Analysis of the average depth in a suffix tree under a Markov model

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1 Analysis of the average depth in a sffix tree nder a Markov model Jlien Fayolle, Mark Daniel Ward To cite this version: Jlien Fayolle, Mark Daniel Ward. Analysis of the average depth in a sffix tree nder a Markov model. Conrado Martínez International Conference on Analysis of Algorithms, 2005, Barcelona, Spain. Discrete Mathematics and Theoretical Compter Science, DMTCS Proceedings vol. AD, International Conference on Analysis of Algorithms, pp.95-04, 2005, DMTCS Proceedings. <hal > HAL Id: hal Sbmitted on 2 Ag 205 HAL is a mlti-disciplinary open access archive for the deposit and dissemination of scientific research docments, hether they are pblished or not. The docments may come from teaching and research instittions in France or abroad, or from pblic or private research centers. L archive overte plridisciplinaire HAL, est destinée a dépôt et à la diffsion de docments scientifiqes de nivea recherche, pbliés o non, émanant des établissements d enseignement et de recherche français o étrangers, des laboratoires pblics o privés.

2 2005 International Conference on Analysis of Algorithms DMTCS proc. AD, 2005, Analysis of the average depth in a sffix tree nder a Markov model Jlien Fayolle and Mark Daniel Ward 2 Projet Algorithmes, INRIA, F-7853 Rocqencort, France. jlien.fayolle@inria.fr 2 Department of Mathematics, Prde University, West Lafayette, IN, USA. mard@math.prde.ed In this report, e prove that nder a Markovian model of order one, the average depth of sffix trees of index n is asymptotically similar to the average depth of tries (a.k.a. digital trees bilt on n independent strings. This leads to an asymptotic behavior of (log n/h + C for the average of the depth of the sffix tree, here h is the entropy of the Markov model and C is constant. Or proof compares the generating fnctions for the average depth in tries and in sffix trees; the difference beteen these generating fnctions is shon to be asymptotically small. We conclde by sing the asymptotic behavior of the average depth in a trie nder the Markov model fond by Jacqet and Szpankoski ([4]. Keyords: Sffix trees, depth, average analysis, asymptotics, analytic methods Introdction The sffix tree is a data strctre that lies at the core of pattern matching, sed for example in the lossless data compression algorithm of Lempel and Ziv [7]. Sffix trees are also tilized in bio-informatics to track significant patterns. A thorogh srvey of the se of sffix trees in compter science can be fond in Apostolico ([]. The average depth in a sffix tree of index n + is the average time (nmber of letters read necessary to insert a ne sffix into a tree of index n; an alternate interpretation is the average time to discriminate beteen the crrent sffix and any previos one. A device called a sorce emits letters randomly, and independently of their emission date. In this report, e focs on Markov sorces of order one: the letter emitted at a given time is a random variable obeying a Markov dependency of order one i.e., the distribtion of each letter depends only on the letter actally emitted immediately beforehand. Increasing the order introdces no ne technical challenges comptations only become more intricate and or proof for a Markovian dependency of order can easily be extended to any sorce ith greater Markovian dependency. We assme that the sorce is stationary throghot this report. We denote the stationary probability vector as π = (π i, the transition probability matrix as P = (p i,j, and the probability the sorce starts emitting a text ith the ord as P(. We also introdce the stationary probability matrix Π hose ros are the stationary probability vector π. We make the sal assmptions that the nderlying Markov chain is irredcible and aperiodic. This paper focses on the asymptotic behavior of the average depth in sffix trees. There are to challenging parts in this stdy: first the sffixes on hich the sffix tree is bilt are mtally dependent. For example, if e jst fond the pattern = 0000 in the text, then it sffices to have the letter 0 next, in order to find once more in the text. The trie (a.k.a. digital tree is mch easier to analyze than the sffix tree, becase the strings in a trie are independent from each other. The second challenge lies in the probabilistic dependency beteen symbols (Markov model. The probability that a pattern occrs in the text depends not only on the pattern itself bt also on hat as previosly seen in the text. In Jacqet and Szpankoski ([4], inclsion-exclsion as sed to obtain the asymptotics of the average depth in an (independent trie ith nderlying Markovian sorce. Therefore, it sffices for s to compare the asymptotic average depths in sffix trees against that of independent tries (here the probability model is Markovian in both cases. We prove that, for a Markovian sorce of order one, the average depth of a sffix tree of index n has a similar asymptotic behavior as a trie bilt on n strings. In section 2, e present the main reslts of this paper. We give the precise asymptotics for the average depth in a sffix tree ith an nderlying Markovian sorce. Afterards, e give a sketch of the proof. In section 3 e prove that the atocorrelation polynomial S (z is approximately ith high probability c 2005 Discrete Mathematics and Theoretical Compter Science (DMTCS, Nancy, France

3 96 Jlien Fayolle and Mark Daniel Ward (for of large length. To prove that sffix trees and independent tries have similar average depths, e first derive bivariate generating fnctions for D n and D t n in section 4. Then in sbsections 5. and 5.2 e analyze the difference beteen the generating fnctions for D n and D t n by tilizing complex asymptotics. Ultimately, e conclde in sbsection 5.3 that the depth in a sffix tree has the same average p to the constant term as the depth in a trie. Or method is motivated by the analytic approach of Jacqet and Szpankoski ([3] for the sffix tree nder a memoryless (a.k.a. Bernolli sorce model. Or reslts are proved in the more general context of Markovian sorces. 2 Main Reslts Consider a tree (say it s binary and it shall be so throghot this paper ith n leaves nmbered from to n. The depth D n (i of the leaf nmber i is defined as the length of the path from the root to this leaf. Pick one of the n leaves at random (niformly; the typical depth D n of the tree is the depth of the picked leaf. The random variable D n informs s abot the typical profile of the tree, rather than its height (i.e., the maximm depth. A trie is defined recrsively on a finite set S of infinite ords on A = {0, } as if S = 0, trie(s = if S =,, trie(s\0, trie(s\ else, here S\α represents the set of ords starting ith the letter α hose first letter has been removed. In order to bild a sffix tree, one needs as inpt an infinite string T on A called text and an integer n. We rite T = T T 2 T 3... and then T (i = T i T i+ T i+2... denotes the ith sffix of T (for example, T = T (, namely the entire string itself, is the first sffix. The sffix tree of index n bilt on T is defined as the trie bilt on the set of the n first sffixes of T (namely, T (,..., T (n. In the context of sffix trees, D n (i is interpreted as the largest vale of k sch that there exists j i ith j n for hich T i+k i = T j+k j. Or discssion in this paper primarily concerns the comparison of the average depths in sffix trees verss independent tries. Throghot this discssion, D n alays denotes the typical depth in a sffix tree. In a trie bilt over n independent strings, e let D t n denote the typical depth. In [4], the asymptotic behavior of the average depth for a trie bilt on n independent strings nder a Markovian sorce of order as established sing inclsion-exclsion. In this paper, e prove the analogos reslt for sffix trees. Namely, e establish the asymptotic average depth for a sffix tree ith an nderlying Markovian sorce. Or proof techniqe consists of shoing that D n and D t n have similar asymptotic distribtions. To do this, e estimate the difference of their probability generating fnctions, and sho that the difference is asymptotically small. In order to prove that D n and D t n have asymptotically similar averages, e first discss the atocorrelation of a string. Since a string can overlap ith itself, the bivariate generating fnction D(z, for D n, here marks the depth and z the size, incldes the atocorrelation polynomial S (z. Fortnately, ith high probability, a random string has very little overlap ith itself. Therefore the atocorrelation polynomial of is close to ith high probability. After discssing atocorrelation, e present the bivariate generating fnctions for both D n and D t n, hich e denote by D(z, and D t (z,, respectively. We have D(z, = (z P( D (z 2 and D t (z, = A A zp( ( z + zp( 2, ( here D(z is a generating fnction. Or goal is to prove that the to generating fnctions are asymptotically very similar; it follos from there that Dn t and D n have the same average asymptotically p to the constant term. In order to determine the asymptotics of the difference D(z, D t (z,, e tilize complex analysis. From (, e see that D t (z, has exactly one pole of order 2 for each. Using Roché s theorem, e prove that, for z ρ (ith ρ > and for sfficiently large, the generating fnction D (z has exactly one dominant root of order 2 for each. Therefore D(z, has exactly one dominant pole of order 2 for each. We next se Cachy s theorem and singlarity analysis to qantify the contribtion

4 Analysis of the average depth in a sffix tree nder a Markov model 97 from each of these poles to the difference Q n ( := ( [z n ](D(z, D t (z,. Or analysis of the difference Q n ( ltimately relies on the Mellin transform. We conclde from there that the averages of the depths D n (in sffix trees and Dn t (in independent tries are asymptotically similar. Therefore, D n has mean h log n pls some flctations. Specifically, Theorem For a sffix tree bilt on the first n sffixes of a text prodced by a sorce nder the Markovian model of order one, the average typical depth is asymptotically E(D n = ( log n + γ + h 2 h 2h H + P (log n + O ( n ɛ, (2 for some positive ɛ, here γ is the Eler constant, h is the entropy of the Markov sorce, h 2 is the second order entropy, H is the stationary entropy and P is a fnction flctating arond zero ith a very small amplitde. In particlar h := π i p i,j log p i,j and H := π i log π i. (3 i,j A 2 i A 2 3 Atocorrelation The depth of the ith leaf in a sffix tree can be constred in the langage of pattern matching as the length of the longest factor starting at a position i in the text that can be seen at least once more in the text. Hence D n (i k means that there is at least another pattern in the text T starting ith T i+k i While looking for the longest pattern in T that matches the text starting in position i, there is a possibility that to occrrences of this longest pattern overlap; this possible overlap of the pattern ith itself cases complications in the enmeration of occrrences of in a text. The phenomenon exhibited hen overlaps ith itself is called atocorrelation. To accont for this overlapping, e se the probabilistic version of the atocorrelation polynomial. For a pattern of size k, the polynomial is defined as: k S (z := [ k i = i+ k ]P(k i+ k k i z i. (4 i=0 For simplicity, e introdce c i = [[ k i = i+ k ] here [.] is Iverson s bracket notation for the indicator fnction. We say there is an overlap of size k i if c i =. This means that the sffix and the prefix of size k i of coincide. Graphically, an overlap of a pattern (hite rectangles looks like this: here the to black boxes are the matching sffix and prefix. For example = has overlaps of sizes 3, 6 and 9; note that (here = 9 is alays a valid overlap since the pattern alays matches itself. We define a period of a pattern as any integer i for hich c i = (so a pattern often has several periods, the minimal period m( of is the smallest positive period, if one exists (if has no positive period for, e define m( = k. We no formlate precisely the intition that, for most patterns of size k, the atocorrelation polynomial is very close to. It stems from the fact that the sm of the probabilities P( over all loosely correlated patterns (patterns ith a large minimal period of a given size is very close to. Lemma There exist δ <, ρ > ith ρδ <, and θ > 0, sch that for any integer k A k [ S (ρ (ρδ k θ ]P( θδ k. (5 Proof: Note that S (z has a term of degree i ith i k if and only if c i =. If c i = the prefix of size i of ill repeat itself flly in as many times as there is i in k, hence knoing c i = and the first i letters of allos s to describe flly the ord. Therefore, given a fixed... i, there,

5 98 Jlien Fayolle and Mark Daniel Ward is exactly one ord i+... k sch that the polynomial S (z has minimal degree i k. We let p i,j = p i, j, π i = π i, and p = max ( p i,j, π i. Then, for fixed j and k, i,j 2 j i= A k [m( = i]p( = j i= j i=,..., i A i π p,2 p i,i,..., i A i π p,2 p 2,3 p i,j p k i, i+,..., k A k i [m( = i] p i,i+ p k,k bt e can factor p k i otside the inner sm, since it does not depend on,..., i. Next, e observe that,..., i A i π p,2 p 2,3 p i,i = ΠP i =, so A k [S (z has minimal degree j ]P( and this holds hen j = k/2. So j i= p k i pk j p, A k [all terms of S (z have degree > k/2 ]P( p k/2 p = θδk. (7 Remark that, if all terms of S (z have degree > k/2, then (6 k S (ρ i= k/2 (ρp i ρ k p k/2 + p = (ρδ k θ. (8 We select δ = p, θ = ( p and some ρ > ith δρ < to complete the proof of the lemma. The next lemma proves that, for sfficiently large and for some radis ρ >, the atocorrelation polynomial does not vanish on the disk of radis ρ. Lemma 2 There exist K, ρ > ith pρ <, and α > 0 sch that for any pattern of size larger than K and z in a disk of radis ρ, e have S (z α. Proof: Like for the previos lemma, e split the proof into to cases, according to the index i of the minimal period of the pattern of size k. Since the atocorrelation polynomial alays has c 0 =, e rite k S (z = + c j P(k j+ k k j z j. (9 j=i We introdce ρ > sch that pρ <. Therefore, if i > k/2, then k S (z c j P(k j+ k k j z j (pρ i pρ, (0 j=i in z ρ. Bt since i > k/2 and pρ <, e get S (z (pρ k/2 pρ. ( We observe that, for some K sfficiently large, any pattern of size larger than K satifies S (z α and this loer bond α is positive. We recall that if c i =, the prefix of size i of ill repeat itself flly in as many times as there is i in k i.e., q := k/i times, the remainder ill be the prefix v of of size r := k k/i i, hence = k/i v. We also introdce the ord v sch that vv = (of length t := i r = i k + k/i i.

6 Analysis of the average depth in a sffix tree nder a Markov model 99 If i k/2, e make the atocorrelation polynomial explicit: S (z = + P(v v k z i + P(v v k z 2i + + P(v q 2 v k z i(q S v (z. (2 Overall, e can rite P(v j v k = A j+ here A = p k,v p v,v 2... p v t,v... p v r,v r is the prodct of i transition probabilities, bt since k = v r e obtain S (z = + Az i + (Az i (Az i q S v (z = (Azi q Az i + (Az i q S v (z. (3 Then e provide a loer-bond for S (z : S (z (Az i q Az i (Az i q S v (z (pρ i(q + (pρ i (pρ i(q S v (z (pρ i(q + (pρ i (pρ i(q pρ. We have pρ < so that (pρ k tends to zero ith k. Since i(q is close to k (at orse k/3 if = v for some K 2 sfficiently large and patterns of size larger than K 2, only the term ( + (pρ i remains. Finally, e set K = max{k, K 2 }. 4 On the Generating Fnctions The probabilistic model for the random variable D n is the prodct of a Markov model of order one for the sorce generating the strings (one string in the case of the sffix tree, n for the trie, and a niform model on {,..., n} for choosing the leaf. Hence, if X is a random variable niformly distribted over {,..., n}, and T a random text generated by the sorce, the typical depth is D n := n D n (i(t [X = i]. i= For the rest of the paper, the t exponent on a qantity ill indicate its trie version. Or aim is to compare asymptotically the probability generating fnctions of the depth for a sffix tree (namely, D n ( := k P(D n = k k and a trie (namely, D t n( := k P(Dt n = k k. We provide in this section an explicit expression for these generating fnctions and their respective bivariate extensions D(z, := n nd n(z n and D t (z, := n ndt n(z n. We first derive D t n(. Each string from the set of strings S is associated to a niqe leaf in the trie. By definition of the trie, the letters read on the path going from the root of the trie to a leaf form the smallest prefix distingishing one string from the n others. We choose niformly a leaf among the n leaves of the trie. Let A k denote the prefix of length k of the string associated to this randomly selected leaf. We say that D t n < k if and only if the other n texts do not have as a prefix. It follos immediately that P(D t n(i < k = A k P(( P( n, conseqently D t n( = A P(( P( n and D t (z, = A for < and z <. The sffix tree generating fnction is knon from [5] and [3]: for < and z < D(z, = zp( ( z + zp( 2, (4 (z P( D (z 2, (5 A here D (z = ( zs (z + z P(( + ( zf (z and for z < P Π, F (z := Π π n 0(P n+ z n = [(P Π(I (P Πz π m, ] m,. (6

7 00 Jlien Fayolle and Mark Daniel Ward 5 Asymptotics 5. Isolating the dominant pole We prove first that for a pattern of size large enogh there is a single dominant root to D (z. Then e sho that there is a disk of radis greater than containing each single dominant root of the D (z s for any of size big enogh bt no other root of the D (z s. Lemma 3 There exists a radis ρ > and an integer K sch that for any of size larger than K, D (z has only one root in the disk of radis ρ. Proof: Let be a given pattern. We apply Roché s Theorem to sho the niqeness of the smallest modls root of D (z. The main condition e need to flfill is that, on a given circle z = ρ, f(z := ( zs (z > P(z k ( + ( zf (z =: g(z. (7 The fnction f is analytic since it is a polynomial, F is analytic for z < P Π (here P Π >, so g is too. For patterns of a size large enogh, P(z k ill be small enogh to obtain the desired condition. The main isse is the bonding from above of F (z on the circle of radis ρ. We note d = min a A π a ; this vale is positive (otherise a letter old never occr and since (P Π n+ = P n+ Π (remember that P Π = ΠP = Π and ΠΠ = Π, e have: F (z d n 0(P n+ Πz n [P n+ ] k, [Π] k, z n (8 d br n+ ρ n br d d n 0 k, n 0 rρ, (9 here b and r are constants (independent of the pattern ith 0 < r < and ρ sch that rρ <. Let K be an integer and ρ some radis satisfying Lemma 2, e ask ρ to be smaller than ρ so that pρ < and S (z α on z = ρ and for K. There exists K large enogh sch that for any k > K e verify the condition (pρ k ( + ( + ρ br d < α(ρ. (20 rρ On a disk of sch radis ρ and for k > max{k, K }, the assmptions of Roché s Theorem are satisfied, conseqently (f + g(z = D (z has exactly as many zeros in the centered disk of radis ρ as f(z, namely one zero, since S (z does not vanish. Frthermore the assmptions of Roché s Theorem are satisfied on z = ρ for any pattern of size larger than K := max{k, K }. So for any ith K, D (z has exactly one root ithin the disk z = ρ. 5.2 Compting resides The se of Roché s Theorem has established the existence of a single zero of smallest modls for D (z, e denote it A. We kno by Pringsheim s Theorem that A is real positive. Let also B and C be the vales of the first and second derivatives of D (z at z = A. We make se of a bootstrapping techniqe to find an expansion for A, B, and C along the poers of P( and e obtain A = + P( S ( + O(P2 (, [ B = S ( + P( k F ( 2 S ( S ( C = 2S ( + P( ] + O(P 2 (, and [ 3 S ( S ( + k(k 2F ( 2kF ( ] + O(P 2 (. We no compare D n ( and Dn( t to conclde that they are asymptotically close. We therefore introdce to ne generating fnctions ( z Q n ( := D 2 (z (D n( Dn( t and Q(z, := nq n (z n = P( n 0 A (2 z ( z( P( 2.

8 Analysis of the average depth in a sffix tree nder a Markov model 0 We apply Cachy s Theorem to Q(z, ith z rnning along the circle centered at the origin hose radis ρ as determined in 5.. There are only three singlarities ithin this contor: at z = 0, at A, and at ( P(. In order to jstify the presence of the third singlarity ithin the circle, e note that the condition (20 implies P(ρ < P(ρ k < (pρ k ( + ( + ρ br d < α(ρ < (ρ, (22 rρ since α is taken smaller than one. Ths ( P( is smaller than the radis ρ. For any of a size larger than K, e have I (ρ, := ( z P( 2iπ z =ρ z n+ D 2 (z z ( z( P( 2 dz = P(f( ( = Res(f(z; 0 + Res(f(z; A + Res f(z; P( ( ( ( = nq n ( + P( Res + Res z z n+ D 2 (z ; A z z n+ ( z( P( 2 ;. P( (23 Since z/( z( P( 2 is analytic at z = A it does not contribte to the reside of f(z at A and can be discarded in the comptation of the reside. The same is tre for the /D 2 (z part in the reside of f(z at z = /( P(. We compte the reside at A sing the expansion e fond for B and C throgh bootstrapping. We set k = to simplify the notation, and by a Taylor expansion near A e obtain ( z k (n+ Res D 2 (z ; A = A n ( (n + B 2 A C B 3. (24 In order to compte the reside at z = ( P(, e se the general formla [z n ]( az 2 = (n + a n ths [z ] z n ( z( P( 2 = [zn ] ( z( P( 2 = n( P(n. (25 Before e proceed to get the asymptotic behavior of Q n (, the folloing technical lemma is very sefl. Lemma 4 For any fnction f defined over the patterns on A, and any y e have A k P(f( y + f max P({ A k : f( > y}, here f max is the maximm of f over all patterns of size k. Proof: For the patterns of size k ith f( > y, f max is an pper bond of f(; the probability of the other patterns is smaller than. We also note that S (ρ ( pρ and D (z = O(ρ k for z ρ. Ths (details are omitted, e obtain the folloing bond for the sm of I (ρ, over all patterns of size k: I (ρ, = O((δρ k ρ n A k There are only finitely many patterns ith < K ; these terms provide a contribtion of at most O(B n to Q n ( for some B >. We have jst proved the folloing Lemma 5 For some β >, and for all β, there exists B > for hich e have Q n ( = ( ( (n + P( A n n B 2 C A A B 3 n( P( n + O(B n. (26

9 02 Jlien Fayolle and Mark Daniel Ward 5.3 Asymptotic behavior of Q n ( Lemma 6 For all β sch that < β < δ, there exists a positive ɛ sch that Q n ( = O(n ɛ niformly for all β. Proof: For n large enogh, the dominant term in eqation 26 is Q n ( = ( A n 2 P( B 2 ( P( n + O(n. (27 A We introdce the fnction [ ] [ A x 2 f (x = B 2 ( P( x A 2 B 2 ] exp( x, P( and perform a Mellin transform (see Flajolet, Gordon, and Dmas [2] for a thorogh overvie of this techniqe on P(f (x. It old have seemed natral to define f (x by its first term bt for simplicity s sake e force a O(x behavior for f (x near zero by sbtracting some vale. By an application of Lemma 4, ith y = ( δ k for some δ <, e prove that the sm is absoltely convergent for β. Hence the Mellin transform f (s, of the sm is defined and e have f (s = Γ(s ( (log A s A 2 B 2 ( log( P( s and f (s, = P( P(f (s. f (s, is analytical ithin an open strip < R(s < c for some positive c. In order to do so e split the sm over all patterns into to parts. For the patterns ith I(sP( small, e se an expansion of f(s and then apply Lemma 4. We make sre these patterns do not create any singlarity. For any given s there are only finitely many patterns ith I(sP( large so their contribtion to the sm f (s,, althogh each is individally large, do not create any singlarity. Therefore, e are able to take the inverse Mellin transform of f (s,, and since there is no pole in the strip e obtain the stated reslt. This last reslt is sfficient to prove or initial claim. By definition Dn( t = D n ( =, ths e have Q n ( = ( (D n( Dn( t Dn ( D n ( = Dt n( D t n(. (28 We have D n( = E(D n and (D t n ( = E(D t n, therefore hen tends to e obtain E(D t n E(D n = O(n ɛ. (29 This means that asymptotically the difference beteen the to averages is no larger than O(n ɛ for some positive ɛ. 6 Conclsion We have shon that the average depths of tries and sffix trees behave asymptotically likeise for a Markov model of order one. This reslt can be extended to any order of the Markov model. An extended analysis shold yield analogos reslts for both the variance and the limiting distribtion of typical depth. A normal distribtion is then expected for sffix trees. In the ftre, e also hope to extend or reslts to the more general probabilistic sorce model introdced by Vallée in [6]. Acknoledgements The athors thank Philippe Flajolet and Wojciech Szpankoski for serving as hosts dring the project. Both athors are thankfl to Wojciech Szpankoski for introdcing the problem, and also to Philippe Flajolet and Mireille Régnier for sefl discssions.

10 Analysis of the average depth in a sffix tree nder a Markov model 03 References [] Alberto Apostolico. The myriad virtes of sffix trees. In A. Apostolico and Z. Galil, editors, Combinatorial Algorithms on Words, volme 2 of NATO Advance Science Institte Series. Series F: Compter and Systems Sciences, pages Springer Verlag, 985. [2] Philippe Flajolet, Xavier Gordon, and Philippe Dmas. Mellin transforms and asymptotics: Harmonic sms. Theoretical Compter Science, 44( 2:3 58, Jne 995. [3] P. Jacqet and W. Szpankoski. Analytic approach to pattern matching. In M. Lothaire, editor, Applied Combinatorics on Words, chapter 7. Cambridge, [4] Philippe Jacqet and Wojciech Szpankoski. Analysis of digital tries ith Markovian dependency. IEEE Transactions on Information Theory, 37(5: , 99. [5] Mireille Régnier and Wojciech Szpankoski. On pattern freqency occrrences in a Markovian seqence. Algorithmica, 22(4:63 649, 998. [6] Brigitte Vallée. Dynamical sorces in information theory: Fndamental intervals and ord prefixes. Algorithmica, 29(/2: , 200. [7] Jacob Ziv and Abraham Lempel. A niversal algorithm for seqential data compression. IEEE Transactions on Information Theory, IT-23: , 977.

11 04 Jlien Fayolle and Mark Daniel Ward