Probabilistic analysis of the asymmetric digital search trees

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1 Int. J. Nonlinear Anal. Appl No. 2, ISSN: electronic Probabilistic analysis of the asymmetric digital search trees R. Kazemi a,, M. Q. Vahidi-Asl b a Department of Statistics, Imam Khomeini International University, Qazvin, Iran b Department of Statistics, Shahid Beheshti University, Tehran, Iran Communicated by M. Madadi Abstract In this paper, by applying three functional operators the previous results on the Poisson variance of the external profile in digital search trees will be improved. We study the profile built over n binary strings generated by a memoryless source with unequal probabilities of symbols and use a combinatorial approach for studying the Poissonized variance, since the probability distribution of the profile is unnown. Keywords: Digital Search Tree; Profile; Functional Operators; Poisson Variance MSC: Primary 05C05; Secondary 60F Introduction Digital trees lie tries or digital search trees are important in many computer science applications lie data compression, pattern matching or hashing. For example, the popular Lempel-Ziv compression scheme is strongly related to digital search trees. Digital trees have been widely studied in the literature. The motivation for studying the profiles of such trees is multifold. Of course, digital trees are used in various applications. For example, the profile represents the number of phrases of length in the Lempel-Ziv 78 built over n phrases. Second, the profile is a fine shape measure closely connected to many other cost measures height, saturation level, depth, path length, etc.. And finally, the related analytical problems are mathematically challenging and lead to interesting distributional results [3]. Digital search trees are intended for the same ind of problems as binary search trees. However, they are not constructed from the total order structure of the eys for the data stored in the internal nodes of the tree but from digital representations or binary sequences which serve as eys. Corresponding author addresses: r.azemi@sci.iiu.ac.ir R. Kazemi, m-vahidi@sbu.ac.ir M. Q. Vahidi-Asl Received: June 2014 Revised: June 2015

2 162 Kazemi, Vahidi-Asl Figure 1: A digital search tree built on eight strings s 1,..., s 8. In a digital search tree strings are directly stored in internal nodes. More precisely, the root contains the first string and the next string occupies the right or the left child of the root depending on whether its first symbol is 0 or 1. The remaining strings are stored in available nodes which are directly attached to nodes already existing in the tree external nodes. A digital search tree with n internal nodes is completed with n 1 external nodes. These external nodes can be seen as those positions the next item can be stored. The resulting tree is then a complete binary tree with the external nodes as leaves. The search for an available node follows the prefix structure of a new string [2]. Figure 1 shows a digital search tree built on eight strings s 1,..., s 8 i.e., s 1 = 0..., s 2 = 1..., s 3 = 01..., s 4 = 11..., etc. with internal ovals and external squares nodes, respectively. In this paper, we are concerned with probabilistic properties of the external profile defined as the number of external nodes at the same distance from the root. It is a function of the number of strings stored in a tree and the distance from the root. We write B n, for the external profile. Actually we study the external profile built over n binary strings generated by a memoryless source with unequal probabilities of symbols asymmetric case, that is, we assume each string is a binary independently and identically distributed i.i.d. sequence with p being the probability of a 1 and q = 1 p 0 < p < q < 1. Definition 1.1. Let X n be a sequence of integer random variable and X N its corresponding Poisson driven sequence, N is a Poisson random variable with mean z. Let Gz, u = Eu X N = n 0 EuXn zn n! e z be its Poisson transform. The Poisson variance is introduced as V z = G uz, 1 G uz, 1 G uz, 1 2, G uz, 1 and G uz, 1 denote respectively the first and the second derivative of Gz, u with respect to u at u = The Previous Results In the following, we first review the main equations and results described in [2]. Let B n, denotes the random number of external nodes at level in a digital search tree built over n strings generated

3 Probabilistic analysis of the asymmetric digital search trees No. 2, by a memoryless source with parameter q > p = 1 q. We recall the initial conditions { 1, for n = 0 B n,0 = 0, for n 1. The probability generating function of the external profile, P n, u = Eu B n,, satisfies the following recurrence relation n n P n1, u = p j q n j P j j, 1 up n j, 1 u, 2.1 j=0 with initial conditions P 0, u = 1 for 1, P 0,0 u = u, P n,0 u = 1 for n 1. The corresponding exponential generating function G x, u = n 0 P n, u xn n! fulfills the following functional recurrence x G x, u = G 1 px, ug 1 qx, u, 1, 2.2 with initial conditions G 0 x, u = u e x 1 and G 0, u = 1 1. By taing derivatives with respect to u and setting u = 1 we obtain for the exponential generating function E 1 x = EB n, xn n! n 0 the following functional recurrence E 1 x = e qx E 1 1 px epx E 1 1 qx, 2.3 with initial conditions E 1 0 x = 1 and E 1 0 = 0 1. The Poisson transform of E1 x, namely 1 x = e x EB n, xn n! = e x E 1 x, 0 n 0 translates recurrence 2.3 into 1 x 1 x = 1 1 px 1 1 qx, 1, 2.4 with initial conditions 1 0 x = e x and 1 0 = 0 1. Similarly, by taing second derivatives with respect to u and setting u = 1 we obtain for the exponential generating function E 2 x = EBn, 2 xn n! n 0 the following functional recurrence E 2 x = e qx E 2 1 px epx E 2 1 qx 2E1 1 pxe1 1 qx, 2.5

4 164 Kazemi, Vahidi-Asl with initial conditions E 2 0 x = 1 and E 2 0 = 0 1. Furthermore, the Poisson transform of x, namely E 2 2 x = e x EBn, 2 xn n! = e x E 2 x, 0 n 0 translates recurrence 2.5 into 2 x 2 x = 2 1 px 2 1 qx w x, 1, 2.6 w x = px 1 1 qx, 2 0 x = e x and 2 0 = 0 1. By induction it is easy to prove that 2 x can be represented as finite linear combinations of functions of the form e pl 1 q l 2 x with l 1, l 2 0, and of products of two of these functions. Hence, the Mellin transform of 2 x, 2 s = 0 2 xxs 1 dx. exists for all s with Rs > 0 see [1]. Since B n, = 0 for > n it follows that E 2 x = Ox as x 0 which ensures that 2 s actually exists for s with Rs >. Let us now express 2 s as 2 s = ΓsF 2 s, Γs is the Euler gamma function. By definition, we now that F 2 s is the finite linear combinations of functions a s with certain values of a. Thus, F 2 s is an entire function. Furthermore 2.6 translates into H 2 s = 1 Γs F s F s 1 = T sf 1 w xx s 1 dx, s H2 s, and F 2 0 s = 1. Note that 2.7 does not only hold for Rs > the Mellin transform exists. Since F 2 s continues analytically to an entire function, 2.7 holds for all s, too. The inhomogeneous part in 2.7 is very large compared to the order of magnitude of the homogeneous equation F s F s 1 = T sf 1 s, 0, 2.8 for the first moment. Since F 1 s behaves geometrically as T s it seems that the term F 1 1 s 1 is negligible compared to the other two terms in 2.8. This phenomenon will also occur for F 2 s and F s. We introduce the so-called Poisson variance 2, V x := 2 x 1 x which should be a good approximation for the variance of the profile [3]. By 2.4 and 2.6, V x satisfies V x V x = V 1 px V 1 qx 1 2, x 2.9

5 Probabilistic analysis of the asymmetric digital search trees No. 2, with initial condition V 0 x = e x 1 e x and V 0 = 0 1. The Mellin transform of V x is then given as V s = 0 V xx s 1 dx, and again, we can use a factorization of the form V s = ΓsF s, V s and F s can be written in terms of 1 s, 2 s, F 1 2 s, and F s respectively. In particular, 2.9 translates into F s F s 1 = T sf 1 s H s, 0, 2.10 H s = 1 Γs 0 1 x 2x s 1 dx and also F 0 s = 1 2 s. We also observe that F r = 0 for > r, since ΓsF s is the Mellin transform of V x that exists for Rs >. We will use this property. The inhomogeneous part of 2.10 i.e., H s does not dependent on p and q in comparison with the inhomogeneous part of the functional-differential equation satisfied by variance of the profile see Section 2.1 in [2] for details: page 6. Thus asymptotic analysis of the variance done there through Poissonized variance is not a precise analysis. In other words, the analytic function arising there Ds, w [2, Lemma 4] is not completely explicit. Let I be an identity operator and A[f]s = j 0 fs jt s j for some function f. Kazemi and Vahidi-Asl [2] showed the following results: Lemma 2.1. The function fs, w = 0 F sw can be represented as fs, w = Ds, wgs, w, the functions gs, w = I wa 1 s and Ds, w is analytic for w < 1/T Rs/ Theorem 2.2. Let V x denote the Poisson variance of the profile in unbalanced digital search trees with underlying probabilities 0 < p < q = 1 p. Let and n be positive integers such that / log n satisfies log 1 p 1 ε / log n log 1 q 1 ε. Then uniformly V n = L ρ n,, log p/q p n p ρ n, q ρ n, n ρ n, 2πβρn, 1 1 O, 2.11 ρ n, = ρ/ log n and Lρ, x is a non-zero periodic function with period 1 in x. The function Lρ, x can be represented as Lρ, x = j Z fρ it jγρ it j e 2jπix, fs has the form fs = gs 1, 1/T sds, 1/T s.

6 166 Kazemi, Vahidi-Asl Lemma 2.3. For every real interval [a, b] there exist 0, γ > 0 and ε > 0 such that F s = fst s 1 O e γ, 2.12 uniformly for all s with Rs [a, b], Is 2lπ logq/p ε for some integer l and 0, fs = Ds, 1/T sgs 1, 1/T s is an analytic function that satisfies f r = 0 for r = 1, 2,... and is bounded in this region. Furthermore, if Is 2lπ logq/p > ε for for all integers l, then we have uniformly for Rs [a, b]. and In Theorem 3.4 below we show that fs = F l rw l T s l h 1s, 1/T s s T r.t A r, 1/T s T s F l rw l T s l h 2s, 1/T s s T r.t B r, 1/T s T s η A r, w = h 1 r, w B r, w = h 2 r, w F s = O T Rs e γ, wt r 1 wt r η h 2 r, w 1 wt r η h 1 r, w. 1 wt r 2.14 In this function all parts are explicit and are analytic for w < T Rs η 1 see [2] for h 1 s, w and Lemma 3.3 for h 2 s, w. We obtain an explicit solution of 2.10 by introducing two proper functional operators: C[f]s = fs j, j 0 D[f]s = wt s j fs j, w 1 wt s j η j 1 for some function f and η > 0. Set R s = A [1]s, H l s = C[H l 1 ]s C[H l 1 ]c, T,l s = A [ H l ]s, Gs, w = H sw, 0 1 wt s η, 1 wt s for all, l 1.

7 Probabilistic analysis of the asymmetric digital search trees No. 2, The New Results Lemma 3.1. Suppose fs, w = 0 F sw. Then for all 1 and complex s, ns, w = gc, wgs, w fs, wgc, w = gs, w ns, w, 3.1 I wa 1 [ H ]s I wa 1 [ H ]c 1 w gc, wi wa 1 [G]s, w Proof. See Appendix A. gs, wi wa 1 [G]c, w, w Remar 3.2. The proof of 3.1 maes use of the fact that F c = 0 for 1. However, we also have F r = 0 for > r [2]. In particular, if we set s = r in 3.1 we find that and consequently f r, wgc, w = g r, w n r, w fs, wg r, w n r, w = gs, w ns, w F l rw l. First of all, we have the following result by convolution of Laplace transform [2]: 1 H s = 1 2x x s 1 dx Γs 0 1 = 1 ci Γs 1 t 1 s tdt 2πi c i C ci dt t s t Γt Γs t 1 T Rt 1T Rs t 1 Γs c i CT c 1 2 Rt = c = Rs t, 2 Rs = CT 1 c = Rs 2 2,, C T Rs η 3.3 for constants C and C and for some η > 0. Lemma 3.3. There exists a function h 2 s, w that is analytic for all w and s satisfying such that wt s n η 1, for all n 1, ns, w = h 2 s, w 1 wt s η. 3.4 Thus, ns, w has a meromorphic continuation w 0 = 1/T s η is a polar singularity.

8 168 Kazemi, Vahidi-Asl Proof. By definition of H s and 3.3, ns, w converges absolutely and represents an analytic function, since T s j T s maxpq j for j 0. Also I wa[ns,.]s = rs, w, rs, w = Gs, w H s H c 1 w 1 Gs, wmc, w Gc, wgs, w. If we substitute ns, w by then For convenience, set V s j, w = h 2 s, w = D[h 2 ]s h 2 s, w 1 wt s η, 1 wt s η rs, w. 1 wt s wt s j wt s η.1, j 1. 1 wt s j η 1 wt s By induction it follows that D [1]s = V s j 1, wv s j 1 j 2, w j 1 0 j 2 0 j 0 V s j 1 j, w Hence, n 1 = n V s n, w. n 1 1 n 1 = 1 n 2 = 2 D [1]s n n 1 1 V s n, w n n 2 2 V s n, w V s n 1, w. n 1 1 n 2 1 V s n 1, wv s n 2, w n 1 =1 V s n 1, wv s n 2, w n 1 1 n 1 1 V s n 1, w Using the fact that T s n = Oq n, it follows directly that the series S := V s n, w = wt s n 1 wt s η 1 wt s n η 1 wt s, n 1 n 1 converges if wt s n η 1 for all n 1 and 1 wt s. Let now 0 be any value such that n 0 V s n, w 1 2,

9 Probabilistic analysis of the asymmetric digital search trees No. 2, Then we have for all 0, 1 2S D [1]s 0. 2 Since rs, w C1 wt Rs η 1 and [ ] h 2 s, w = D [1]s 0 thus h 2 s, w 22S 0 C for constant C. 1 wt s η rs, w, 1 wt s Theorem 3.4. For every real interval [a, b] there exist 0, γ > 0 and ε > 0 such that F s = fst s 1 O e γ 3.5 uniformly for all s with Rs [a, b], Is 2lπ logq/p ε for some integer l and 0, fs is an analytic function that satisfies f r = 0 for r = 1, 2,.... Furthermore, if Is 2lπ logq/p > ε for for all integers l then we have uniformly for Rs [a, b] fs described in Proof. Let h 1 s, w = j 1 1/1 wt s j [2]. We have fs, w = = = F s = O T s e γ. 3.6 F l rw l gs, w ns, w g r, w n r, w F l rw l gs, w g r, w n r, w F l rw l ns, w n r, w n r, w F l rw l h 1s, w 1 wt r A r, w 1 wt s F l rw l h 2s, w 1 wt r η B r, w 1 wt s η = f 1 s, w f 2 s, w. 3.7 Suppose first that s > r 1 for some integer r 0 but s is not a positive integer. The function h 1 s, w is analytic for w < 1/T s 1. It also follows that h 1 s, w is non-zero for real 0 < w < 1/T s 1 and that h 1 r, w is analytic and non-zero for 0 < w < 1/T r 1. Hence, w 0 = 1/T s is a singular point of f 1 s, w. Since F s = V s/γs it follows that all values F s have the same sign. Hence, the radius of convergence of the series fs, w equals w 0 = 1/T s. In a next step we show that f 1 s, w has no other singularities on the radius of convergence w = 1/T s. Since all terms in f 1 s, w, that is, r F l rw l, h 1 s, w, A r, w, 1 wt r, and 1 wt s are analytic for w < 1/T s ε, a singularity of f 1 s, w can only be induced by a zero of A r, w. Suppose first that A r, w has a zero w 1 with w 1 < 1/T s. Since A r, w 0

10 170 Kazemi, Vahidi-Asl for 0 < w < 1/T r 1 it follows that w 1 1/T r and w 1 1/T r η. If we assume that r F l rw l 1 0, then w = w 1 must be a zero of h 1 s, w. We slightly decrease s to s γ for some γ > 0 such that s γ is not a positive integer such that h 1 s γ, w 1 0. Then the zero w = w 1 of A r, w would induce a singularity w 1 of f 1 s, w with w 1 < 1/T s although its radius of convergence is 1/T s γ > 1/T s > w 1. This leads to a contradiction. Hence, if A r, w 1 = 0 for some w 1 with w 1 < 1/T s, then we also have r F l rw l 1 = 0. Actually, it also follows that the order of the zeroes are the same. The above considerations also show that if w = w 1 is a zero of A r, w with w 1 < 1/T r 1, then w 1 is also a zero of r F l rw l 1 = 0 of the same order. Namely, if w 1 < 1/T r 1 then there exists a non-integral real number s > r 1 with w 1 < 1/T s and we proceed as above. This property shows that the only singularity of f 1 s, w is given by w = 1/T s if s > r 1 is real but not an integer. This singularity is a polar singularity of order 1. Hence, by using Cauchy s formula for a contour of integration on the circle w = e γ /T s and the residue theorem it follows that [1] [w ]f 1 s, w = f 1 st s O T se γ, 3.8 f 1 s = F l rt s l h 1s, 1/T s 1 T r. A r, 1/T s T s These estimates are uniform for s contained in a compact interval [a, b] r 1, r for some non-negative integer r or in a compact interval [a, b] contained in the positive real line. Furthermore, we get the same result if s is sufficiently close to the real axis. Thus, if a Rs b and Is ε for some sufficiently small ε > 0, then we obtain 3.8. Here we have also used the fact that f 1 s 0 in this range. Next, suppose that s is real or sufficiently close to the real axis and close to a negative integer r, say r γ s r γ for some γ > 0. Here we use the representation f 1 s, w = = F l rw l h 1s, w 1 wt r A r, w 1 wt s F l rw h 1s, w A r, w 1 wt r A r, w 1 wt s F l rw l1 T s T r F l rw. 1 wt s Now if we subtract the finite sum r F l rw, then we can safely multiply by Γs that is singular at s = r and obtain Γs Γs h 1 s, w A r, w F sw = F l rw A r, w >r =0 1 wt r 1 wt s Γs l1 F l rw T s T r. 1 wt s

11 Probabilistic analysis of the asymmetric digital search trees No. 2, We again use the fact that the function r F l rw l /A r, w is analytic for w < 1/T r 1 and observe that w = 1/T s is a polar singularity. By applying Cauchy s formula we obtain for > r similar to the above Γs[w ]f 1 s, w = F l rt s l 1 T r T s Γs h 1 s, 1/T s A r, 1/T s A r, 1/T s F l rt s l 1 Γs T s T r O T se γ. Thus, we have actually proved 3.8 for > r and we also observe that f 1 r = 0. Since T s2πil/ logq/p = T s for integer l, it follows that w = 1/T s is a polar singularity of f 1 s, w if Is 2πil/ logq/p < ε for some integer l. Thus, 3.8 follows also for s in this range. Finally, if Is 2πil/ logq/p > ε for some integer l, then there exists γ > 0 such that T s < e 2γ T Rs. Hence it follows that f 1 s, w is regular for w < e 2γ /T Rs. Thus, for a contour of integration on the circle w = e γ /T Rs in Cauchy s formula we obtain [w ]f 1 s, w = O T Rs e γ. It should be clear that this estimate is uniform if Rs varies in a finite interval [a, b]. Since f 2 s, w only has a polar singularity w = 1/T s η of order 1, thus for a contour of integration on the circle w = e γ /T s η, f 2 s = [w ]f 2 s, w = f 2 st s O T s ηe γ, If Is 2πil/ logq/p > ε, then [w ]f 2 s, w = O T Rs η e γ. F l rt s l h 2s, 1/T s 1 T r η. B r, 1/T s T s η It is obvious that w 0 = 1/T s is a dominant singularity of fs, w. Thus for every real interval [a, b] there exist 0, γ > 0 and ε > 0 such that F s = [w ]fs, w = fst s 1 O e γ, fs = f 1 s f 2 s = F l rw l T s l h 1s, 1/T s s T r.t A r, 1/T s T s F l rw l T s l h 2s, 1/T s s T r.t B r, 1/T s T s η

12 172 Kazemi, Vahidi-Asl uniformly for all s with Rs [a, b], Is 2lπ logq/p ε for some integer l and 0, fs is an analytic function that satisfies f r = 0 for r = 1, 2,.... Furthermore, if Is 2lπ logq/p > ε for all integers l then we have uniformly for Rs [a, b]. F s = O T Rs e γ By the above discussion, we now that F s and V s = ΓsF s behave asymptotically as T s. Thus we are in a situation similar to the analysis of the previous article [2] but here the analytic function fs introduced in the above theorem is completely explicit. References [1] F. Flajolet and R. Sedgewic, Analytic Combinatorics, Cambridge University Press. Cambridge, [2] R. Kazemi and M.Q. Vahidi-Asl, The Variance of the Profile in Digita Search Trees, Discrete Math. Theor. Comput. Sci [3] W. Szpanowsi, Average Case Analysis of Algorithms on Sequences. John Wiley, New Yor, Appendix A Proof.[Proof of Lemma 3.1] It is obvious that and 1 ns, w = R l c H l sw j 1 Thus it is enough to show that 1 T l,l s T l,l c w R j ct l j,l s R j st l j,l c fs, wgc, w = 0 w F l sr l cw. 1 1 F s = R s F l sr l c R l c H l s 1 T l,l s T l,l c 2 j 1 R j ct l j,l s R j st l j,l c. We will prove this relation by induction. Certainly, it is satisfied for = 0. Now suppose that it holds for some 0. Thus 1 F 1 s = R 1 s R 1 c R l cf l1 s 1 R l c H 1 l1 s R l ca[ H 1 l ]s R l ca[ H l ]c 1 T l1,l s T l1,l c

13 Probabilistic analysis of the asymmetric digital search trees No. 2, R 1 st l,l c R 1 ct l,l c j 1 j 1 j 1 = R 1 s R j ct l j1,l s R j1 st l j,l c 2 j 1 R j1 ct l j,l c H 1 s R 1 l cf l s R j ct l j1,l c R l c H 1 l s 1 R l ca[ H 1 l ]s R l ca[ H l ]c T l1,l s T l1,l c 1 1 R 1 st l,l c R 1 ct l,l c A[ H ]s A[ H ]c 1 j 1 R j ct l j1,l s R l ca[ H l ]s 2 1 R 1 ct l,l c R l ca[ H l ]c 1 l=2 j 1 R j st 1 l j,l c R 1 st l,l c. Now by removing the same expressions the proof is completed.