A New Binomial Recurrence Arising in a Graphical Compression Algorithm

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1 A New Binomial Recurrence Arising in a Graphical Compression Algorithm Yongwoo Choi, Charles Knessl, Wojciech Szpanowsi To cite this version: Yongwoo Choi, Charles Knessl, Wojciech Szpanowsi. A New Binomial Recurrence Arising in a Graphical Compression Algorithm. Broutin, Nicolas and Devroye, Luc. 3rd International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms AofA 1), 01, Montreal, Canada. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AQ, 3rd Intern. Meeting on Probabilistic, Combinatorial, and Asymptotic Methods for the Analysis of Algorithms AofA 1), pp.1-1, 01, DMTCS Proceedings. <hal > HAL Id: hal Submitted on 11 Sep 015 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 AofA 1 DMTCS proc. AQ, 01, 1 1 A New Binomial Recurrence Arising in a Graphical Compression Algorithm Yongwoo Choi 1, Charles Knessl and Wojciech Szpanowsi 3 1 J. Craig Venter Institute, USA Department of Mathematics, Statistics & Computer Science, University of Illinois at Chicago, USA 3 Department of Computer Science, Purdue University, USA In a recently proposed graphical compression algorithm by Choi and Szpanowsi 009), the following tree arose in the course of the analysis. The root contains n balls that are consequently distributed between two subtrees according to a simple rule: In each step, all balls independently move down to the left subtree say with probabilityp) or the right subtree with probability1 p). A new node is created as long as there is at least one ball in that node. Furthermore, a nonnegative integer d is given, and at level d or greater one ball is removed from the leftmost node before the balls move down to the next level. These steps are repeated until all balls are removed i.e., after n + d steps). Observe that whend = the above tree can be modeled as a trie that storesnindependent sequences generated by a memoryless source with parameter p. Therefore, we coin the name n,d)-tries for the tree just described, and to which we often refer simply as d-tries. Parameters of such a tree e.g., path length, depth, size) are described by an interesting twodimensional recurrence in terms of n and d) that to the best of our nowledge was not analyzed before. We study it, and show how much parameters of such a n,d)-trie differ from the corresponding parameters of regular tries. We use methods of analytic algorithmics, from Mellin transforms to analytic poissonization. Keywords: Digital trees, Mellin transform, poissonization, graph compression 1 Introduction In 1] an algorithm was described to compress the structure of a unlabeled) graph. The main idea behind the algorithm is quite simple: First, a vertex of a graph, say v 1, is selected and the number of neighbors of v 1 is stored in a binary string. Then the remaining n 1 vertices are partitioned into two sets: the neighbors of v 1 and the non-neighbors of v 1. This process continues by selecting randomly a vertex, say v, from the neighbors of v 1 and storing two numbers: the number of neighbors of v among each of the above two sets. Then the remaining n vertices are partitioned into four sets: the neighbors of both v 1 andv, the neighbors ofv 1 that are non-neighbors ofv, the non-neighbors ofv 1 that are neighbors of This author was supported by NSA Grant H This wor was supported in part by the NSF Science and Technology Center for Science of Information Grant CCF , NSF Grants CCF and DMS , AFOSR Grant FA , NSA Grant H , and the MNSW Grant N c 01 Discrete Mathematics and Theoretical Computer Science DMTCS), Nancy, France

3 Yongwoo Choi, Charles Knessl and Wojciech Szpanowsi Fig. 1: A 6,1)-trie with six balls and d = 1, in which the deleted ball is shown next to the node where it was removed. v, and the non-neighbors of bothv 1 andv. This procedure continues until all vertices are processed. In the Erdős-Rényi model, a random graph has any pair of vertices connected by an edge with probability p. It is proved in 1] that for large n our algorithm optimally compresses any unlabeled) graph generated by the Erdős-Rényi model and, in fact, it wors well in practice even for graphs not generated by the Erdős-Rényi model). To establish this asymptotic optimality result, an interesting tree was used in the construction described next. The root of such a tree contains n balls vertices of the underlying graph) that are consequently distributed between two subtrees according to a simple rule: In each step, all balls independently move down to the left subtree say with probabilityp) or the right subtree with probability1 p), and a new node is created as long as there is at least one ball in that node. Finally, a non-negative integer d is given so that at level d or greater one ball is removed from the leftmost node before the balls move down to the next level. These steps are repeated until all balls are removed i.e., after n+d steps). Of interest are such tree parameters as the depth, path length sum of all depths), size, and so forth. This is illustrated in Figure 1 in which the deleted ball is shown next to the node from where it was removed. The tree just described falls between two digital trees, namely tries and digital search trees 3, 1, 14, 19]. In fact, when d = the tree can be modeled as a trie that stores n independent sequences generated by a memoryless source with parameter p. Hence, we coin the term n, d)-trie or simply d-trie) for the tree just described. In 1] lower and upper bounds were proved for parameters of interest, by using nown results for tries and digital search trees 3, 19]. In this paper, we establish precise asymptotic results. In particular, we show by how much the path length of a d-trie differs from the path length of the corresponding regular trie. Many parameters of a n,d)-trie can be described by the following two dimensional recurrence

4 A New Binomial Recurrence Arising in a Graphical Compression Algorithm 3 an,d) = fn)+ and the boundary equation an+1,0) = fn)+ p q n a,d 1)+an, +d 1)], d 1, 1) p q n a,0)+an,)], ) for 0 < p < 1, q = 1 p, and a nown additive term fn). For example, when fn) = n, then an,d) represents the path length. Recurrence ) is equivalent to the following boundary condition an,1) = an+1,0). For d = recurrence 1) becomes a traditional recurrence arising in the analysis of tries 19] whose solutions exact and asymptotic) are well nown. Thus, it is natural to study the difference ãn, d) := an,d) an, ), and that is our objective. In passing, we should point out that recurrence ) resembles the one used to analyze another digital search tree, nown as a digital search tree. In this paper we prove, however, that a n,d)-trie more closely resembles a trie, rather than a digital search tree. Our main interest lies in solving recurrence 1) for fixedd. For graph compression we only needd = 0, and we focus on this case. In particular, for fn) = n that is, for the path length in a d-trie) we prove that the excess quantity ãn, d) becomes asymptotically, as n and d = O1), 1 hlogp log n+ d h logn+ 1 h + 1 hlogp γ +1+ h h +Ψlog pn) )] logn where Ψ ) is the periodic function when log p/ log1 p) is rational, and h is the entropy rate. Digital trees such as tries and digital search trees have been intensively studied for the last thirty years, 3, 5, 7, 11, 1, 13, 16, 17, 18, 19]. However, our two-dimensional recurrence seems to be new and harder to analyze. It somewhat resembles the profile recurrences for digital trees, which were studied for tries in 15] and digital search trees in 4], and which are nown to also be challenging. The paper is organized as follows. In the Section we precisely formulate our problem and analyze it for fn) = n. Some proofs are presented in Section 3, while further details are provided in our journal version of this paper. Problem Statement In this section, we first formulate some recurrences describing n, d)-tries, then summarize our main results and discuss some extensions..1 Main Results Let us consider a n, d)-trie with n balls and parameter d 0. First, we analyze the average path length bn, d). It satisfies the following recurrence equations bn+1,0) = n+ p q n b,0)+bn,)], forn, 3)

5 4 Yongwoo Choi, Charles Knessl and Wojciech Szpanowsi and bn,d) = n+ p q n b,d 1)+bn, +d 1)], forn,d 1. 4) Recurrence 3) follows from the fact that starting withn+1 balls in the root node, and removing one ball, we are left withnballs passing through the root node. The root contributesn since each time a ball moves down it adds1 to the path length. Thosenballs move down to the left or the right subtrees. Let us assume balls move down to the left subtree the other n balls must move down to the right subtree); this occurs with probability n ) p q n. At level one, one ball is removed from those balls in the root of the left subtree. This contributesb,0). There will be no removal fromn balls in the right subtree until all balls in the left subtree are removed. This contributes bn, ). Similarly, for d > 0 we arrive at recurrence 4). Here0 < p < 1 andq = 1 p, and we also use the boundary conditions b0,d) = b1,d) = 0, d 0; b,0) = 0. 5) By setting d = 1 in 4) and comparing the result to 3) we can replace 3) by the simpler boundary condition bn,1) = bn+1,0), forn 0. 6) We are primarily interested in estimatingbn,0) for largen. If we letd in 4) and assume that bn,d) tends to a limit bn, ), then 4) becomes bn, ) = n+ p q n b, )+bn, )], 7) with b0, ) = b1, ) = 0. This is the same as the recurrence for the mean path length in a trie, which was analyzed, for example, in 1, 19]. One form of the solution is given by the alternating sum bn, ) = 1) l l l 1 p l ql, 8) l= and an alternate form is given by the integral 19] bn, ) = n! πi) Br z s ] Γs+1) e z 1 p s q sds zn+1dz, 9) where Γ ) is the Gamma function, Br is a vertical Bromwich contour on which < Rs) < 1 and thez-integral is over a small loop aboutz = 0. The asymptotic expansion of 9) as n may be obtained by a combination of singularity analysis and depoissonization arguments see 7, 9, 19]) and we obtain bn, ) = 1 h nlogn+ 1 γ + h ] h h +Φlog p n) n+on), 10)

6 A New Binomial Recurrence Arising in a Graphical Compression Algorithm 5 where h = plogp qlogq, h = plog p+qlog q, γ is the Euler constant, and Φx) is the periodic function Φx) = Γ πir ) e πrix, 11) logp =, 0 provided thatlogp/logq = r/s is rational, withr andsbeing integers withgcdr,s) = 1. Iflogp/logq is irrational, then the term with Φ is absent from the On) term of 10). We shall later use 10) to analyze the behavior ofbn,d) forn and a fixedd. Next we set bn,d) = bn,d) bn, ) 1) so that bn,d) measures how the path lengths in the d-trie differs from those in a trie. From 4) and 7), we then obtain bn,d) n n = )p q n b,d 1)+ bn, ] +d 1), forn,d 1, 13) which unlie 4) is a homogeneous recurrence. Then from 6) and 1) we have the boundary condition bn+1,0) bn,1) = bn, ) bn+1, ). 14) From 5) and 7) we also have b0,d) = b1,d) = 0 ford 0. We further defineb n,d) to be the solution of b n,d) = p q n b,d 1), forn,d 1, 15) and b n+1,0) b n,1) = bn, ) bn+1, ). 16) Note that 15) differs from 13) in that the former neglects the term involving bn,+d 1). We will show that this term in 13) is asymptotically negligible forn with fixedd, so that bn,d) b n,d). The recurrence 15) is much easier to solve by transform methods 7, 19] than is 13). We summarize our main result below. In Section 3 we establish Theorem 1 along with some other exact and asymptotic results for 3)-6) and 13)-16). Theorem 1 Forn andd = O1) we have bn,d) = Olog n). More precisely 1 bn,d) = hlogp log n+ d h logn+ 1 h + 1 γ +1+ h )] hlogp h +Ψlog pn) logn+o1), 17) where Ψ ) is the periodic function Ψx) = 1+ πir ] Γ πir ) e πirx 18) logp logp =, 0 and logp/logq = r/t is rational, as in 11). If logp/logq is irrational, the term involving Ψ in 17) is absent.

7 6 Yongwoo Choi, Charles Knessl and Wojciech Szpanowsi We see thatbn,d) bn, ) = Olog n), which shows that then,d)-tries studied in 1] are in some sense more similar to tries than to digital search trees DST). In 1], it was shown thatbn,0) was bounded above by average path lengths in tries and below by average path lengths in DST s. It was also conjectured thatbn,d) bn, ) ison), but our result shows that this difference is in fact much smaller.. Related Recurrence Equations The method presented in the next section allow us to analyze a class of recurrences of the type 3) with inhomogeneous terms other than n. For example, suppose we define an, d) by an,d) = fn)+ p q n a,d 1)+an, +d 1)] 19) where fn) is a given sequence that grows algebraically or logarithmically for n. The boundary condition is again of the type 3), or equivalently, an,1) = an+1,0), 0) and we have a0,d) = a1,d) = 0. Also, let an, ) satisfy 19) with the second argument of a, ) replaced by infinity. This recurrence can be solved by generating functions and Mellin transforms, and we can then establish that an, d) an, ) ãn, d), will satisfy and ãn,d) = p q n ã,d 1)+ãn, +d 1)] 1) ãn+1,0) ãn,1) = an, ) an+1, ). ) The asymptotic behavior of ãn,d) for d = O1) and n can be obtained in a manner completely analogous to the case fn) = n, discussed in the next section. For example, the case fn) = logn+1) arose in analyzing the compression algorithm in 1]. In 1] it was shown that an, ) has the asymptotic form an, ) = n h A 1)+on), n 3) where A s) = log +1) Γ +s).! iflogp/logq is irrational. Iflogp/logq = r/s is rational, the constanta 1) in 3) must be replaced by the oscillatory function A 1)+ =, 0 A 1+ πir ) e πirlog p n. 4) logp

8 A New Binomial Recurrence Arising in a Graphical Compression Algorithm 7 By analyzing 1) and ) for n we can show that the difference an,d) an, ) is Ologn), and more precisely ãn,0) = an,0) an, ) A 1) hlogp logn. Again if logp/logq is rational we must replacea 1) by the Fourier series in 4). 3 Analysis We first give an intuitive derivation of the asymptotics of bn,d) for fixed d 0 and n, and in particular of bn, 0). We use the fact that for algebraically or logarithmically varying smooth f) for ) we have see 6, 10] for rigorous proofs) p q n f) = fnp)+onf np)), n. 5) Starting from 13) we argue that the second sum is negligible forn. Then if bn,d) varies wealy with n, we use 5) to approximate the first sum, which will be asymptotic to bnp,d 1) so that 13) becomes bn,d) bnp,d 1), n 6) and, in particular, bn,1) bnp,0), n 7) which when added to 14) leads to bn+1,0) bnp,0) bn, ) bn+1, ). 8) The right side of 8) may be estimated from 10) or by 9). Using 9) we can show that term by term differentiating of the asymptotic series in 10) is permissible, and thus 8) becomes, for n, 1 bn+1,0) bnp,0) = h logn 1 γ +1+ h ) 1 h h h ψlog pn)+o1), 9) whereψ ) is the periodic function ψx) = =, 0 1+ πir ] Γ πir ) e πirx, 30) logp logp where we note that, in view of 11),ψx) = Φx)+logp) 1 Φ x). Now 9) suggests that bn,0) admits an asymptotic expansion of the form bn,0) = Alog n+blogn+c +o1), n 31) and then bn+1,0) bnp,0) = Alogp)logn Alog p Blogp+o1). 3)

9 8 Yongwoo Choi, Charles Knessl and Wojciech Szpanowsi Comparing 9) to 3) we conclude thata = hlogp) 1 and then B = 1 h + 1 γ +1+ h ] hlogp h +ψlog pn). 33) We have thus formally derived the result in Theorem 1 for bn,0). For any fixedd > 0 we can extend this argument by asymptotically solving 6) by an expansion of the form bn,d) = Ad)log n+bd)logn+o1) 34) to find from 6) thatad) = Ad 1) andbd) = Bd 1)+logpAd 1). Then using 34) in 8) or 9) we find thatad) = A0) = hlogp) 1 andbd) Bd 1) = logpad 1) = h 1 so that Bd) = B0)+h 1 d, whereb0) = B is as in 33). We proceed to provide a more rigorous derivation of the theorem. We first inductively establish the bound bn,d) A0 n ν+ǫ p +q ) d ; n,d 0 35) where ν = logp +q )/logp), and this holds for all ǫ > 0. When n = we have exactly) b,d) = 1 pq )p +q ) d 1 so 35) clearly holds. Assuming that 35) holds for all N,D) withn +D < n+d, we can estimate the first sum in 13) by p q n b,d 1) n p q n A 0 ν+ǫ p +q ) d 1 A 0 np) ν+ǫ p +q ) d 1 = A 0 n ν+ǫ p ǫ p +q ) d. Using the inductive assumption in 35) we then estimate the second sum in 13) by p q n bn, n +d 1) A0 n ) ν+ǫ p +q ) +d 1 p q n n pp A 0 n ν+ǫ p +q ) d 1 +q ) ] q n = A 0 n ν+ǫ p +q ) d 1 q +pp +q ) ] n which is smaller than the first sum, by an exponentially small factor. Combining these estimates and recalling that ǫ can be made arbitrarily small leads to the conclusion that 35) holds by induction. By subtracting 15) from 13) and using the estimate in 35) to bound the second sum in the right side of 13), we conclude that bn,d) b n,d) = O1) for n. We proceed to analyze 15), with 16), and thus re-establish Theorem 1. Introducing the exponential generating function B dz) = n= b n,d) zn n! = ez A d z), 36)

10 A New Binomial Recurrence Arising in a Graphical Compression Algorithm 9 whereb n,d) is defined from 15), we find that B dz) = B d 1pz)e qz 37) or, since A d z) = B d z)e z, This can be solved by iteration to yield A d z) = A d 1 pz). 38) Then setting and noting that 16) leads to n=1 A d z) = A 0 p d z). 39) G z) = n= bn, ) zn n! 40) b n+1,0) zn n! = d dz B 0z), 41) d dz B 0 z) B 1 z) = G z) G z). 4) If G z) = e z Gz), from the integral representation in 9) we conclude that the Mellin transform of Gz) is Gz)z s 1 Γs+1) dz = 1 p s q s, 43) Using 37), 39), and the definitions ofa d ) and G ), 4) becomes We introduce the Mellin transform ofa 0 z) 0 A 0z)+A 0 z) A 0 pz) = G z). 44) Ms) = and use 44) to obtain the functional equation 0 A 0 z)z s 1 dz 45) s 1)Ms 1)+1 p s )Ms) = s 1)Γs) 1 p 1 s q1 s. 46) Next we set with which 46) becomes Ms) = Γs)Ns) 47) Ns 1)+1 p s )Ns) = s 1 1 p 1 s q1 s. 48)

11 10 Yongwoo Choi, Charles Knessl and Wojciech Szpanowsi To solve 48) we let and then 48) becomes Ns) = N 1 s) N 1 s 1) = ] 1 p + N 1 s) 49) =1 1 p s ] 1 p s s 1 1 p +1 1 p 1 s q1 s. 50) Now, for s the right side of 50) behaves as s 1) =1 1 p+1 ) 1, with an exponentially small error. Letting the equation forn ) becomes N s) N s 1) = N 1 s) = ss 1) =1 1 1 p +1 s 1 1 =1 1 p+1 ) 1 p 1 s q 1 s ) +N s) 51) ] 1 p s ) 1 whose right hand side is, unlie that of 50), exponentially small fors. The solution to 5) is N s) = N )+ i=0 =1 5) ] =1 1 p s+i ) s 1 i 1 p 1+i s 1 q1+i s =1 1 p+1 ). 53) Note thatn ) exists, since the summand in 53) decays uniformly exponentially iniass. From 36) we see that A d z) = Oz ) as z 0 so that Ms) in 45) must be analytic at s = 1. From 47) we then conclude that N 1) = 0. From 49) we have N 1 1) = 0 and from 51) and 53) we thus obtain an expression forn ): ] N ) 1 p +1 =1 )+1 i+) 1 p+i+1 ) 1 p +i q +i 1 = 0. 54) =1 i=0 We have thus obtained the final expression forms) in 47) as Γs) ss 1) ] ) Ms) = =1 +β + s i 1) 1 p s+i ) L=0 1 pl s ) 1 p 1+i s 1, 55) q1+i s where i=0 β = N ) 1 p +1 ) =1 can be computed from 54). Inverting the transforms in 36) and 45) we obtain b n,d) = n! e z ] 1 πi z n+1 p d z) s Ms)ds dz. 56) πi Br

12 A New Binomial Recurrence Arising in a Graphical Compression Algorithm 11 The final step is to expand b n,d) bn,d)) for n with d fixed. The integral over z can be asymptotically evaluated by a standard depoissonization argument, which corresponds to replacing z by n in the inners-integral. The functionms) in 55) has a triple pole at s = 0, and there are other double poles on the imaginary s-axis if 1 p 1 s q 1 s has zeros there, which occurs only if logp/logq is rational, say r/t where r and t are integers 8] cf. also 3, 19]). First we compute the contribution from s = 0. Using the expansionγs) = 1 γs+os )]/s as s 0, with γ being the Euler constant, 55) becomes Ms) = 1 s 1 γs+os )]1 p s ) 1 1 p L s ) 1 L=1 s 1 1 p 1 s q 1 s 1 p s ) s 1)+ ss 1) +β =1 ] ) =1 + s i 1) 1 p s+i ) 1 p 1+i s 1. 57) q1+i s i=1 Now 1 p s = slogp 1 s logp) +Os 3 ) and 1 p 1 s q 1 s = hs h s +Os 3 ). Also, using the expression in 54) to compute β + 1 the expansion of 57) for s 0 becomes Ms) = 1 s 3 1 γs logp = 1 s 3 1 hlogp + 1 s 1+ s ] { 1 s logp+os ) h γ hlogp 1 hlogp 1 h 1+ h h It follows that the integrandp ds z s Ms) in 56) has the residue { Res s=0 p ds z s Ms) } = 1 log z hlogp + d h ] h s+os ) ) + 1 h ] +O } +Os ) 1 s ). 58) 1 γ +1 logz +logz + h ) logp h h 1 ] +O1) 59) h where the O1) refers to terms that are O1) for z, and these can be evaluated by explicitly computing theos 1 ) terms) in 58). Then the expansion of bn,d) b n,d) follows by settingz = n in 59), and we have thus regained the formula in 17). If logp/logq is rational we must also compute the contribution from the double poles along the imaginary axis at such points p s = q s = 1 and p 1 s + q 1 s = 1. These poles lead to the oscillatory terms in 17), as can be seen by computing their residues from 55). We have thus established 17) more rigorously, though the intuitive derivation in 6) 34) is much simpler, and more revealing of the basic asymptotic structure of the equations 13) and 14).

13 1 Yongwoo Choi, Charles Knessl and Wojciech Szpanowsi References 1] Y. Choi and W. Szpanowsi, Compression of Graphical Structures: Fundamental Limits, Algorithms, and Experiments, IEEE Transaction on Information Theory, 58), , 01. ] L. Devroye, A Study of Trie-Lie Structures Under the Density Model, Annals of Applied Probability,, , ] M. Drmota, Random Trees, Springer, New Yor, ] M. Drmota and W. Szpanowsi, The Expected Profile of Digital Search Trees, J. Combin. Theory, Ser. A, 118, , ] P. Flajolet, X. Gourdon, and P. Dumas, Mellin Transforms and Asymptotics: Harmonic sums, Theoretical Computer Science, 144, 3 58, ] P. Flajolet, Singularity Analysis and Asymptotics of Bernoulli Sums, Theoretical Computer Science, 15, , ] P. Flajolet and R. Sedgewic, Analytic Combinatorics, Cambridge University Press, Cambridge, ] P. Flajolet, M. Roux, and B. Vallee, Digital Trees and Memoryless Sources: from Arithmetics to Analysis, AofA 10, Vienna, Proceedings DMTCS, 33 60, ] P. Jacquet, and W. Szpanowsi, Analytical Depoissonization and Its Applications, Theoretical Computer Science, 01, 1 6, ] P. Jacquet and W. Szpanowsi, Entropy Computations via Analytic Depoissonization, IEEE Trans. Information Theory, 45, , ] C. Knessl, and W. Szpanowsi, Asymptotic Behavior of the Height in a Digital Search Tree and the Longest Phrase of the Lempel-Ziv Scheme, SIAM J. Computing, 30, , ] D. Knuth, The Art of Computer Programming. Sorting and Searching, Vol. 3, Second Edition, Addison-Wesley, Reading, MA, ] G. Louchard, Exact and Asymptotic Distributions in Digital and Binary Search Trees, RAIRO Theoretical Inform. Applications, 1, , ] H. Mahmoud, Evolution of Random Search Trees, John Wiley & Sons Inc., New Yor, ] G. Par, H.K. Hwang, P. Nicodeme, and W. Szpanowsi, Profile of Tries, SIAM J. Computing, 8, , ] B. Pittel, Asymptotic Growth of a Class of Random Trees, Annals of Probability, 18, , ] B. Pittel, Path in a Random Digital Tree: Limiting Distributions, Advances in Applied Probability, 18, , ] W. Szpanowsi, A Characterization of Digital Search Trees From the Successful Search Viewpoint, Theoretical Computer Science, 85, , ] W. Szpanowsi, Average Case Analysis of Algorithms on Sequences, Wiley, New Yor, 001.