwors: Gr os, Ctln numrs, ln prnthss, inr trs, omintoril nrtion, loopr lorithm. 1 Introution Bln prnthsis strins r on o th most importnt o th mn isrt struturs tht r ount th Ctln numrs, C n ( ) 2n n /(n + 1). Th Ctln numrs n th ojts ount thm r tnsivl isuss in Stnl (1999). Th onlin supplmnt lists 149 istint isrt struturs ount th Ctln numrs (Stnl (27)). Binr trs n orr trs r lso ount th Ctln numrs; ths tr struturs r o prmount importn to omputr sintists. Thr is lr numr o pprs lin with th unmntl prolm o hustivl listin n rnkin inr trs. In this ppr w vlop n lorithm tht hs numr o ttrtiv n uniqu turs s ompr with istin lorithms. Lt B t,s th st o ll itstrins ontinin t 1s n s s n stisin th onstrint tht th numr o 1s in n pri is t lst s lr s th numr o s. For mpl, B 3,2 {111, 111, 111, 111, 111}. In prtiulr, B t,s is mpt i t < s. Furthrmor, i t s thn B t,s n thouht o s th st o ll ln prnthsis strins mppin 1 to lt prnthsis n to riht prnthsis. In this s, w somtims rop th s rom th nottion; B n B n,n. Copriht 28, Austrlin Computr Soit, In. This ppr ppr t th Fourtnth Computin: Th Austrlsin Thor Smposium (CATS28), Univrsit o Wollonon, Nw South Wls, Austrli. Conrns in srh n Prti in Inormtion Thnolo (CPIT), Vol. 77, Jms Hrln n Prhu Mnm, E. proution or mi, not-or proit purposs prmitt provi this tt is inlu. I 1, 2,..., t+s is mmr o B t,s, thn th k-th pri shit is th strin 1, k, 2,..., k 1, k+1,..., t+s. Not tht th irst it, 1 is not prt o this inition; this is nturl sin 1 is lws 1. Furthrmor, it is impossil to nrt B t,s i 1 is inlu in th shits (.., 1 t s is th onl vli shit o oth 1 t 1 s 1 n 1 t 1 s 1 1). In orr to nti th rr into rin urthr, low w show th simpl itrtiv rul, whos sussiv pplition will nrt B t,s usin pri shits. Itrtiv sussor rul: Lot th ltmost 1 n suppos tht its 1 is in position k. I th (k + 1)- st pri shit is vli ( mmr o B t,s ), thn it is th sussor; i it is not vli thn th k-th pri shit is th sussor. Th onl strin without 1 is 1 t s, whih is th inl strin. Th initil strin is 11 t 1 s 1. Applin th rul to B 3,2 ivs th squn 111, 111, 111, 111, 111. This is th irst ppr tht onsirs whthr ln prnthss n nrt pri shits. It is known tht B t,s n nrt trnsposin pir o its (usk & Proskurowski (199)), pir o its with onl s in twn (Bultn & usk (1998)), or trnsposin on or two pirs o jnt its (Vjnovszki & Wlsh (26)). In nrl it is impossil to nrt B t,s trnsposin onl on pir o jnt its (usk & Proskurowski (199)). Our lorithm will shown to nrt B t,s trnsposin on or two pirs o its, ut thos its r not jnt in nrl. An lorithm or nrtin omintoril ojts is si to looplss i onl onstnt mount o omputtion is us in trnsormin th urrnt strutur into its sussor. Looplss lorithms r known or vrious lsss o isrt struturs tht r ount th Ctln numrs. S, or mpl, th pprs olnts (1991), Korsh, LFoltt, & Lipshutz (23), Mtos, Pinho, Silvir-Nto & Vjnovszki (1998), Vjnovszki & Wlsh (26) n Tkok & Violih (26). Thr is ppr tht shows tht inr trs in thir onvntionl rprsnttion o no with two pointrs n iintl nrt onl mkin onstnt numr o pointr hns twn sussiv trs (Lus, olnts, & usk (1993)). This lorithm n implmnt looplssl n is prsnt in Knuth (26). Th urrnt ppr ivs th sis or nothr suh lorithm. Th pproh tkn in this ppr ws initit in th pprs o usk & Willims (25, 28) or nrtin omintions tht r rprsnt itstrins in th usul w. Thr th itstrins r lso nrt pri shits. It is rmrkl how mn

2 o th rsults o thos pprs hv los nlous with th rsults o th urrnt ppr. Th orrin o omintions in (usk & Willims 25, 28) ws ll ool-l orr us o its los onntion with th wll-known ol orr o omintions. In similr spirit, w hv u our orr CoolCt orr us o its los onntions with ool-l orr n with th Ctln numrs. ltiv to list o ojts, th rnk o prtiulr ojt is th position tht it oupis in th list, ountin rom zro. To summriz, our mtho hs th ollowin proprtis: 1. Eh sussiv strin irs rom its prssor th rottion o pri o th strin. Furthrmor, th list o strins is irulr in th sns tht th irst n lst lso ir pri rottion. 2. Eh sussiv strin irs rom its prssor th intrhn o on or two pirs o its. 3. It hs simpl rursiv sription. This sription os not involv th rvrsl o sulist, s is usull th s or Gr os. Th unrlin rph is irt rph; tht is, i 1 irs rom 2 pri rottion, thn in nrl it is not th s tht 2 irs rom 1 pri rottion. 4. It hs rmrkl simpl itrtiv sussor rul. This rul ws stt ov. 5. Th itrtiv sussor rul n implmnt s looplss lorithm. Also, th sussor rul n trnslt to looplss lorithm or nrtin inr trs. No prvious listin o ln prnthsis strins is simultnousl Gr o or th strins n or th orrsponin inr trs. 6. It hs rnkin lorithm tht uss O(n) rithmti oprtions. No prvious Gr o or ln prnthsis strins hs this proprt. 2 Gnrtin Binr Trs To iv th rr lvor o how usul th itrtiv sussor rul is, in this stion w trnslt th rul so tht it pplis to inr trs, s tritionll implmnt on omputr. Th rsult is looplss lorithm tht mks t most 16 pointr upts twn sussiv trs. An implmnttion o this lorithm is vill rom th uthors. Th stnr ijtion twn B n,n n tn inr trs with n intrnl nos is to ssoit h intrnl no with 1 n h l with n thn o prorr trvrsl o th tr, inorin th inl l. I z is no in inr tr, thn w us l(z) n r(z) to not th pointrs to th lt n riht hilrn o z. Unortuntl, w lso n to mintin th prnt o h intrnl no; this is not p(z). To upt th tr w mintin thr pointrs:, th irst no tht is not on th ltmost pth o intrnl nos;, th prnt o ; n, th root o th tr. Th ssinmnts low rprsnt prlll utions, so tht, or mpl, [, ] [, ] swps th two vlus n. Th lorithm trmints whn oms nil. Aorin to th itrtiv sussor rul thr r thr ss to onsir: () th strin is o th orm 1 p q 11α, () th strin is o th orm 1 p q 1α, with p > q, n () th strin is o th orm 1 p p 1α. Blow w show th upts tht r nssr in h p h q r Fiur 1: Th trs orrsponin to This is n mpl o Cs (). o th thr ss. Importnt not: Th upts to th prnt il r not shown pliitl low, ut vr tim tht n upt is on to r(.) or l(.), thn n upt must on to p(.). E.., i th upt is r(v) w, thn it shoul ollow with i w nil thn p(w) v. Cs (): Th nw strin is 1 p+1 q 1α. This s ours whn l() nil. Th orrsponin upt to th inr tr is [r(), r(), l(), l()] [r(), l(), l(), ] ; r(); Cs (): Th nw strin is 11 p 1 q 1α. This s ours whn l() nil n. Th orrsponin upt to th inr tr is [l(p()), r(p()), l(), r(), l(), r()] [l(),, r(), r(p()), nil, ]; [, ] [, r()]; Cs (): th nw strin is 1 p+1 p+1 α. This s ours whn l() nil n. Th orrsponin upt to th inr tr is [l(), r()] [, nil]; [,, ] [,, r()]; Atr this upt th lorithm trmints i nil (i.., i α is th mpt strin). Ths thr ss r illustrt in Fiurs 1, 2, n 3. Cirls r us or intrnl nos, squrs r us or lvs, n th trinls rprsnt sutrs whos strutur is not spii (ut whos prorr orr must prsrv). 3 ursiv Strutur In this stion w min th rursiv strutur o th CoolCt orrin on ln prnthsis. In prtiulr, w provi two rursiv ormul n prov tht th prou lists tht r intil to thos prou th itrtiv rul. A orollr to this rsult is tht th itrtiv rul nrts vr strin in B t,s. For omprison purposs w lso provi th rursiv strutur or o-liorphi, or ol orrin. W in this stion ivin orml inition o th itrtiv rul. p r q h

3 Th CoolCt itrtiv rul mps inr strin B t,s to nothr inr strin σ() B t,s. Whn os not ontin n 1 or 11 s sustrin thn it is sist to in σ() usin th ollowin two spil ss, whih simpl mov th rihtmost smol into th son ltmost position. σ() { 11 i j i 11 i j (1) 111 i j i 11 i j 1 (1) r Othrwis, w n ssum tht 11 i j 1z or som smol z n som (possil mpt) strin. q r σ() { 111 i j z i i j (2) 1z1 i j 1 i i > j (2) p p q W inutivl lt σ () n σ k () σ(σ k 1 ()) or k >, so tht w n in n itrtiv list t,s tht uss σ. t,s, σ(), σ 2 (),..., σ k 1 () (3) Fiur 2: Th trs orrsponin to This is n mpl o Cs (). whr 1 t s n k B t,s. W ll lso in it usul to strt th itrtiv pross t th sussor o, n in t our irst rursiv strutur will qul this sonr listin. Inst o strtin th itrtiv pross t th sussor o, this sonr listin n lso sn s th rsult o pplin σ to h strin in t,s. S t,s σ(), σ 2 (),..., σ k () (4) σ( t,s ) (5) h h To ttr illustrt our irst rursiv ormul, lt us in minin th rursiv strutur o th ol list L 4,4 n thn omprin it to th CoolCt list S 4,4. Th trm ol rrs to th t tht th strins in B t,s r in inrsin liorphi orr whn h strin is r rom riht to lt. Th ol list L 4,4 n uilt rursivl rom th smllr lists L 3,i or i 3. Eh o ths lists pprs s olumn within Fiur 4. Noti tht in h olumn th suis innin with 1 r unrlin, n ll o th strins with ivn unrlin sui ppr onsutivl. In th s o L 4,4 (whr t s) th suis innin with 1 r 1, 1, 1, n 1. Noti tht thr is no sui 1 sin thr is no strin in B 4,4 with tht sui. Howvr, th sui 1 os ppr in L 3,2 (whr t > s) sin thr is strin with tht sui in B 3,2. Finll, in h s th suis r orr rsin numr o zros. In nrl h o ths osrvtions hols tru, n it ls to th ollowin rursiv ormul or L t,s Fiur 3: Th trs orrsponin to This is n mpl o Cs (). { Lt 1, 1 s, L t 1,1 1 s 1,..., L t 1,s 1 1 i t s L t 1, 1 s, L t 1,1 1 s 1,..., L t 1,s 1 i t > s. To ompt prssions o this kin w introu to omin short lists o strins into lrr lists, n w rstt th rursiv ormul or L t,s s ollows s 1 L t 1,i 1 s i i t s (6) i L t,s s L t 1,i 1 s i i t > s. (6) i Now w turn our ttntion to th rursiv strutur o W 4,4 tht is illustrt in Fiur 5. As in ol th suis innin with 1 r unrlin n

4 L 3, L 3,1 L 3,2 L 3,3 L 4, Fiur 4: Th rursiv strutur o ol. W 3, W 3,1 W 3,2 W 3,3 W 4, Fiur 5: Th irst rursiv strutur o CoolCt. th strins with ivn unrlin sui ppr onsutivl within h list. Howvr, in this s th suis innin with 1 r orr rsin numr o zros, pt or th sui 1 s tht pprs lst inst o irst. O ours, thr is onl sinl strin in B t,s tht hs th sui 1 s, nml 1 t s. Amzinl, th ltrnt plmnt o this sinl strin ull pturs th irn twn th rursiv strutur o CoolCt n ol. W in th list W t,s s ollows, n w prov tht it is qul to S t,s in Thorm 1 s 1 W t 1,i 1 s i, 1 t s W t,s s W t 1,i 1 s i, 1 t s i t s (7) i t > s.(7) Sin th rursiv strutur o W t,s is rorrin o th strins in L t,s w hv th ollowin rmrk. mrk 1. W t,s ontins h strin in B t,s tl on. An importnt stp towrs provin Thorm 1 is th ollowin lmm, tht pliitl intiis th irst n lst strins tht ppr in W t,s whn s >. Lmm 1. For s > irst(w t,s ) 11 t 1 s 1 (8) lst(w t,s ) 1 t s. (9) Proo. Th vlu o lst(w t,s ) ollows immitl rom (7). To trmin th vlu o irst(w t,s ) w hv th ollowin irst(w t,s ) irst(w t 1,1 )1 s 1 irst(w t 2,1 )11 s 1 irst(w t 3,1 )111 s 1... irst(w 1,1 )1 t 1 s 1 11 t 1 s 1. Now w r in position to prov th min rsult o this stion. Thorm 1. S t,s W t,s. Proo. To prov th rsult w n to show tht within W t,s th irst strin in h sulist is otin pplin σ to th lst strin o th prvious sulist. Th sulists in W t,s r slihtl irnt pnin on whthr t s (7) or t > s (7), so w pro in two ss. First w prov th rsult whn t > s. For th lst trnsition w hv σ(lst(w t 1,s 1)) σ(1 t 1 s 1) 1 t s whih ollows rom Lmm 1 n th inition o σ (1). For th rminin trnsitions w hv, or 1 i s 1, σ(lst(w t 1,s i 1 i )) σ(1 t 1 s i 1 i ) 11 t 2 s i 1 i 1 irst(w t 1,s i+1 1 i 1 ) whih ollows rom Lmm 1 n th inition o σ (2). In prtiulr, (2) pplis hr sin t > s n i 1 impl tht t 1 > s i. Nt w prov th rsult whn t s. For th lst trnsition w hv σ(lst(w t 1,s 1 1)) σ(1 t 1 s 1 1) 1 t s whih ollows rom Lmm 1 n th inition o σ (2). In prtiulr, (2) pplis hr sin t s. For th rminin ss w hv, or 1 i s 2, σ(lst(w t 1,s i 1 i )) σ(1 t 1 s i 1 i ) 11 t 2 s i 1 i 1 irst(w t 1,s i+1 1 i 1 whih ollows rom Lmm 1 n th inition o σ (2). In prtiulr, (2) pplis hr sin t s n i 2 impl tht t 1 > s i. Thorm 1 llows us to show tht th itrtiv inition o CoolCt prous lists tht r irulr. Tht is, in oth t,s n S t,s, th irst strin n otin pplin σ to th lst strin. Mor nrll w hv th ollowin orollr. Corollr 1. For n B t,s n k B t,s σ k (). Proo. W n prov this rsult showin tht th list S t,s is irulr. This provs th sttmnt o th orollr n lso provs tht t,s is irulr (4)

5 n (3). W omplish our ol throuh th ollowin hin o qulitis tht rrn Thorm 1, Lmm 1, n (1) σ(lst(s t,s )) σ(lst(w t,s )) σ(1 t s ) 11 t 1 s 1 irst(w t,s ) irst(s t,s ). Thorm 1 lso llows us to prov tht th itrtiv inition o CoolCt nrts vr strin in B t,s. Corollr 2. t,s n S t,s ontin h strin in B t,s tl on. Proo. Th rsult or S t,s ollows rom mrk 1 n Thorm 1. Th rsult or t,s ollows rom th t tht σ k (1 t s ) 1 t s or k B t,s Corollr 1, n thus t,s is rorrin o S t,s (3) n (4). Althouh th rursiv inition o W t,s hs its nits, somtims it is mor onvnint to work with rursiv inition tht ontins wr trms. For mpl, in Stion 5 w rnk th orr o th strins within CoolCt utilizin th ollowin inition K t,s 1 i t s K t,s K t 1,s 1, 1 t 1 1 i s 1 K t,s 1, K t 1,s 1, 1 t 1 s 1 i 1 < s < t. (1) In Thorm 2 w prov tht K t,s is intil to W t,s pt tht it is missin th strin 1 t s. Th proo is involv, so w provi n illustrtion or h o th thr ss o (1) in Fiur 6. In h olumn th ovrlin n unrlin strins not whthr th numr o zros or ons r in rursivl rs, rsptivl. Strins without n ovrlin or unrlin r o th orm 1 t 1 s 1 n r not involv in th nt lowr lvl o rursion, whil th strins low th horizontl lin r o th orm 1 t s n rprsnt th uniqu strin tht is in W t,s ut is not in K t,s. For th sk o svin sp w onl prou th olumns with smllr numr o zros, until th numr o zros quls on. K 3,1 K 4,1 K 4,2 K 4,3 K 4, Fiur 6: Th son rursiv strutur or CoolCt. Thorm 2. W t,s K t,s, 1 t s. Proo. W prov th rsult oul inution. Th irst inution will on th numr o zros, n th son inution will on th numr o ons. For th s s o th irst inution w hv s 1 n it is s to vri tht W t,1 t 1 i 1 t i t 1 1 i 1 t i, 1 t K t,1, 1 t. Now suppos tht s k > 1 n tht th thorm hols or ll s < k. At this point w strt our son inution. For th s s o th son inution w hv t k. In othr wors th numr o ons is qul to th numr o zros, whih is th minimum possil numr o ons. W hv th ollowin prssion or W k,k k 1 k 1 ( k 1 W k 1,i 1 k i, 1 k k W k 1,i 1 k 1 i, 1 k k 1 (W k,k 1 ) W k 1,i 1 k 1 i, 1 k k 1 ) (K k,k 1, 1 k k 1 ) K k,k 1, 1 k k K k,k, 1 k k. Now to ontinu with th son inution w suppos tht t k + j, or som j >, n tht th thorm hols or ll t < k + j. In othr wors, w r supposin tht thr r j mor ons thn zros, n tht th thorm hols whn thr r wr thn j itionl ons. Thn w hv th ollowin prssion or W k+j,k k W k+j 1,i 1 k i, 1 k+j k k 1 ( k 1 W k+j 1,i 1 k i, W k+j 1,k 1, 1 k+j k W k+j 1,i 1 k 1 i ), W k+j 1,k 1, 1 k+j k. Th rkt prout hs wr thn k zros n quls W k+j,k 1 pt tht it is missin 1 k+j k 1 s its lst strin. Thror, th irst inution K k+j,k 1, W k+j 1,k 1, 1 k+j k. Th son trm hs wr thn k+j ons. Thror, th son inution w ontinu s ollows K k+j,k 1, (K k+j 1,k, 1 k+j 1 k )1, 1 k+j k K k+j,k 1, K k+j 1,k 1, 1 k+j 1 k 1, 1 k+j k K k+j,k, 1 k+j k. This omplts th inutiv s o th son inution, n so th thorm is tru or s k n ll t k. This omplts th inutiv s o th irst inution, n so th thorm is tru or ll s 1.

6 Bor losin this stion w pliitl stt th irst n lst strins o t,s sin it will usul in th nt stion. Lmm 2. For s > irst( t,s ) 1 t s { 1 lst( t,s ) t 1 s 1 1 i t s 1 t 1 s 1 i t > s. 4 Alorithm In this stion w prsnt n lorithm to nrt t,s. Tht is, w prsnt n lorithm tht itrtivl visits h sussiv strin in th CoolCt orrin strtin with 1 t s. Th lorithm is rmrkl iint in trms o tim n stor. In prtiulr it is looplss in th sns tht h sussiv strin is nrt in O(1) tim, n it is onstnt tr-sp in th sns tht it uss O(1) stor whn luin th rr tht hols th inr strin. Th rr uss 1-s inin, so [1] is th irst vlu in th rr. For proposition P, th nottion [[P ]] mns 1 i P is tru n i P is ls. As in Stion 2, th vril is us to rprsnt th position o 1 in th ltmost 1. Howvr, th vril is now us to rprsnt th position o th ltmost. Th initil vlus o n o not o this rul, n th r hosn simpl or th sk o th irst itrtion. Th initil vlu 1 t s is visit th visit() ommn on lin 5, whil ll othr vlus o ontin ltmost 1 n r visit on lin 21. W s tht h itrtion strts t th whil sttmnt on lin 6. Durin h itrtion thr r thr possil routs throuh th i sttmnts n ths thr routs orrspon tl to th ss rom Stion 2. I 1 p q 11α (s ()) t th strt o n itrtion thn th outr i sttmnt on lin 11 is not ntr. I 1 p p 1α (s ()) thn th innr i sttmnt on lin 12 is ntr. I 1 p q 1α or p > q (s ()) thn th innr ls sttmnt is ntr. Th nrl i o th lorithm is to mintin n n to us thir vlus, n th vlus o [] n [], to trmin how ns to hn rom on itrtion to th nt. CoolCt(t, s) quir: t s > 1: n t + s 2: rr(1 t s ) 3: t 4: t 5: visit() 6: whil < n [t s]] o 7: [] 8: [] 1 9: + 1 1: : i [] 12: i 2 2 { Cs () } 13: : ls 15: [] 1 { Cs () } 16: [2] 17: 3 18: 2 19: n 2: n { ls Cs () } 21: visit() 22: n To prov th orrtnss o th lorithm w trk th vlus o th thr vrils rom on visit ll to th nt visit. W lt 1, 2,... rprsnt th vlus tkn vril t h susqunt visit, n w us th sm onvntion or n. For mpl, 1 will th irst n onl vlu o visit t lin 5, whil 2 will th irst vlu o visit t lin 21. Whn i is th smllst vlu whr i [ i ] thn w will s tht i is orrt. Likwis whn i is th smllst vlu whr i [ i ] 1 n i > i thn w will s tht i is orrt. For onvnin w lso lt V t,s 1, 2,..., k whr k is th lst vlu o tht is visit or th prorm trmints. Ultimtl w will show tht th prorm os in t trmint, n tht V t,s t,s (Thorm 3). W rr to th urrnt vlus o,, n s th urrnt oniurtion. From lins 2-4 w s tht 1 1 t s, 1 t n 1 t, so th initil oniurtion or ntrin th whil loop is 1 t s t t. B Lmm 2, irst( t,s ) 1 t s so is initiliz to th orrt vlu. Th prorm trmints on n (t s) (lin 6), whr (t s) quls on i t s, n zro othrwis. In othr wors, i t s thn CoolCt trmints on n 1, n othrwis it trmints on n. ll tht this onition hos th two ss o (7). Finll, w point out CoolCt s pliit rquirmnt tht t s >. Th nt two lmms will rss th irst two itrtions o th lorithm. Lmm 3. V t,s t,s whn t 2. Proo. It is s to vri tht V 1,1 1, V 2,1 11, 11, n V 2,2 11, 11. In th irst s th prorm os not ntr th whil loop n in th lst two ss th prorm trmints tr th whil loop s irst itrtion. Lmm 4. I t > 2 thn 2 σ( 1 ), 2 3, n 2 2. Proo. Whn t > 2 th prorm ntrs th whil loop n tr lins 7-1 w hv th ollowin oniurtion 1 t s t + 1 t + 1. Sin [] [t + 1] th prorm ntrs th outr i sttmnt on lin 11. Sin t > 2 it os not ntr th innr i sttmnt on lin 12 n so lins 15 n 16 r ut to iv th ollowin oniurtion 11 t s t + 1 t + 1. Atr lin 17 n lin 18 w hv th ollowin oniurtion 11 t 1 s Sin th nt lin to ut is visit sttmnt w hv 2 11 t 1 s 1. Thror, w hv provn th rsult sin 1 1 t s n σ(1 t s ) 11 t 1 s 1 (1). Th nt lmm plins how th lorithm trmints (th vlus or lst( t,s ) r rll rom Lmm 2). Lmm 5. I t > 1, vr i B t,s, n i is orrt thn { 1 lst(v t,s ) lst( t,s ) t 1 s 1 1 i t s 1 t 1 s 1 i t > s

7 Proo. Whn t s, th onition on th whil loop is < n 1. I k 1 t 1 s 1 1 n k is upt orrtl thn k n 1, so on k is visit th prorm will trmint. Furthrmor, (6) w hv tht i < n 1 or ll i k sin th ssumption ll i B t,s. Whn t > s, th onition on th whil loop is < n. I k 1 t 1 s 1 n k is upt orrtl thn k n, so on k is visit th prorm will trmint. Furthrmor, (6) w hv tht i < n or ll i k sin th ssumption ll i B t,s. Now tht th trm ss o CoolCt hv n ount or, w n ous on th nrl hvior o th lorithm. In prtiulr, 1 t s n 1 t 1 s 1 hv n lt with in Lmm 4 n Lmm 5 rsptivl, so w n onl onsir th hvior o th lorithm on inr strins tht ontin ltmost 1 n t lst on itionl smol ollowin it. In othr wors, w ssum tht 11 p q 1z... whr z {, 1}. From Stion 3 w rll our itrtiv inition or σ() { 111 p q z... i p q n z (11) 1z1 p q 1... i p > q or z 1. (11) Noti tht whn z 1 thn th lt si o (11) n (11) r intil. Thror, w n intrhn thir rols whn th onition o z 1 is stisi. Thus, th onitions in (11) n (11) n quivlntl stt s p q n p > q, rsptivl. In t, th onitions wr oriinll stt this w in (2) n (2); w mk th hn hr sin it optimizs th loi o th rsultin prorm. Anothr w o sttin th quivln is tht i 11 p p thn it os not mttr i w mov th (2p + 3)r smol or th (2p + 4)th smol sin oth r qul to 1. W now r l to omplt this stion with thr lmms. Th irst two lmms orrspon to (11) (ss () n () rsptivl), whil th thir lmm orrspons to (11) (s ()). Lmm 6. Suppos z 1, so tht i 11 p q I i n i r orrt, thn i+1 σ( i ) n i+1 n i+1 r orrt. Proo. From th sttmnt o th lmm, w n ssum tht th urrnt oniurtion pprs low n th prorm just stisi th onition o th whil loop 11 p q p + 2 p + q + 3. Atr utin lins 7-1 th urrnt oniurtion oms 11 p 1 q 1... p + 3 p + q + 4. Sin [] 1 th prorm os not ntr th i sttmnt on lin 11 n so i+1, i+1, n i+1 r qul to thir rsptiv vlus ov. From (11), σ( i ) i+1. Furthrmor, th vlus o i+1 n i+1 r orrt. Lmm 7. Suppos p > q n z, so tht i 11 p q 1... with p > q. I i n i r orrt, thn i+1 σ( i ) n i+1 n i+1 r orrt. Proo. From th sttmnt o th lmm, w n ssum tht th urrnt oniurtion pprs low n th prorm just stisi th onition o th whil loop 11 p q 1... p + 2 p + q + 3. Atr utin lins 7-1 th urrnt oniurtion oms 11 p 1 q... p + 3 p + q p q... Sin [] th prorm ntrs th i sttmnt on lin 11. Sin 2 2 woul impl tht p+q +4 2p + 4 n thus p q, thn th i sttmnt on lin 12 is not ntr. Atr utin lins 15 throuh 18 th oniurtion oms 11 p q At this point th prorm mks th nt visit in lin 21, so i+1, i+1, n i+1 r qul to thir rsptiv vlus ov. From (11), σ( i ) i+1. Furthrmor, th vlu o i+1 is orrt. Finll, th vlu o i+1 is lso orrt sin p >. Lmm 8. Suppos p q n z, so tht i 11 p p I i n i r orrt, thn i+1 σ( i ) n i+1 n i+1 r orrt. Proo. From th sttmnt o th lmm, w n ssum tht th urrnt oniurtion pprs low n th prorm just stisi th onition o th whil loop 11 p p 1... p + 2 2p + 3. Atr utin lins 7-1 th urrnt oniurtion oms 11 p 1 p... p + 3 2p + 4. Sin [] th prorm ntrs th i sttmnt on lin 11. Sin 2 2 th prorm ntrs th i sttmnt on lin 12. Atr utin lin 13 th urrnt oniurtion oms 11 p 1 p... p + 3 2p + 5. At this point th prorm mks th nt visit in lin 21, so i+1, i+1, n i+1 r qul to thir rsptiv vlus ov. From (11), σ( i ) i+1. Furthrmor, th vlu o i+1 is orrt. Howvr, n w rtin tht th vlu o i+1 is orrt? Noti tht th pliitl ispl portion o in th ov oniurtion ontins n qul numr o 1s n s. Hn, th nt smol must 1, n so th vlu o i+1 is lso orrt. Th rsult o Lmms 3-8 is tht CoolCt(t, s) orrtl visits n upts irst( t,s ), n thn orrtl visits n upts vr othr strin in t,s up to n inluin lst( t,s ) tr whih it trmints. Thror, w hv th ollowin thorm. Thorm 3. For ll t s >, w hv V t,s t,s. 5 nkin In this stion w vlop rnkin lorithm tht uss O(n) rithmti oprtions. W will n to know th numr o lmnts in K t,s, whih w not K t,s B t,s 1. Tl 1 shows K t,s or s t 8. Thorm 4. For ll s t, K t,s + 1 t s + 1 ( ) ( t + s t + s t + 1 t t ) ( ) t + s. t + 1

8 Tl 1: Th Ctln( trinl. Th row t, olumn s ntr is K t,s t s+1 t+s ) t+1 t Proo. Ths r wll-known proprtis o th Ctln trinl (Knuth (26), Stnl (1999)). Lt 2 t+s 1 B t,s. W us ρ() to not th rnk o in th list K t,s. Hr is rursiv sription o th rnkin pross; it ollows irtl rom (1). Lt 2 t+s 2. ρ( ) i t+s 1 ρ() K t,s 1 i 1 t 1 s 1 K t 1,s + ρ( ) othrwis. For mpl, ρ(1111) K 4,2 + ρ(111) 8 + ρ(111) 8 + K 3,1 + ρ(11) ρ(11) 1 + K 2,1 1 1 (12) Not tht (12) inors trilin s; th rnk thror pns onl on th positions o th 1s. I 1, 2,..., t r th positions oupi th 1s n q is th minimum vlu or whih q > q, thn (12) n itrt to otin ρ( t ) K q,q q 1 + t jq+1 K j,j j 1. (13) W now show tht thr is ni w to viw th rnkin pross s wlk on rtin intr ltti. r to Fiur 7. Th wlk strts t th uppr lt; h 1 is vrtil stp own n h is horizontl stp to th riht. Th vrtil s r ll, whr th t-th row o vrtil s (ountin rom 1) ts ll s ollows rom lt-to-riht: (no ll), K t,, K t,1,..., K t,t 1. Th ll urthst to th riht in h row is not on n. Fiur 7 illustrts th pth or th itstrin Th squr mrks th npoint o th prt o th pth tht ns t th ltmost 1; i., th strin 1111 in th mpl itstrin. Th rnk o th itstrin is otin summin th lls on th pth tr th squr, in th ll on th to th riht o th on tht prs th squr (th irl ll in th iur), n thn sutrtin 1. Thus ρ( ) To unrnk w rvrs th pross. W us ρ 1 t,s (m) to not th strin B t,s whos rnk in K t,s is m. Suppos, or mpl, tht w wnt th rnk 212 itstrin with t 8 n s 6; i.., ρ 1 (8,6)(212). W strt whr th mpl pth ns. W mov to th lt so lon s th lls th rminin rnk, Fiur 7: nkin thn mov up n rpt. Arrivin t th ol squr, w r t n impss; th rminin rnk is 7, so w hv t to nountr th squr. So w so up n th rnk oms 4, whih is wht rmins i w mk th urrnt lotion (on mov ov th ol squr) th nw squr. Thus ρ 1 (8,6)(212) W lv it to th rr to turn this sription into n lorithm. Wht is th runnin tim o th rnkin lorithm? Lt n t+s. Not tht (12) n (13) involv O(n) itions n othr oprtions. W n voi omputin th ntir tl onl omputin th vlus n lon th pth. First omput K t,s, whih tks O(n) rithmti oprtions. Thn mk us o th ollowin rltions whih n hk usin Thorm 4: 1 + K t 1,s 1 + K t,s 1 (t + 1)(t s) (t s + 1)(t + s) (1 + K t,s) n s(t s + 2) (t s + 1)(t + s) (1 + K t,s). O ours, i mn rnkin/unrnkin oprtions r in prorm thn it will ttr to pr-omput th K t,s tl. 6 Finl mrks For utur rsrh, it woul intrstin to trmin whthr th rsults o this ppr n tn to th nturl /1 rprsnttion o k-r trs, or to orr trs with prsri r squn (Zks & ihrs (1979)). W thnk th rrs or rull rin this ppr n pointin out numr o tpos n pls whr th position oul improv. rns B. Bultn & F. usk (1998), An Es-MK Alorithm or Wll-Form Prnthsis Strins, Inormtion Prossin Lttrs, 68, pp Donl E. Knuth (25), Th Art o Computr Prormmin, Volum 4: Gnrtin ll Comintions n Prtitions, Fsil 3, Aison-Wsl, 15 ps.

9 Donl E. Knuth (25), Th Art o Computr Prormmin, Volum 4: Gnrtin ll Trs; Histor o Comintiontoril Gnrtion, Fsil 4, Aison-Wsl, 12 ps. J. Korsh, P. LFoltt, & S. Lipshutz (23), Looplss Alorithms n Shrör Trs, Intrntionl Journl o Computr Mthmtis, 8, pp J. Lus, D. olnts, n F. usk (1993), On ottions n th Gnrtion o Binr Trs, Journl o Alorithms, 15, pp A.. Mtos, F.A.A. Pinho, A. Silvir-Nto & V. Vjnovszki (1998), On th Looplss Gnrtion o Binr Tr Squns, Inormtion Prossin Lttrs, 68, pp D. olnts (1991), A Looplss Alorithm or Gnrtin Binr Tr Squns, Inormtion Prossin Lttrs, 39, pp F. usk (1979), Simpl omintoril Gr os onstrut rvrsin sulists, 4th ISAAC (Intrntionl Smposium on Alorithms n Computtion), Ltur Nots in Computr Sin, #762, pp F. usk n A. Proskurowski (199), Gnrtin Binr Trs Trnspositions, Journl o Alorithms, 11, pp F. usk & A. Willims (25), Gnrtin Comintions B Pri Shits, Computin n Comintoris, 11th Annul Intrntionl Conrn, COCOON 25, Kunmin, Chin, Auust 16-29, 25, Proins. Ltur Nots in Computr Sin 3595, Sprinr-Vrl. F. usk n A. Willims (28), Th Coolst w to Gnrt Comintions, Disrt Mthmtis, to ppr, 28..P. Stnl (1999)Enumrtiv Comintoris, vol. 2, Cmri Univrsit Prss, Nw York/Cmri, 1999, ii ps..p. Stnl (27), Ctln Anum, vrsion o 2 Jun 27; 61 ps, T. Tkok (1999), O(1) Tim Alorithms or Comintoril Gnrtion Tr Trvrsl, Th Computr Journl, vol. 42, no. 5, pp T. Tkok & S. Violih (26), Comintoril Gnrtion Fusin Looplss Alorithms, In Pro. Twlth Computin: Th Austrlsin Thor Smposium (CATS26), Hort, Austrli. C- PIT, 51. Gumunsson, J. n J, B., Es., ACS V. Vjnovszki & T. Wlsh (26), A looplss twolos Gr-o lorithm or listin k-r Dk Wors, Journl o Disrt Alorithms, Vol. 4, No. 4, pp Wlsh, A Simpl Squnin An nkin Mtho Tht Works On Almost All Gr Cos, Unpulish srh port, Dprtmnt o Mthmtis n Computr Sin, UQAM P.O. Bo 8888, Sttion A, Montrl, Qu, Cn H3C 3P8, 68 ps. T.. Wlsh (23), Gnrtin Gr os in O(1) worst-s tim pr wor, Ltur Nots in Computr Sin 2731, Proins o th 4h Intrntionl Conrn, Disrt Mthmtis n Thortil Computr Sin 23, Dijon, Frn, Jul 7-12, 23, Sprinr-Vrl, Nw York, (23), S. Zks & D. ihrs (1979), Gnrtin Trs n Othr Comintoril Ojts Liorphill, SIAM J. Computin, 8, pp