Combinatorics and Random Generation. Dominique Gouyou-Beauchamps

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1 Algorithms Semiar , F. Chyzak ed., INRIA, 2003, pp Available olie at the URL Combiatorics ad Radom Geeratio Domiique Gouyou-Beauchamps LRI, Uiversité Paris-Sud Frace March 18, 2002 Summary by Nicolas Boicho Abstract We preset a overview of differet techiques to radomly ad uiformly geerate combiatorial objects. 1. Motivatios ad Hypotheses Oe of the goals of a combiatorist is to recogize, to eumerate, ad to geerate objects of differet combiatorial classes. Here we preset several methods to radomly ad uiformly geerate objects of a give class. This has applicatios i simulatio i geeral: image sytheses, statistical physics, geomic, program testig, algorithms aalyses, etc. I geeral, we wat to geerate a object of size such that the probability that a object appears is the same for all objects of size. There are several ways to measure the complexity of a geeratig algorithm. The first oe is to cout the umber of calls to the RANDOM fuctio. This fuctio returs a floatig-poit umber betwee 0 ad 1. Aother way to measure the complexity is to cout the umber of arithmetic operatios o floatig-poit umbers or o itegers a call of the RANDOM fuctio is cosidered as a arithmetic operatio. This measure is called the arithmetic complexity. Sice the geerated objects ca be huge up to 10 7 or 10 8 ad the maipulated umbers are as large as a or! hece coded with O or O log bits, it also makes sese to cout the umber of operatios o sigle bits. This is the bit complexity. I order to compute some objects of large size, the time complexity of efficiet algorithms is usually O or O log. 2. The Predecessors Nijehuis ad Wilf [7] were the first oes to propose two types of geeratio algorithms: NEXT: with a total order o objects of size ad a give object of the family, compute the ext oe i the order. Geerally, these algorithms have a costat average time complexity; RANDOM: we select radomly ad uiformly objects of size of the family. Here are two examples of these types of algorithms. Here ad throughout the remaider of the text, [] deotes the set {1,..., }. Example Permutatios of [], algorithms NEXTPER ad RANPER [7]. NEXTPER uses the otio of sub-exceedig fuctio: a fuctio f from [] to [] is sub-exceedig is ad oly if for each i [], 1 fi i. It is obvious that the umber of sub-exceedig fuctios from [] to [] is! oe possibility for f1, two for f2, ad so o.

2 178 Combiatorics ad Radom Geeratio A clever order o sub-exceedig fuctios allows us to trasform a permutatio ito the ext oe with few operatios. The average cost of a trasformatio is O1 steps. Iput: σ = σ 1, σ 2,..., σ, a permutatio of S ad its sigature s i.e., the umber of couples i, j such that σ i > σ j ad i < j. Output: ext permutatio. if s = 1 the s 1; switch σ 1 ad σ 2 ad exit else s 1; i 0; t 0 loop d 0; i i + 1 for j from 1 to i do if σ j > σ i+1 the d d + 1 ed if ed for t t + d if t is odd ad d < i the fid i σ = σ 1, σ 1,..., σ i the largest umber less tha σ i+1 ; switch this umber with σ i+1 ad exit ed if if t is eve ad d > 0 the fid i σ = σ 1, σ 1,..., σ i the smallest umber greater tha σ i+1 ; switch this umber with σ i+1 ad exit ed if ed loop ed if Figure 1. Algorithm NEXTPER. To compute a radom permutatio of [], the algorithm is quite simple see Algorithm 2. This algorithm is icremetal. This meas that after m steps, for each m, the algorithm geerates a radom permutatio of [m]. Cosiderig this algorithm, we ca see that the umber of calls of the fuctio RANDOM is. The arithmetic complexity is also liear. Sice this algorithm works with itegers less tha, the bit complexity is O log. σ 1 1 for i from 2 to do σ i i k RANDOM i switch σ k ad σ i ed for Figure 2. Algorithm RANPER. Example Subsets of size k of a set of size, algorithms NEXTKSB ad RANKSB. I this case, RANKSB is more difficult tha NEXKSB. For NEXTKSB, it is possible to use the lexicographic order. If k < /2, less tha 2 operatios are ecessary to obtai the ext subset with k elemets.

3 D. Gouyou-Beauchamps, summary by N. Boicho 179 If k > /2, we apply the algorithm o subsets of k elemets. For RANDKSB, it is more complicated because k memory cells are eeded to store the subsets with k elemets. For this purpose, there exists a rejectio algorithm with a Ok average complexity [7]. 3. Ad Hoc Algorithms For some classes of objects, geeral methods do ot work or are ot efficiet. Hece, it is ecessary to develop ad hoc algorithms. I this sectio we preset several algorithms that geerate radom complete biary trees with 2 edges Rémy s Algorithm. Rémy s Algorithm [8] uses the fact that complete biary trees are i bijectio with well-formed paretheses words or Dyck words o the alphabet A = {x, x}. The equatio of the o-commutative geeratig series of this laguage is D = ɛ + xd xd. The Dyck words of legth 2 are eumerated by the Catala umbers: D A 2 = =: C. Hece C eumerates the complete biary trees with + 1 leaves, ier vertices, ad 2 edges. For a complete biary tree T with 2 edges, we have ways to choose oe edge if we admit that there is a virtual edge that goes to the root. The, we ca choose a orietatio left or right 2 choices. We place a ew vertex i the middle of the chose edge ad we add a ew edge to this vertex o the left or o the right depedig o the chose orietatio. If it is the virtual edge, we place above the root a reversed chevro, so two edges, liked to the root by the right leaf or the left leaf depedig of the chose orietatio see Figure 3. We obtai a complete biary tree T with edges with a poited leaf the ew added leaf. This tree T has + 2 leaves. So, there are + 2 ways to poit it, i other words, there are + 2 ways to obtai it from a tree with + 1 leaves with the described process. 22+1C 2+2C +1 Figure 3. Rémy s costructio. We just proved bijectively ad gave a combiatorial iterpretatio of the obvious recurrece relatio C = + 2C +1. We also proved that this process geerates after m steps a radom tree of size m i the class of trees of size m. By recursio, the probability of T is 1/C. The probability of a poited tree T is 1/ C. If we call T the tree obtaied from T while forgettig the poitig, the the probability of the tree T to be geerated is +2/ 22+1C ad so 1/C +1. Note that this algorithm is icremetal. Aother advatage of this algorithm is that it maipulates umbers of order O. Moreover it computes i liear time ad memory for fixed-size arithmetic operatios.

4 180 Combiatorics ad Radom Geeratio 3.2. Algorithm based o the cyclic lemma. There are other algorithms that ca be built from a combiatorial iterpretatio of the idetity 2 + 1C = 2+1 satisfied by Catala umbers. For this idetity we use the cyclic lemma or Raey s lemma [6, p ]: Lemma 1. A word f o the alphabet A = {x, x} composed of letters x ad + 1 letters x has oly oe factorizatio f = f f with f ɛ i the possibilities such that f f represets a complete biary tree with 2 edges i.e., a Dyck words followed with a letter x. I this case, we start from a radom word composed of letters x ad + 1 letters x it is easy to build such a word sice it correspods with a subset of elemets of a set of elemets. The we look for the uique possible factorizatio. Oe ca remark that this algorithm is ot icremetal. Aother idetity we ca use is + 1C = 2, proved by the Catala factorizatio [3] Step-by-step radom geeratio of Dyck words. Let L be a laguage o a alphabet A. Let L be the set of words of L of legth. For a word w i L, a letter a i A, ad a iteger, let us defie pw, a, as the ratio of the umber of words i L begiig with wa over the umber of words i L begiig with w. Usig this fuctio, it is possible to geerate a word uiformly: w ɛ while w < do a a radom letter with probability pw, a, w wa ed while Figure 4. Algorithm to compute a uiform radom word. This method ca be efficietly applied to geerate Dyck words, ad therefore complete biary trees. For this purpose, let us assume that we have geerated a left factor w of a Dyck word o the alphabet A = {x, x}. This word is composed of p letters x ad q letters x with p+q = 2 m 2, p q = h 0 ad such that m ad h have the same parity. The umber of Dyck words begiig with p letters x ad q letters x is equal to the umber of left factors of Dyck words with p letters x ad q letters x times the umber of left factors of Dyck words of legth m with h more x tha x: F h,m = h + 1 m + 1 m + 1 m h/2 By iductio we suppose that w is selected such that all Dyck words ww have probability 1/C to appear. The probability of w is F,m /C. Now a letter x is selected with probability h+2 m h h+1 2m ad a letter x is selected with probability h m+h+2 h+1 2m. With such probabilities, the probability of the left factor wx is equal to the probability of the left factor w to appear times the probability of the letter x: h+2 m h h+1 m+1 h+2 m h+1 2m m+1 m h/2 m m h 2/2 = = F h+1,m 1 C C C Similarly, the probability for the left factor w x is equal to the probability of w to be selected times the probability of the letter x to be selected:. h m+h+2 h+1 m+1 h m h+1 2m m+1 m h/2 m m h/2 = = F h 1,m 1. C C C We set the expected probabilities. This proves the uiformity of the distributio.

5 4. Rejectio Algorithms D. Gouyou-Beauchamps, summary by N. Boicho 181 Assume we wat to uiformly geerate a object of a set S 1. The idea is to uiformly geerate a object e i a set S 2 such that S 1 S 2. If e S 1 the we keep e, otherwise we reject e ad we try to select aother oe. This method assumes tha we kow how to select efficietly objects i S 2, that the ratio S 2 / S 1 is ot too big, ad that we ca test membership to S 1 efficietly Left factor of Motzki words. Here is a example of rejectio algorithm that computes left factor of Motzki words [2]. Motzki s laguage M is composed of words f o the alphabet A = {x, x, a} such that the subset of f composed oly of letters x ad x is a Dyck word. The idea is to geerate a word, letter by letter, with probability 1/3 for each letter util it reaches the legth or util there is more letters x tha letters x. I this last case, the partially built word is rejected ad the algorithm is started agai. To evaluate the complexity of this algorithm, we eumerate the average umber of letters geerated before a word of legth i F is obtaied. The laguage M satisfies the equatio M = ɛ + am + xm xm. So, the geeratig fuctio Mt of M satisfies the equatio Mt = 1 + tmt + t 2 Mt 2 ad is equal to 1 t 1 + t1 3t /2t 2. The laguage F of left factors of Motzki words satisfies the equatio F = M +MxF. So, the geeratig fuctio F t 1+t of F satisfies the equatio F t = Mt + tmtf t ad is equal to t /2t. Let R be the laguage of rejected words by the algorithm ad let F p be the laguage of words of F of legth less or equal to p. Oe ca remark that R = F 1 A F ad that lim R = M x. The algorithm geerates words of the laguage G = F + R G. The geeratig fuctio Gt of G is equal to Gt = ft 1 Rt where R t is the geeratig fuctio of R ad where F t is the geeratig fuctio of F. Let P G t the probability geeratig fuctio of G, i.e., the geeratig fuctio where each word is weighed by its probability; the average legth γg of words of G is P G 1. I formulas: P G t = Gt/3 = f t 3 1 R t/3 ad P Gt = f t R t/3 + f t R t/ R t/3 2. Oe ca remark that A = F i=1 Ri A i where R i = R A i. If we ote r i the cardiality of R i, the 3 = f + i=1 ri 3 i ad so f /3 = 1 R 1/3 ad we get P G = + R 1/3. But, as R 1/3 = i=1 iri 3 i+1 λf 1 F = i=0 i kf i 1 f i [t ] 1 P G1 1 t F t/3 = [t ] F t/3 3 i =, [t ] F t/3 = = kλr, we get P G 3 1 = + f λr where λr = i=0 i f i f 3 i 3. Usig both equatios above, we obtai: 3 π + O 1 1, [t ] 1 t F t/3 = O1. π Therefore whe goes to ifiity, the average umber of selected letters coverges to 2. Alai Deise has exteded this rejectio method to the f g-laguages [4] Motzki words. Let us cosider a Motzki word of legth. This is a word of paretheses o {x, x} of legth 2i with 2i letters a itertwied. The Motzki words of legth are eumerated by Motzki umbers m = /2 i=0 2i Ci where C i is a Catala umber. To select a Motzki word of legth uiformly, the problem is to decide the umber i of letters x i the word. The the problem is easy to solve, as we kow how to select a Dyck word legth 2i ad we kow how to select 2i positios i possible oes where the letters of this word will be iserted. f

6 182 Combiatorics ad Radom Geeratio The probability that a word has i letters x ad i letters x is 2i Ci /m. To geerate i with the appropriate distributio, usig the formula, it is ecessary to maipulate huge umbers. The idea of Lauret Aloso [1] is to approximate this distributio by a larger distributio, easy to simulate. This idea ca also be foud i the Luc Devroye s book [5]. Assume that we have v + 1 boxes umbered from 0 to v. Box i cotais N i black balls N = v i=0 N i. This is the iitial distributio. For each i = 0, 1,..., v, we add B i white balls ito box i. Globally, there are D balls D = v i=0 D i, D i = N i + B i. This is a easy distributio to compute. We select box i with probability D i /D. The we cosider that this choice is correct with probability N i /D i the probability to select a black ball i the box i. If this choice is ot correct, we choose aother box. Otherwise the iteger i is defiitively selected. The probability to select the box i with such process is N i /N ad the average umber of trials before a box is defiitively selected is D/N. For Motzki s laguage, L. Aloso [1] takes v = /3, ad N i = D i = {! i 1! i! 2i+2! for i [ 1, 1 + /2 ], 0 otherwise,! + 1/3! i! + 1 i + 1/3!. Note that D i is maily a biomial coefficiet times a costat. We ca show that whe i [ 1, 1 + /2 ], we have N i /D i = a c / b c 1 where the values of a, b, ad c deped oly of the positio of i from + 1/ The choice of box i with probability D i /D ca be doe by geeratig a sequece of /3 bits ad cosiderig the sum of geerated bits. The validity test of the choice of a box with a probability N i /D i = a c / b c ca be doe by choosig c itegers i the iterval [ 1, b ] ad verifyig that they are less tha a. We ca compute that D /2 π, M /2 π. This implies the result D/M 3. Theorem 1. [1] The average complexity of the radom geeratig algorithm of Motzki words is liear. Bibliography [1] Aloso Lauret ad Schott Reé. Radom geeratio of trees. Kluwer Academic Publishers, Bosto, MA, 1995, x+208p. Radom geerators i computer sciece. [2] Barcucci E., Pizai R., ad Sprugoli R.. The radom geeratio of directed aimals. Theoretical Computer Sciece, vol. 127, 2, 1994, pp [3] Chotti Lauret ad Cori Robert. Ue preuve combiatoire de la ratioalité d ue série géératrice associée aux arbres. RAIRO Iformatique Théorique, vol. 16, 2, 1982, pp [4] Deise Alai. Méthodes de géératio aléatoire d objets combiatoires de grade taille et problèmes d éumératio. Thèse, Uiversité Bordeaux I, [5] Devroye Luc. Nouiform radom variate geeratio. Spriger-Verlag, New York, 1986, xvi+843p. [6] Lothaire M.. Combiatorics o words. Addiso-Wesley, Readig, Mass., 1983, Ecyclopedia of Mathematics ad its Applicatios, vol. 17, xix+238p. Collective work uder a pseudoym. [7] Nijehuis Albert ad Wilf Herbert S.. Combiatorial algorithms. Academic Press, New York, 1978, secod editio, Computer Sciece ad Applied Mathematics, xv+302p. For computers ad calculators. [8] Rémy Jea-Luc. U procédé itératif de déombremet d arbres biaires et so applicatio à leur géératio aléatoire. RAIRO Iformatique Théorique, vol. 19, 2, 1985, pp

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