Phase changes in random m-ary search trees and generalized quicksort

Transcription

1 Phase changes in random m-ary search trees and generalized quicksort Hua-Huai Chern Department of Mathematics and Science Education Taipei Municipal Teacher College Taipei Taiwan Hsien-Kuei Hwang 2 Institute of Statistical Science Academia Sinica Taipei 5 Taiwan July, 2 Abstract We propose a uniform approach to describing the phase change of the limiting distribution of space measures in random m-ary search trees: the space requirement, when properly normalized, is asymptotically normally distributed for m 26 and does not have a fixed limit distribution for m > 26. The tools are based on the method of moments and asymptotic solutions of differential equations, and are applicable to secondary cost measures of quicksort with median-of-2t + ) for which the same phase change occurs at t = 58. Both problems are essentially special cases of the generalized quicksort of Hennequin in which a sample of mt + ) elements are used to select m equi-spaced ranks that are used in turn to partition the input into m subfiles. A complete description of the numbers at which the phase change occurs is given. For example, when m is fixed and t varies, the phase change occurs at m, t) = {2, 58), 3, 9), 4, ), 5, 6), 6, 4), }. We also indicate some applications of our approach to other problems including bucket recursive trees. A general framework on asymptotic transfers of the underlying recurrence is also given. Introduction The aim of this paper is to develop analytic tools for characterizing the limiting distributions of the space requirement of random m-ary search trees and that of the secondary cost measures like the number of partitioning stages, the number of stack pushes or pops, etc.) of the generalized quicksort of Hennequin see below and [5]) of which quicksort with median-of-2t + ) is a special case. Our main results state that for all these structures the limit distributions exhibit a phase change from asymptotic normality to non-existence at some point depending on the parameters. Given a set K of n keys, the m-ary search tree m 2) associated with K is constructed as follows. If n < m then the m-ary search tree contains only a root node with all keys in it arranged in increasing order); if n m then place and sort the first m keys in increasing order in the root; these m keys are used to direct the remaining n m + keys in the m subtrees of the root, which are themselves m-ary search trees: keys with values lying between the i )-st and the i-th Supported by National Science Council of ROC under the Grant NSC M The work of this author was partially supported by National Science Council of ROC under the Grant NSC M--32. Part of his work was done while he was visiting School of Computer Science, McGill University; he thanks the School for hospitality and support.

2 4,,2 7,8 4,5 3 5,6 9, 2,3 4,7,,2,3 5,6 8,9, 2,4,5 3 Figure : The ternary left) and quaternary right) search trees constructed from the sequence {, 4, 7, 5, 8,, 4, 9, 5,, 2, 2, 3, 6, 3}. keys of the root go to the i-th subtree from left to right) for 2 i m. The first the last) subtree contains, respectively, keys less than larger than) the smallest largest) key in the root. The m-ary search trees represent one of the many generalizations of binary search trees. The ternary and quaternary search trees corresponding to the sequence {, 4, 7, 5, 8,, 4, 9, 5,, 2, 2, 3, 6, 3} are depicted in Figure. Assume that all n! permutations of n elements are equally likely. Given a random permutation, the m-ary search tree associated with it is called a random m-ary search tree of size n. Several cost measures on random m-ary search trees have been studied in the literature. Let X n denote the total number of nodes leaf and internal) used by a random m-ary search tree of n elements. Mahmoud and Pittel [3] derived the asymptotics of the first two moments of X n and proved that X n is asymptotically normally distributed for m 5. This range was later extended to m 26 in Lew and Mahmoud [25]. For m > 26, it is generally believed that the limiting distribution of X n would not be normal, although a rigorous proof is still lacking; see Mahmoud and Smythe [32] and Smythe [42]. Let N, ) denote a normal random variable with zero mean and unit variance. Let H m = j m j. The following result delineates the phase change of the limit laws of X n and is the first main result in this paper. 2

3 Theorem. For 3 m 26, X n µn σ n d N, ), ) d where denotes convergence in distribution, µ = /2H m 2), and σ = σ m > is a constant depending only on m; see 9). For m > 26 and k, ) Xn µn k E = Φ k β log n) + o)), 2) n α where α and β > denote, respectively, the real and imaginary parts of the zeros of the indicial equation xx + ) x + m 2) m! with the second largest real part, and Φ k x) = Φ k,m x), k, are bounded 2π-periodic functions. We will indeed prove convergence of all moments when 3 m 26, which is stronger than convergence in distribution. Corollary. For m > 26, the limiting distribution of the random variables X n µn)/n α does not exist. Our starting point is, as in Mahmoud and Pittel [3], the bivariate generating function of X n F z, y) := n Ey Xn )z n, which satisfies the nonlinear differential equation see [3, 28]) m z m F z, y) = m )!yf m z, y); F z, y) = + yz + yz yz m ) Except for m = 2 and m = 3 see Section 7), this differential equation seems hard to be solved exactly. The approach by Mahmoud and Pittel for the mean of X n consists first in taking derivative with respect to y and then substituting y = on both sides of 3), yielding an ordinary differential equation of Euler-type, which is then solved by the method of undetermined coefficients. The same approach applies to the variance with more calculations. They proved the asymptotic normality via an inductive approximation approach, which has proved useful for a number of recursive structures since [3]; see Pittel [36]. We use consistently the method of moments to first re-prove that the asymptotic normality of X n holds for 3 m 26 and then to prove that the distribution function of X n µn)/n α does not approach a fixed distribution in the limit. The hard part is, as usual, to find means of handling the excessive cancellations caused by shifting the mean in computing higher central moments. The crucial step is to shift X n by its asymptotic mean by considering either the nonlinear differential equations satisfied by the bivariate generating function G of X n µn + ) see )) Gz, y) := y µ F zy µ, y) = E y Xn µn+)) z n, n 3

4 or by considering the recurrence relation P n y) := E y Xn µn+)) = y n ) m i + +i m=n m+ i,...,i m P i y) P im y) n m), with P y) = y µ and P j y) = y µj+) for j m. This step, although minor, incorporates succinctly the cancellation effect in the associated differential equations. For example, With the introduction of this shifting-the-mean step, we can compute directly the leading term of the variance without cancellation, which occurs if one computes first an expansion for the second moment; see Section 3.3. For higher moments, the cancellations avoided are much more significant. Another important ingredient of our approach is to apply the method of linear operator for Eulertype differential equations; see Ince [22], Greene [3], Hennequin [5], Sedgewick [4]. Although the other two approaches, the method of undetermined coefficients and the method of variation of parameters or of Legendre), work equally well for Euler equations, the approach by linear operator requires less calculations for higher moments. We discuss more examples in Sections 6. We also summarize this approach in Section 8 by proving a general result on asymptotic transfers between the toll cost and the total cost in the underlying recurrence of the coefficients. In addition to many other linear measures on m-ary search trees see Baeza-Yates [], Lew and Mahmoud [25]), our general tools can also be applied to unveil the phase change at t = 58 in quicksort with median-of-2t + ), t, although almost all practical versions use median-of-three t = ); see Bentley and McIlroy [3]. This phase change has long been neglected because most effort in the literature has been concentrated in either the total cost like the number of comparisons or exchanges, which has mean of order n log n, or the recursion depth, which has logarithmic mean; in contrast, we focus on cost measures whose mean is linear like the number of times the median-of- 2t + ) is used, the number of stack pushes or stack pops, the number of comparisons and insertions used by the final stage insertion sort namely, if smaller subfiles are left unsorted in quicksort and then sorted by a one-run insertion sort; see Sedgewick [4]). Alternatively, these and many other linear measures can also be described in terms of the associated fringe-balanced binary search trees see Mahmoud [29]). The linear mean seems to play a special role for the phase change since no such phenomenon occurs or can be produced for n log n- and log n-measures. Roughly, for cost measures in quicksort with median-of-2t + ) with linear mean, the variances are linear for t 58, but between linear and quadratic for t > 58, the periodic oscillation terms becoming dominating in this latter case. The origin of such a periodic oscillation is discussed in Section 4. On the other hand, the n log n- cost measures like the path length) have quadratic variance, the periodic fluctuations being never predominant for any fixed t; the log n-measures like the depth) have also logarithmic variance, periodic terms playing an asymptotically negligible role. We can encapsulated the above two structures into a more general framework by studying the generalized quicksort of Hennequin see [4, 5]). In this version, a sample of τ := mt + ) random elements are chosen in which the t + )-st, 2t + )-st,..., and the m )t + )-st smallest elements are then determined. These m elements are used to partition the original input into m subfiles as in the construction of the m-ary search tree. Then the same procedure recurs for each subfiles until the size of the subfile is less than τ. A final insertion sort will then complete the remaining sorting task. We call this sorting scheme quicksort with m-tiles-of-τ or simply the Quicksort is ranked among the top ten algorithms with the greatest influence on the development and practice of science and engineering in the 2th century; see [23] 4

5 generalized quicksort. Without going into implementation details, we assume that the randomness is preserved at each partitioning stage. Such a sorting scheme has a natural interpretation in terms of m-ary search tree in which the usual m-ary search tree corresponds to t =. For t, the construction is almost the same as t = but instead of randomly choosing m partitioning keys, we take first randomly mt+) elements from which the m equi-spaced with step t + ) ranks are selected as the partitioning keys, so that the resulting m-ary search tree is more balanced. Such m-ary search trees will be referred to as the fringe-balanced m-ary search trees or the generalized m-ary search trees. When given a random permutation or a sequence of independent and identically distributed continuous random variables, we refer to the associated tree as a random generalized m-ary search tree. To simplify the presentation, we consider the following special case in detail. Given a random permutation of n elements, let ξ n denote the number of times the m-tiles-of-τ is used by the generalized quicksort to sort these elements. In terms of trees, it denotes the number of internal nodes in a random generalized m-ary search tree. Define fz, y) := n Eyξn )z n. Then f satisfies the nonlinear differential equation see Hennequin [5, p. 69]) mt+) mt + ) )! t fz, y) = zmt+) t! m y fz, y) = + z + z z mt+) 2 +, fz, y) zt where m 2 and t. The m-ary search tree corresponds up to initial conditions) to t =, and quicksort with median-of-2t + ) corresponds to m = 2. Exactly solvable cases of this equation are m, t) = 2, ) see [34]) and m, t) = 3, ) see Section 7). In order to describe the phase changes, we define the following set of integer pairs S := {m, t) : t 58, 2 + δ t m ς t }, ) m ; where δ ab denotes the Kronecker symbol and the sequence ς t is given in Table. t ,6 7,...,,...,9 2,..., 58 ς t Table : The sequence ς t, the upper bound of m for which the asymptotic normality subsists. 4) As in the case of X n, the phase change of the limit laws for ξ n can be explicitly described as follows. Theorem 2. For m, t) S, ξ n µ n σ n d N, ), where µ = /τ + )H τ+ τ + )H t+ ), and σ = σ m, t) > is a constant depending only on m and t. For integers m, t) S, where m 2, t and m + t > 2, ) ξn µ n k E = Ψ k b log n) + o)) k ), n a where a and b > denote, respectively, the real and imaginary parts of the zeros of the indicial equation xx + ) x + τ ) xx + ) x + t )mτ!/t! with the second largest real part, and Ψ k x) are bounded 2π-periodic functions dependent on m. 5

6 Corollary 2. For integers m, t) S, where m 2, t and m + t > 2, the distribution functions of the random variables ξ n µ n)/n a do not approach a fixed distribution function. More concrete examples having the same phase changes will be given in Section 6. Why phase change? From an analytic point of view, this is caused by the second largest zeros in real parts of the associated indicial equation. But it is not obvious how this can be explained in more intuitive terms. Since phase change occurs when we vary either m or t, there are at least two processes at play: the outdegree of each internal node and the choice of partitioning keys. The periodicity seems the origin of the phase change see Section 4 for discussions on the connection of periodicity and limit law); on the other hand, it is also closely related to the exact solvability of differential equations of the form 4). This is because the underlying recurrence for the moments is of the type a n = b n + m n τ) t j n m )t+) n j m )t + ) ) j t ) a j n τ), 5) with suitable initial conditions and the general solution to the recurrence 5) contains no periodic terms the associated indicial equation has only real zeros) when m, t) = 2, ), 3, ) and 2, ), which are exactly the three cases for which 4) can be exactly solved. Thus the problem more or less becomes: why periodicity? Even this problem does not seem to have a simple intuitive interpretation. For example, if we fix m = 2 and vary t, and take b n = in 5), then the solution to a n involves periodic terms for t 2. Why two? Problems of this sort are very interesting but unclear at this stage. Note that although periodic solutions exact or asymptotic) appear often in recurrences issuing from algorithmics, a classification distinguishing periodic and non-periodic solutions in the spirit of the huge amount of work in nonlinear difference equations and dynamic systems) seems hard at this stage. This paper is organized as follows. We first group in the next section a number of simple lemmas that will be used throughout this paper. Then we prove Theorem as well Corollary in Section 3. A more detailed discussion on periodicity and existence of limit laws is given in Section 4. The proof of Theorem 2 and Corollary 2 is then sketched in Section 5 due to similarity. We indicate more applications of our approach in Section 6. Exactly solvable cases are discussed in Section 7. We then extend the method of linear operator in Section 8 to derive more general asymptotic transfers, which can be applied to many problems; in particular, we apply the transfers to refine a result of Rösler [38] on the second-order term of the mean number of comparisons used by quicksort using median-of-2t + ). We conclude with several problems of future study in Section 9. Notations. Throughout this paper, we use the symbol [z n ]φz) to denote the coefficient of z n in the Taylor expansion of φ. The falling and rising factorials are denoted by x k := j k x j) and x k := j k x + j), respectively. The generic symbols c and c always stand for constants whose values vary from one occurrence to another; small positive quantities are denoted generically by ε. Define an extended region C of the unit disk as follows. The symbol τ always represents mt + ). 2 Lemmas C := {z : z + ε, z [, + ε]} ε > ). We group some useful lemmas in this section that will be repeatedly applied. Let ϑ denote the operator z)d/dz). For convention, we also use ϑ freely as a complex number. 6

7 Lemma. The function is the general solution to the equation Corollary 3. Let Y z) = c z) r z) s + A s r Y z) = s r), ϑ r)y z) = A z) s A C). A z) s s r ) s r k ) + j k e j z) r j, where s and the r j s are all distinct complex numbers and the e j s are constants. Then Y z) is the general solution to the equation ϑ r ) ϑ r k )Y z) = A z) s A C). The following lemma will be applied to handle more general functions in the non-homogeneous part and the case of multiple zeros. Lemma 2. The function Y p z) = z) r x) r hx) dx is a particular solution to the first order differential equation ϑ r)y z) = hz), provided that the integral converges. Furthermore, is the solution with initial value Y ). Y z) = Y ) z) r + z) r x) r hx) dx Proposition. Assume that φ satisfies the non-homogeneous Euler equation z z) n dn dz n φz) + a n dn z) dz n φz) + + a nφz) = hz) n ), 6) where h satisfies the following conditions: i) h is analytic in z < ; ii) h is analytically continuable to C ; iii) h has at most a singularity at z = for z C {} and hz) = A + o)) z) s, as z in C, where A, s C. Let r C be the largest zero in real part of the indicial equation 2 ϖϑ) := ϑ n + j<n a n jϑ j =. Furthermore, assume that r is a simple zero of ϖϑ). Then as z in C A + o) z) s, if Rs) > Rr); φz) = ϖs) c + o)) z) r, if Rs) < Rr); c + o)) z) r + c + o)) z) s, if s r and Rs) = Rr); A + o) ϖ z) r log z), if s = r. r) 2 In general, the indicial equation of a Cauchy-Euler equation characterizes all types of the fundamental solutions; see, for example, Coddington and Levinson [6] for details. 7) 7

8 Proof. Write hz) = A z) s + o z Rs) ) and divide the proof into two parts: the = -version and the o -version. The desired result will follow from linearity of linear differential equations. First of all, by the relation z) j dj dz j = ϑϑ + ) ϑ + j ) = ϑj j ), 8) we can rewrite the equation 6) in the form ϖϑ)[φ] = h. Arrange the zeroes of ϖϑ) in descending order of their real parts as Rr) Rr 2 )... Rr m ), and let d j be the multiplicity of the zero r j, 2 j m. Consider first the case when hz) = A z) s. If s r, then the particular solution of ϖϑ)[φ] = h is of the form A ϖs) z) s, if ϖs), φ p z) = ) d j A dj d j! ϖs) z) r j log z)), if s = r j. Also the general solution ϖϑ)[φ] = h is of the form where φ j z) span φz) = φ p z) + c z) r + 2 j m φ j z), { z) r j, z) r j log z),..., z) r j log z)) d j }, 2 j m). From this expression, the first three assertions in 7) for hz) = A z) s follow. When s = r, the function A ϖ r) z) r log z) is a particular solution of ϖϑ)[φ] = h since r is simple and we used Lemma 2. The remaining proof is similar to the case s r. Now assume hz) = o z Rs) ) in C. We need to show that φz) = { o z max{rs),rr)} ), if s r; o z Rr) log z ), if s = r, 9) which follows from our analytic assumptions on h and Lemma 2. Similarly, an O-version of the Proposition by replacing the two o s in 9) by O) also holds. Finally, by the superposition principle of linear differential equations, we complete the proof. 8

9 3 Phase change in random m-ary search trees 3. Differential equations Recall that Gz, y) := y µ F zy µ, y). Let Xn := X n µn + ). From 3), G satisfies m z m Gz, y) = m )!ygm z, y), Gz, y) = y µ + y µj+) z j For k, let G k z) = k Gz, y) yk = y= n j m 2 E X nx n ) X n k + )) z n ) denote the generating function of the k-th factorial moment of X n. By definition, G z) := / z). Lemma 3. For k, G k x) satisfies G k ) = µ) k and { L[Gk ]z) = k!m )! z) m Q k z), G j) k ) = )k j! µj + ) ) k, j m, where and Q k z) := j + j m=k j,...,j m<k L[G] := z) m dm G j z) G jm z) j! j m! G m!g, dzm + j + j m=k j,...,j m<k G j z) G jm z). j! j m! Proof. First observe that by definition G k z) = k![w k ]Gz, + w). By substituting y = + w and by taking coefficient of w k on both sides of ), we obtain L[G k ]z) = k!m )! z) m Q k z), where Q k is given by Q k z) = [w k ] + w) m G j z) w j m j! k! G z) m G k z). j The desired expression follows from straightforward expansion. By using the Bell polynomials B k,l defined by we have, for k, B k,l x,..., x k l+ ) := Q k z) = k! 2 l min{k,m} + k )! i +2i 2 + =k i +i 2 + =l i,i 2, k!x i x i 2 i!i 2!!) i 2!) i 2 k, l ), ) m G z) m l l! B k,l G z),..., G k l+ z)) l ) m G m l z)l! B k,l G z),..., G k l z)). ) l l min{k,m} 9

10 3.2 Mean For a better presentation of the approach used in this paper and for completeness, we give a selfcontained derivation of the solution see 3)) of G z). Although this requires the a priori knowledge of the asymptotically dominant term of EX n ), our approach also applies when this is not available since both differential equations differ only in the initial conditions. On the other hand, this dominant information up to the growth order) can be easily obtained by Proposition below. By Lemma 3, the generating function G z) of EX n) satisfies G ) = µ and L[G ]z) = G j) m )! z, By the relation 8), we can express 2) as ) = j!µj + ) ), j m. 2) Λϑ)G z) = m )! z, where Λϑ) = is the indicial equation of the equation L[G]Y = : Λϑ) = ϑ m m! = k m ϑ λ k ), the λ k s, k m, being the zeros of Λϑ). These zeros are simple and Λ2) =. More properties of this polynomials can be found in Mahmoud [28]. Arrange these zeros by descending order of their real parts: 2 = λ Rλ 2 ) Rλ 3 ) Rλ m ). See Figure 2 for a plot of the distribution of the zeros Figure 2: Distribution of the zeros of Λϑ) for m from 5 to 4; zeros with positive real parts and the vertical line Rϑ) = 3/2 which may be called the phase-change line ) are shown on the right figure. The zeros are distributed in an extremely regular way.

11 We now show that see [3, 28]) where for 2 k m G z) = A k = 2 k m A k z) λ k λ k λ k )H m λ k ), H sx) := For convenience, define A := /m ) and A := µ. To that purpose, let so that By Lemma, we obtain Υ k ϑ) := m ) z), 3) j s Λϑ) ϑ λ k 2 k m ), Λϑ)G z) = ϑ λ k )Υ k ϑ)g z). Υ k ϑ)g z) = x + j. κ z) λ + m )! k λ k ) z), 4) for some constant κ depends on λ k, which can be determined by the initial conditions of G z) as follows; see Greene [3] and Hennequin [5]. First, by evaluating both sides at z =, By the identity and Λλ k ) = λ m k Then we obtain, by 2), κ = Υ k ϑ)g ) + m )! λ k. ϑ m m = ϑ m j λ k + m 2) j + λ kλ k + ) λ k + m 2), ϑ λ k ϑ λ κ = m! = m! j= = m!, k m, we can rewrite Υ k as Υ k ϑ) = j m 2 j m 2 Λϑ) ϑ λ k = m! j m 2 G j) ) m )! + λ k λ k + j) λ k j! λ k λ k + j) µm! ϑ j λ k λ k + j). 5) j m 2 j + )! m )! + λ k λ k + j) λ k.

12 By expressing the terms in the sums as beta integrals, we obtain m! j m 2 j! λ k λ k + j) = m! = y) λ k 2 y y m ) dy m! m )! λ k λ k ) λ k, which is a priori valid when Rλ k ) >. But the same result also hold for other values of λ k by using, say the following contour integral representations for beta function see [8,.6]) Bλ k, j + ) = 2 cschπiλ k) for λ k not an integer and all j >. Similarly, m! j m 2 +) j + )! λ k λ k + j) =. v λ k v ) j dv, Thus κ = m! λ k λ k ) 2 k m ). Applying Corollary 3 to 4), we obtain A k = κ Υ k λ k ) = λ k λ k )H m λ k ) 2 k m ). Finally, the coefficient of / z) in 3) is obtained again by Corollary 3: m )! )! = m = λ 2 ) λ m ) Λ) m, as desired. Although the preceding approach is essentially the same as the method of undetermined coefficients and the method of variation of parameters, it requires only beta integrals and simple combinatorial sums; also it is readily amended for asymptotics of higher moments that we discuss in the next section. 3.3 Variance of X n We solve asymptotically in this section the equation { L[G2 ]z) = R 2 z), G j) 2 ) = j! µj + ) )2, j m, 6) with G 2 ) = µµ + ), where, by 3), R 2 z) := m!m ) z)g 2 z) + 2m!G z) ) 2 m!m ) z) A 2 z) λ 2 + A 3 z) λ m! 3 m ) z), 7) 2

13 There are basically two different cases: i) if 2α < 2 or 3 m 26), then R 2 z) = o z 2 ), and ii) if 2α > 2 or m > 26), then R 2 z) z 2. In the first case, since the dominant zero of the polynomial Λϑ) is 2, G 2 z) is asymptotic to σ 2 z 2 ; in the second case, the non-homogeneous part is dominating, implying, by Proposition, that G 2 z) z 2α+. Our approach is different from that used by Mahmoud and Pittel [3]. An advantage of our approach is that we can easily prove that σ >. Variance is linear for 3 m 26. We show that G 2 z) σ 2 z) 2 z, z C ), where σ 2 > is a constant to be determined by the same approach as for G. From this estimate, we deduce, by singularity analysis see Flajolet and Odlyzko []), that By 3) and 6), VarX n ) [z n ]G 2 z) σ 2 n. m! L[G 2 ]z) = Λϑ)G 2 z) = m ) z) + m!m ) 2 j,k m A j A k z) λ j+λ k. Again we separate the factor ϑ 2 from Λϑ) by writing Λϑ) = ϑ 2)Υϑ). Note that Rλ j ) < 3/2 for 3 m 26 and 2 j m, implying that Rλ j + λ k ) < 2. Thus the function Υϑ)G 2 z) can be solved as in the last section using Lemma ); the result is Υϑ)G 2 z) = υ z) 2 + m! m ) z) + m!m ) where υ is a constant to be determined. As in 5), Υ can be decomposed as Υϑ) = m! j m 2 2 j,k m ϑ j j + 2)!. A j A k λ j + λ k 3 z) λ j λ k +, 8) Evaluating both sides of 8) at z =, we have recalling that A = /m ) and A = µ) υ = m! G j) 2 ) j + 2)! + A A j A k + m ) 3 λ j λ j m 2 k 2 j,k m = m! A 2 + A2 m ) + A 2 + m ) A j A k. 3 λ j λ k 2 j,k m Finally, by 8) and Corollary 3, we obtain G 2 z) σ 2 z) 2, as z in C, where σ 2 = υ Υ2) = A A + 2A 2 A j A k m ) + 2A + 2m ), 9) 3 λ j λ k 3 2 j,k m

14 for 3 m 26. We now prove that σ 2 >. Observe first, by 3), that α = Rλ 2 ) < 3/2 for 3 m 26, or, equivalently, G z) = o z 3/2+ε ) z, z C ). Thus 2 j,k m A j A k 3 λ j λ k = x) 2 G x) A ) 2 dx >. x On the other hand, it is easily shown that A + 2A 2 m ) + 2A > for all m 3. This proves the positivity of σ 2 for 3 m 26. Variance is larger than linear for m 27. G 2 z) m!m ) 2 j,k 3 We prove that A j A k Λλ j + λ k ) z) λ j λ k + z, z C ); 2) from this and singularity analysis []), we obtain EX n µn) 2 [z n ]G 2 z) m!m ) = Φ 2 β log n)n 2α 2, 2 j,k 3 A j A k n λ j+λ k 2 Λλ j + λ k )Γλ j + λ k ) where A 2 2 Φ 2 x) := 2m!m ) Λ2α )Γ2α ) + R A 2 )) 2 e2ix. Λ2λ 2 )Γ2λ 2 ) We apply Proposition to prove 2). First, by 3) and 7), R 2 z) m!m ) A j A k z) λ j λ k +. 2 j,k 3 Then since Rλ j + λ k ) > 2 for m > 26, we are in the first case of 7) and 2) follows. Note that VarX n ) Φ 2 β log n) Φ 2 β log n) ) n 2α 2, where Φ x) := A 2 Γλ 2 ) eix + A ) 3 A2 Γλ 3 ) e ix = 2R Γλ 2 ) eix. A priori, Φ 2 x) Φ 2 x) > for all real x, but we have not found a direct proof for all m > 26. 4

15 3.4 Asymptotic normality: 3 m 26 We use the method of moments to prove the asymptotic normality of X n µn)/σ n) by applying Proposition case ). Recall that G k z) is the generating function for the k-th order factorial moment of Xn := X n µn + ). Proposition 2. If 3 m 26, then as z in C, { G2k z) = o z k /2 ); G 2k z) 2k)!2 k σ 2k z) k, for k. Corollary 4. The central moments of X n satisfy for k. ) Xn µn 2k E σ = o); n ) Xn µn 2k E σ = 2k)! n 2 k k!, The first part of Theorem follows from the Corollary. Proof of Proposition 2. We use induction. The proposition holds for k = as proved above. Analytic continuation of Q k to C follows from induction and ) because all G k s are expressible as finite sums of powers of z. Note that for l 3 Q l z) R l z) := j + +j m=l j,...,j m<l If l = 2k is odd, then one of the j i s must be odd. Thus R 2k z) s k G 2s z) j 2 + +j m=2k s) j,...,j m<2k s) j,...,j m : even G j z) G jm z). j! j m! G j2 z) G jm z) j 2! j m!. By induction, for fixed s, j 2 + +j m=2k s) j,...,j m<2k s) j,...,j m : even G j2 z) G jm z) j 2! j m! = O z k+s m+), and G 2s z) = o z s /2 ). It follows that z) m Q 2k z) z) m R 2k z) = o z k /2 ) z, z C ). By 9), G 2k z) = o z k /2 ). 5

16 If l = 2k is even, then R 2k z) = j + +j m=2k j,...,j m<2k j,...,j m : even 2i + +2i m=2k i,...,i m<k G j z) G jm z) j! j m! G 2i z) G 2im z). 2) 2i )! 2i m )! By 2) and the asymptotic growth of G 2s z), s 2k 2, we have Q 2k z) R 2k z) 2 i i m σ 2i + +2i m z) i i m m 2i + +2i m=2k i,...,i m<k w 2j = σ 2k z) m [w 2k ] 2 j z) j m2 k σ 2k z) m k j ) ) m + k = σ 2k 2 k m z) m k. k Consequently, by Proposition, G 2k z) 2k)! ) ) m + k Λk + ) 2 k m )!σ 2k m z) k. k But Λk + )/m )! = ) m+k k m, this proves the desired assertion. 3.5 Higher moments of X n for m > 26 We apply again Proposition to derive asymptotics of the moments EX n µn) k. Proposition 3. If m 27 and k, then, as z in C, G k z) g k j,..., j k )A j A jk z) λ j λ jk +k, 22) 2 j,...,j k 3 where g 2) = g 3) = and for k 2 with g k j,..., j k ) = k!m )! Λλ j + λ jk k + ) i +2i 2 + =k i +i 2 + =l m 2 l min{m,k} ) m l! l T i,..., i k l+ ; j,..., j k ) i!i 2!!) i 2!) i 2, T i,..., i k l+ ; j,..., j k ) := g j s ) g 2 j i +2s, j i +2s) s i s i 2 g k l+ j i + +k l)i k l +k l)s )+s,..., j i + +k l)i k l +k l+)s). s i k l+ 6

17 Proof. The cases k =, 2 satisfy 22) by 3) and 2). We show that 22) is valid for k = 3 and the general k follows by induction. By ), z) m Q 3 z) = Again, by 3) and 2), we have z) m Q 3 z) ) m 3 2 j,j 2,j 3 3 s 3 ) m 3 g j s ) + G 3 z) z) 2 + m 2 + ) m G z) G 2 z) z) 2 ) G 2 z) z) + m 2 G 2z). ) m g j )g 2 j 2, j 3 ) A j A j2 A j3 z) λ j λ j2 λ j3 +2, 2 as z in C. Here we can express the terms enclosed by the larger parentheses in the last summation by introducing the functional T ). Finally, using Corollary 3 again, we prove the proposition for k = 3. For k 4, observe that if G k satisfies 22) then for l G l k z) 2 j,...,j kl 3 By this, ) and induction, we have z) m Q k z) for z in C, where χ k j,..., j k ) = g k j,..., j k ) g k j kl )+,..., j kl )A j A jkl z) λ j λ jkl +kl l. 2 j,...,j k 3 2 l min{m,k} χ k j,..., j k )A j A jk z) λ j λ jk +k, ) m l l! i +2i 2 + =k i +i 2 + =l T i,..., i k l+ ; j,..., j k ) i!i 2!!) i 2!) i 2. Since for m 27, Rλ j λ jk k + ) > 2, where j,..., j k {2, 3}, we obtain, by Proposition, the desired expression 22). Recall that β = Iλ 2 ). Corollary 5. For k ) Xn µn k E = Φ k β log n) + o)), n α where Φ k x) = Φ k,m x) is a periodic function of period 2π defined by Φ k x) = 2 j,...,j k 3 g k j,..., j k ) Γλ j + + λ jk k + ) A j A jk e ix )j + + ) jk ). 7

18 Note that Φ k x) can be expressed as Φ k x) = k/2 j= where u k,j and v k,j are functionally independent of n. u k,j λ 2, λ 3 ) cos k 2j)x) + v k,j λ 2, λ 3 ) sin k 2j)x)), Lemma 4. Assume that ϱ > and N is a positive integer. Let Sn) := u j cos k j ϱ log n) + v j sin k j ϱ log n)), j N where u j, v j R, j N and k < k 2 < < k N are nonnegative integers. Then Sn) = for all n sufficiently large if, and only if u j = v j =, j N. Proof. It suffices to show that Sn) = for n sufficiently large implies u j = v j = for j N. For any positive integer l sufficiently large and a positive irrational number δ, we have Define 2πδl ϱ e 2πδl/ϱ ϱ log e2πδl/ϱ 2πδl. θ l ϱ log e 2πδl/ϱ mod 2π). Then the sequence {θ l } l forms a dense subset in [, 2π]. Let S x) := j N u j cos k j x) + v j sin k j x)). If Sn) = for all n sufficiently large, then S x) for x 2π. Hence u j = v j =, for j N. From the lemma, it follows that Φ k β log n) = for all sufficiently large n if, and only if u k,j = v k,j =. It is easily checked that Φ x) and Φ 2 x) are nontrivial periodic functions. We now prove the nonexistence of the limit distribution of X n µn)/n α for m > 26. Proof of Corollary. For simplicity, let X n := X n µn)/n α. Let < w < be fixed. Choose a subsequence {n j } j such that {β log n j } w, as j, where {x} denotes the fractional part of x. By Corollary 5, E X k n j ) Φ k w) k ). By the moment convergence theorem of Frechet and Shohat see Loève [26]), we can find a subsequence, say {n j } j of {n j } j such that for some random variable X w, and P Xn j < x ) P X w < x) ) E Xk n w j k := 8 j ), x k dp X w < x).

19 Similarly, we can find another subsequence {n j } j of N + such that ) P Xn < x P X j z < x) j ), for < z < w < and E Xk n j ) z k := x k dp X z < x). Since w k z k, we have X w X z. Thus the limit distribution of Xn does not exist. Indeed, there are infinitely many subsequences of { X n } n each of which weakly converges to a different law. Roughly, we may say that the lack of simple patterns between the periodic functions Φ k x) leads to the non-existence of the limit law. 4 Periodicity and limit law We give a deeper look at the periodicity and the non-existence of the limit law in this section. The negative result for the existence of a limit law of X n when m 27 naturally suggests several questions. 4. Nodes of specified types? First, since we count the total number of nodes used by a random m-ary search tree, what happens if we count only the number of nodes with a specified number of keys? By the same analytic approach, it can be shown that the same phase change phenomenon subsists no matter what type of nodes we are counting; see also Lew and Mahmoud [25] for more details. 4.2 Normalized by exact mean? Second, what happens if we shift X n by its exact mean instead of the asymptotic mean? It can be shown that there is no essential difference since the periodic constants for the moments cannot be smoothed out. We roughly sketch the arguments. Let µ n := EX n ) and σn 2 := VarX n ). Then σn 2 satisfies the recurrence σ 2 n = m n ) m m k n m+ ) n k σk 2 m 2 + r n,2 with the initial conditions σ n = for n m, where r n,2 := n ) m j + +j m=n m+ n m), + µ j + + µ jm µ n ) 2 n m). [This formula is derived by taking second derivative on the recurrence of the moment generating function of X n µ n.] This is a special case of the recurrence 5) studied in Section 8. Since m 27, we deduce, by 3), that r n,2 w j n 2α 2+2ijβ, j 9

20 for some w j that can be explicitly computed. Applying term by term the asymptotic transfers see 36)), we obtain σ 2 n n 2α 2 j w j n 2ijβ m!γ2α+2ijβ ) Γ2α+2ijβ+m 2) This means that the periodic oscillation still remains. Similarly, higher central moments have similar periodic behaviors. 4.3 Limit law of a weighted functional. Third, the second question somehow suggests that the origin of the non-existence of the limit law of X n for m 27 is simply because Φ β log n) is periodic. Suppose that µ n = µn + cn α + on α ), for some nonzero constant c independent of n. Then the variance would be asymptotic to c n 2α 2 for some constant c since roughly the non-homogeneous part, which is proportional to z 2α+, in the associated differential equation is dominating α > /2). Similarly, the periodicity in higher moments would all disappear and we would expect the existence of a limit law. We explore this idea in this section by considering the random variables Z n := j m v jx n j), where v,..., v m are some numbers and X n j) denotes the number of nodes containing exactly j keys in a random m-ary search tree of n nodes. We show that if the vector v,..., v m ) is chosen in a very special way, then the periodicity can be wiped out and we can derive the limit law of Z n. The problem, although not very interesting, is used to clarify a few points: First, this problem further reflects the intricacy of the periodicity in the random tree structure; indeed, the inherent periodicity in the random variables X n j) and X n is very robust and cannot be smoothed out if the v j s are taken to be independent of n; second, the example shows that the existence of the limit law boils down to the removal of the periodic term in the second order term in the asymptotic expansion of the mean. Let Z n y) := Ee Zny ). Then see Lew and Mahmoud [25]) Z n y) = ev m y ) n m j + +j m=n m+ Z j y) Z jm y) with Z n y) = e vny for n m, where for convenience, v :=. n m), Mean of Z n. By the same approach as above, we can show that, defining U z) := n EZ n)z n, from which it follows that U z) = EZ n ) = j m j m A j z) λ j v m m z), A j Γλ j ) λj + n n ) v m m, 2

21 where A j = H m λ j ) l m 2 v l l! λ j λ j + l) + v m j m ). mλ j ) In particular, A = v m H m m + j m 2 v j, j + )j + 2) and the mean is asymptotic to EZ n ) = A n + A 2 Γλ 2 ) nα +iβ + A 3 Γλ 3 ) nα iβ + on α ). Smoothing out the periodic coefficient. A 2 Γλ 2 ) niβ = 2 j m r j + s j i)v j, Note that we can write A 3 Γλ 3 ) n iβ = 2 j m r j s j i)v j, where r j and s j are real numbers. Thus EZ n ) = A n + n α r j v j + on α ). j m Let v j := x j + y j i, where the x j, y j ) s are chosen to satisfy the equations r j x j =, r j y j = ; j m j m so that EZ n ) = A n + n α + on α ), and U z) = A z) 2 + Γα) z) α + o z α ) z C ). Note that and We may take j!n iβ ) r j = R Γλ 2 )H m λ 2 )λ 2 λ 2 + j) r m = R n iβ mh m λ 2 )Γλ 2 )λ 2 ) j m 2), ). x j = r j j m r2 j and y,..., y m ) to be any vector perpendicular to x,..., x m ). 2,

22 Higher moments. Since the second order term in the asymptotic expansion of EZ n ) is aperiodic, by the same method of proof as above for X n, we deduce that where g = g =, and for k 2 g k := EZ n A n) k g k n kα k k 2), k!m )! Γkα ) + )Λkα ) + ) j + +j m=k j,...,j m<k g j g jm j! j m!. Here Λx) = x x + m 2) m!. Observe first that the sum on the right-hand side contains at most k m terms, and then, by Stirling s formula, that m )!k m m!kα ) m+ Γkα ) + )Λkα ) + ) Γkα ) + ) k ). From this we deduce, by choosing a large enough constant K > and by induction, that g k k!k k k ). This implies, by the Frechet-Shohat moment convergence theorem, that the sequence { g k } k is a moment sequence, and that the limit law is uniquely characterized by the moments. We conclude that Z n A n n α d Z m 27), where EZ k ) = g k for k. See Mahmoud and Smythe [32, Theorem 4] for a similar example on bucket recursive trees, and Mahmoud [3] for an urn model approach to the space of random trees. 5 Phase change in generalized quicksort The analytic approach developed in previous sections can be applied to discover more phase change phenomena in generalized quicksort. In this section, we prove Theorem 2 on limit laws of the random variable ξ n, the number of times the m-tiles-of-τ is used, in generalized quicksort when given a random permutation. 5. Differential equations Recall that the bivariate generating function fz, y) of ξ n satisfies the nonlinear differential equation see [5, p. 69]) τ ) τ! t m fz, y) = zτ t! m y fz, y) m 2, t ), zt with the initial conditions f j), y) = j! for j < τ. Let f k z) := k / y k )fz, y) y= for k. Then f z) = fz, ) = / z). Define the operator L[f k ]z) := z) τ dτ dz τ f kz) mτ! z) t dt t! dz t f kz), 23) 22

23 which, using ϑ := z)d/dz, can be rewritten as L[f k ]z) = Ωϑ)f k z), where Ωϑ) := ϑ τ mτ!ϑ t /t!. Then the function f z) satisfies L[f ]z) = Ωϑ)f z) = τ! z, 24) with f j) ) = for j < τ. Set ν = m )t + ). Let η j, j ν, be the zeroes of the polynomial ϑ + t) ϑ + τ ) mτ!/t!. Arrange them by descending order in real parts as 2 = η > Rη 2 ) Rη 3 ) > Rη ν ). The other zeros of Ωϑ) are {,,..., t + }. For convenience, set η =. All these zeroes are simple; see Hennequin [5] for more properties of the polynomial Ωϑ). 5.2 Mean of ξ n The equation 24) can be solved explicitly as follows. f z) = d j z) j + where C = /m ) and j t k ν C k z) η, 25) k C k = t! mη k )η k η k + t )H τ η k ) H t η k )) k ν). 26) as where The proof uses again the same method of separated) linear operator by first decomposing Ωϑ) V k ϑ)ϑ t = Then by Lemma 3 j<ν Ωϑ)f z) = ϑ η k )V k ϑ)ϑ t f z) = τ! z, τ + η k ) ν j ϑ t+j = mτ! t! j<ν ϑ t+j η k + t) η k + t + j). 27) V k ϑ)ϑ t f z) = κ z) η k + τ! η k ) z). Thus κ = j<ν τ + η k ) ν j f t+j) ) τ! η k = τ! η k, and 26) follows from the relation V k η k )η t k = mτ! t! η t k H τ η k ) H t η k )), 23

24 and Corollary 3. In particular, Eξ n ) µ n, where µ := C = τ + )H τ+ H t+ ). 5.3 Shifting-the-mean and variance Define Then Mz, y) satisfies Mz, y) = y µ fzy µ, y). τ ) τ! t m Mz, y) = zτ t! m y Mz, y), zt and the initial conditions j / z j )Mz, y) z= = j!y µ j+), j τ. Note that Mz, ) = / z). Set M k z) = k / y k )Mz, y) y=, k. Lemma 5. For k, M k z) satisfies L[M k z)] = z) τ k! τ! t! m Y kz), with the initial conditions M j) k ) = j! )k µ j + )) k, j τ, where Y k z) := = k! j + j m=k j,...,j m<k + 2 l min{k,m} k )! the B k,l s being Bell polynomials. By 25), for z, z C, and by Lemma 5 M t) j z) M t) j m z) j! j m! + j + j m=k j,...,j m<k ) m M t) l z)m l l! B k,l l min{k,m} M z) = f z) C z) 2 M t) ) m M t) l z)m l l! B k,l M t) j z) M t) j m z) j! j m! ) t) z),..., M k l+ z) M t) ) t) z),..., M k l z), = C z) + C 2 z) η 2 + C3 z) η 3 + smaller order terms, Ωϑ)M 2 z) = R 2 z), 24

25 where R 2 z) := z) τ τ! t! m 2mM t) z)m M t) = mm ) τ! t! 2 2 j,k 3 = O z 2Rη 2 )+ + z ). z) + mm )M t) η t jη t k C jc k z) η j η k + z)m 2 M t) mτ! m ) z) z)2) Let a = Rη 2 ). We observe that when m is fixed, there exists a t such that a < 3/2 for t t and a > 3/2 for t t. Symmetrically, if t is fixed, there exists an m such that a < 3/2 for m m and a > 3/2 for m m. The values of these m s are given in Table. When a < 3/2, R 2 z) = o z 2 ), and the dominant term of M 2 comes from the homogeneous part; on the other hand, when a > 3/2, R 2 z) z 2, and the non-homogeneous part is dominating. Thus the line 3/2 is again the phase-change line. By the same approach as for the variance of X n, we have σ 2 z) 2, if a < 3/2; M 2 z) mm ) τ! ηj tηt k C jc k z) η j η k + t! 2, if a > 3/2, Ωη j + η k ) 2 j,k 3 where t + ) H τ+ H t+ ) σ 2 = m )t + )µ 2 + µ τ + m + m ) t! 2 x) 2t+2 M t) x) C t! 2 x) t+ dx, the integral being convergent for a < 3/2. Consequently, { σ 2 Varξ n ) n, if a < 3/2; Ψ2 b log n) Ψ 2 b log n)) n 2a 2, if a > 3/2, where Ψ x) = 2RC k e ix /Γη 2 )), and Ψ 2 x) = 2mm ) τ! t! 2 2 η2 t C 2 2 Ω2a )Γ2a ) + R η2 2) tc e 2ix. Ω2η 2 )Γ2η 2 ) 5.4 Higher moments Our condition that a < 3/2 is equivalent to m, t) S. Proposition 4. Assume m, t) S. Then for k as z in C. { M2k z) = o z k /2 ); M 2k z) 2k)!2 k σ 2k z) k, 25

26 The proof is similar to Proposition 2 and is omitted. Corollary 6. The normalized moments of ξ n satisfy for k. ) ξn µ n 2k E = o); σ n ) ξn µ n 2k E = 2k)! σ n 2 k k!, Therefore, the asymptotic normality of ξ n when a < 3/2 can be proved in the same way as for X n. The case when a > 3/2 is equivalent to m 2, t and m + t > 2. Proposition 5. If m, t) S, m 2, t and m + t > 2, then for k M k z) 2 j,...,j k 3 as z in C, where ω 2) = ω 3) = and ω k j,..., j k )η t j η t j k C j C jk η j + + η jk k + ) t z) η j η jk +k, ω k j,..., j k l+ ) k!τ! ) m := l!t! l Ωη j + + η jk k + )) l 2 l min{m,k} ω j s ) ω 2 j i+2s, j i+2s) i!! i 2! 2! s i s i 2 i +2i 2 + =k i +i 2 + =l i k l+! s i k l+ Recall that b = Iη 2 ). Corollary 7. For k where Ψ k x) = 2 j,...,j k 3 is a periodic function of period 2π. ω l j i + +k l)i k l +k l)s )+s,..., j i + +k l)i k l +k l+)s) k l + )!. ) ξn µ n k E = Ψ k b log n) + o)), n a ω k j,..., j k )η t j η t j k C j C jk η j + + η jk k + ) t Γη j + + η jk k + ) eix )j + + )jk ) Theorem 2 follows from the two corollaries. Thus the limit distribution of ξ n µ n)/n a does not exist. 26

27 6 Examples and applications The preceding proof can be extended in several ways. The simplest one is to consider the differential equation τ ) τ! t m fz, y) = zτ t! m py) fz, y), 28) zt with the initial conditions j / z j )fz, y) z= = j!q j y), j < τ, where py) and q j y) are all probability generating functions. Then the same phase change as ξ n of the last section subsists with µ = q j ) H τ+ H t+ j + )j + 2) + p ). τ + t j<τ and H τ+ H t+ )σ 2 = t j<τ q j ) 2µ j + )q j ) + µ j + )) 2 + m t! 2 t + ) j + )j + 2) 2 k ν t j<τ + m )p ) mp ) 2 m )τ + ) j!q j ) η k +t) η k +j) + t!p ) mη k ) H τ η k ) H t η k ) x) η k + 2 dx, provided that σ 2. We briefly mention some examples having the same phase changes as ξ n ; see [2] for more possible examples. Number of nodes of certain types In terms of fringe-balanced m-ary search trees, ξ n counts the number of internal nodes. The same phase changes subsist if we count nodes of any type like the number of leaves or the number of nodes in which the number of keys belongs to some given set. For example, the bivariate generating function of the total number of leaves satisfies 28) with py) = and q k y) = y for k τ. More examples can be found in Lew and Mahmoud [25], Baeza-Yates [] and Cunto and Gascon [5]. Cost of insertion sort If we use insertion sort to sort subfiles of sizes τ, then the bivariate generating function of the number of key insertions is described by 28) with py) = and q k y) = y + y ) k τ); j j k and the same for the number of key interchanges with q k y) = j k + y + + y j j k τ). See [4] and [5] for analyses of the mean. 27

28 Costs of selecting the partitioning keys The cost required for selecting the m partitioning keys among the τ random keys also satisfy 28) with py) = and q k a polynomial of y whose degree depends on m and t. In terms of trees, the above quantity can also be regarded as the number of rotations used in fringe-balanced m-ary search trees. The case when m = 2 and t = was studied previously by Mahmoud [29] and Panholzer and Prodinger [34]. Their approaches, based on martingale argument and Weierstrass elliptic functions, respectively, seem more limited as far as limit distributions are concerned, although they prove stronger results like convergence rate. Our result states that no matter how we arrange the subtrees of sizes < τ, the number of rotations is always linear and exhibits the same phase change as ξ n. Number of stack pushes We consider only the binary case m = 2. If we always sort the smaller subfiles first and then use insertion sort for subfiles of sizes 2t +, then the probability generating function, denoted by W n y), of the number of stack pushes satisfies W n y) = for n 4t + 4 and for n > 4t + 4, W n y) = y j n j n j ) t) t) W j y)w n j y) + 2 y) n 2t+ j 2t+ Thus the bivariate generating function fz, y) := n W ny)z n satisfies fz, y) = y 2t + )! z2t+ 2t+ j n j ) t) t) W n j y). n 2t+ ) t 2 fz, y) + 2 y)π t!t! zt t z) t z t fz, y) 2 y)ϖ tz), where π t and ϖ t are polynomials, the initial condition being fz, y) = + z j +. j 4t+4 This equation, although different from what we have discussed up to now, does not change the analytic nature of the phase change. We still have a phase change at t = 58. The generalized quicksort can also be considered similarly. Bucket recursive trees Recursive trees are trees in which the numbers on all paths form an increasing sequence when traversed from root to leave. Bucket recursive trees are m-ary tree extension of recursive trees; see Mahmoud and Smythe [32] for details. For cost measures or structural parameters with linear mean, the underlying differential equation for the bivariate generating function is of the form m z m fz, y) = py)emfz,y). Again, this form is also manageable by our approach. The phase change is, as already observed in [32], at m = 26 asymptotic normality for m 26 is proved in [32]). 28

29 7 Exactly solvable equations We consider again 28). Three cases are exactly solvable: m, t) = 2, ), 2, ) and 3, ). The case m, t) = 2, ) is easily solved as fz, y) = py)z, and thus the underlying random variable, say ξ n, can be decomposed as the sum of n independent and identically distributed random variables each with the probability generating function equal to py). The other two cases are similar; we consider only the case m, t) = 3, ); see Panholzer and Prodinger [34] for the case m, t) = 2, ). More exactly solvable equations of the form y = c x c y c y ) c 2 y ) c 3 can be found in Polyanin and Zaitsev [37, 3.2]. Exact solution for m, t) = 3, ). The differential equation in question is 2 z 2 fz, y) = 2py)f 3 z, y), with the initial condition fz, y) = + q y)z +. For convenience, we write qy) instead of q y). The equation is solved in a similar way as in Panholzer and Prodinger [34]. Let Ff) = / z)fz, y). Then It follows that 2 z 2 f = df dz = df f df z = Ff)dF df. 2py)f 3 z, y)df = FdF, and, accordingly, by integrating and by taking into account the initial condition, Solving this equation, we obtain Thus f can be solved implicitly as py) f 4 z, y) ) = F 2 f) q 2 z). z fz, y) = pz)f 4 z, y) + q 2 y) py) ) /2. z = fz,y) du py)u 4 + q 2 y) py). 29) Asymptotic solution. Although f can be expressed in terms of Weierstrass s -function, and then analytic properties of the -function can be used to derive asymptotic approximations of [z n ]fz, y), we use instead a direct approach without resort to elliptic function to derive the following asymptotic expression. 29

30 Proposition 6. The probability generating function [z n ]fz, y) satisfies for any K > and y near the unity, where [z n ]fz, y) = py) /2 ζy) n + On K ) ), 3) ζy) := du py)u 4 + q 2 y) py). 3) From this result, we can deduce several results on the underlying random variable, say ξ n having probability generating function [z n ]fz, y), by the results in [7, 8, 9]; we give only the result for convergence rate, details as well as other finer results being omitted here. Theorem 3. The random variable ξ n is asymptotically normally distributed ) sup P ξn µ n < x x e u2 /2 du σ n 2π n = O /2), <x< provided that σ, where µ := q ) + 2p ), 5 σ 2 := p ) + q ) + q ) + 2p ) ) 28p ) 2 + 7q ) 2 + 8p )q ) ) The mean and variance satisfy for n 4 Eξ n ) = µ n + q ) p ) 5, VarX n ) = σ 2 n + 9p ) 2 4q ) 2 6p )q ) 5 p ) 2q )+p ) 2q ). Proof of Proposition 6. Observe first that fz, ) = / z). From 29), we deduce that for y near the unity, say y ε, fz, y) as z. Since the integral in 3) converges, we can rewrite 29) as := ζy) z = fz,y) du py)u 4 + q 2 y) py). The right-hand side can be asymptotically approximated by writing for simplicity p = py), q = qy) and f = fz, y)) fz,y) du py)u 4 + q 2 y) py) = p /2 f q2 p p 3/2 f 5 + p2 2pq 2 + q 4 p 5/2 f 9 +, 24 as f, from which we invert the expansion, giving f = p /2 q2 p p/2 3 + q2 p) 2 p 3/2 7 +, 32) 6 for z in C. This process is purely formal, but can be easily justified analytically by observing that all functions involved are analytic functions and by applying the inverse function theorem. Indeed the precise error term in 3) depends on the radius of convergence of the associated inversion series. The proposition follows by applying the singularity analysis []. In particular, if py) = y and qy) = y, then ξ n represents the total number of nodes in a random ternary search tree of n keys and from the above results, we have µ = 3/5 and σ 2 = 2/75. Thus in this special case, the above theorem is stronger than Theorem. 3