Dependence and phase changes in random m-ary search trees

Dependence and phase changes in random m-ary search trees

Hua-Huai Chern

Department of Computer Science National Taiwan Ocean University

Keelung 202 Taiwan

Michael Fuchs

Department of Applied Mathematics National Chiao Tung University

Hsinchu 300 Taiwan Hsien-Kuei Hwang

Institute of Statistical Science Academia Sinica

Taipei 115 Taiwan

Ralph Neininger

Institute for Mathematics

Goethe University 60054 Frankfurt a.M.

Germany February 26, 2016

Abstract

We study the joint asymptotic behavior of the space requirement and the total path length (either summing over all root-key distances or over all root-node distances) in ran- domm-ary search trees. The covariance turns out to exhibit a change of asymptotic be- havior: it is essentially linear when 3 6 m 6 13 but becomes of higher order when m > 14. Surprisingly, the corresponding asymptotic correlation coefficient tends to zero when3 6m 626but is periodically oscillating for largerm, and we also prove asymp- totic independence when36m626. Such a less anticipated phenomenon is not excep- tional and we extend the results in two directions: one for more general shape parameters, and the other for other classes of random log-trees such as fringe-balanced binary search trees and quadtrees. The methods of proof combine asymptotic transfer for the underlying recurrence relations with the contraction method.

AMS 2010 subject classifications. Primary 60F05, 68Q25; secondary 68P05, 60C05, 05A16.

Key words.m-ary search tree, correlation, dependence, recurrence relations, fringe-balanced binary search tree, quadtree, asymptotic analysis, limit law, asymptotic transfer, contraction method.

∗Partially supported by the Ministry of Science and Technology, Taiwan under the grant MOST-103-2115-M- 009-007-MY2.

†This author’s research stay at J. W. Goethe-Universit¨at was partially supported by the Simons Foundation and by the Mathematisches Forschungsinstitut Oberwolfach.

‡Supported by DFG grant NE 828/2-1.

arXiv:1501.05135v3 [math.PR] 25 Feb 2016

1 Introduction

The m-ary search trees are a class of data structures introduced by Muntz and Uzgalis [35]

in 1971 in computer algorithms to support efficient searching and sorting of data; see the next section for more details. When constructed from a random permutation ofnelements, the space requirement (total number of nodes to store the input) S_n of suchrandomm-ary search trees (m > 3) is known to exhibit a phase change phenomenon: its distribution is asymptotically Gaussian for largenwhen the branching factormsatisfies36m 626but does not approach a limit law when m > 27; see [8, 22, 30, 31] and the references therein. On the other hand, it is also known that the total key path length K_n(the sum over all distances from the root to anykey) does not change its limiting behavior whenmvaries, and tends asymptotically, after properly centered and normalized, to a limit law for eachm>3. Another closely related shape measure, the total node path lengthN_n(summing over all distances from the root to anynode) also follows asymptotically a very similar behavior.

Our motivating question was “how does K_n or N_n depend on S_n?” Surprisingly, despite the strong dependence of the definition of N_n on S_n (see (2)), we show that the correlation coefficientρ(S_n, N_n)satisfies

ρ(Sn, Nn)∼

(0, if36m626;

F_ρ(βlogn), ifm>27, (1) where F_ρ(t)is a 2π-periodic function and β = β_m is a structural constant depending on m.

The same type of results also holds for ρ(S_n, K_n). In words, N_n and S_n are asymptotically uncorrelated for36m 626and their correlation fluctuates (between−1and1) form>27;

see Figure1for an illustration.

Figure 1: The periodic functions F_ρ(2πt)form = 27, . . . ,100(left) andF_ρ(βlogn)form = 27,54, . . . ,270(right).

One reason why the above result (1) may seem less or even counter-intuitive is because of the seemingly strong dependence of N_n onS_n in the recursive equations satisfied by both random variables

(S_n=^d S_I⁽¹⁾₁ +· · ·+S_I^(m)_m + 1,

N_n=^d N_I⁽¹⁾₁ +· · ·+N_I^(m)_m +S_I⁽¹⁾₁ +· · ·+S_I^(m)_m , (2) where the(S_i^(r), N_i^(r))’s are independent copies of (S_i, N_i), respectively, also independent of (I₁, . . . , I_m), and

P(I₁ =i₁, . . . , I_m =i_m) = 1

n m−1

, (3)

wheni₁, . . . , i_m > 0andi₁ +· · ·+i_m = n−m+ 1. Intuitively, we expect, from the above relations, that the node path lengthN_nwould have a strong correlation withS_n.

While one might ascribe this seemingly less intuitive result to the possibly nonlinear de- pendence betweenN_nandS_n, we enhance such an uncorrelation by a stronger joint limit law for(S_n, N_n)for36m626, which further accents the asymptotic independence betweenN_n andS_n; form>27, they are asymptotically dependent and we will derive a precise character- ization of their joint asymptotic distributions. See Section4for a more precise description of the joint asymptotic behaviors of(S_n, N_n)and(S_n, K_n).

Letα denote the real part of the second largest zero (in real parts) of the indicial equation Λ(z) = 0, where

Λ(z) = z(z+ 1)· · ·(z+m−2)−m!. (4) Then α < 1 for m < 14and 1 < α < ³₂ for 14 6 m 6 26; see Table 1. Also α → 2 as m → ∞; see [30, Sec. 3.3] for more properties ofα. The main reason thatρ(S_n, N_n)→ 0for

m 3 4 5 6 7 8 9 10

α −3 −2.5 −1.5 −0.768 −0.260 0.101 0.366 0.568

m 11 12 13 14 15 16 17 18

α 0.726 0.852 0.955 1.040 1.112 1.173 1.226 1.272

m 19 20 21 22 23 24 25 26

α 1.313 1.348 1.380 1.409 1.435 1.458 1.479 1.499 Table 1:Approximate numerical values ofα =α_m for36m626.

3 6 m 6 26is roughly that their covariance is of order max{nlogn, n^α} (see Theorem2.3 below), while the standard deviations forS_nandN_nare of orders√

n andn, respectively. So that

ρ(S_n, N_n) =





 O

n⁻¹² logn

, if36m613;

O

n^−32+α

, if146m626,

which tends to zero in both cases. Briefly, the large quadratic variance of N_n is the major cause of the asymptotic independence betweenS_nandN_nfor36m626.

Such a change from being asymptotically independent to being asymptotically dependent under a varying structural parameter is not an exception. We will extend our study to fringe- balanced binary search trees and quadtrees; a typical related instance states that: the number of comparisons (or exchanges) used by the median-of-(2t+ 1)quicksort is asymptotically inde- pendent of the number of partitioning stages when06t 658, but is asymptotically dependent fort >59.

2 M -ary search trees

We briefly introducem-ary search trees in this section and then describe the random variables we are studying in this paper.

Anm-ary treeis either empty or comprises of a single node called the root, together with an orderedm-tuple of subtrees, each of which is, by definition, anm-ary tree. Given a sequence

6

2 8

1 4 7 10

3 5 9

2,6

1 4,5 7,8

3 9,10

2,4,6

1 3 5 7,8,9

10

Figure 2: Threem-ary search trees for the sequence{6,2,4,8,7,1,5,3,10,9}: m = 2(left), m= 3(middle), andm = 4(right).

of numbers, say{x₁, . . . , x_n}, we construct anm-ary search tree by the following procedure, m > 2. If 1 6 n < m, then all keys are stored in the root. If n > m the first m − 1 keys are sorted and stored in the root, the remaining keys are directed to them subtrees, each corresponding to one of themintervals formed by them−1sorted keys in the root node; see Figure2for an illustration (the rectangular nodes denote yet empty subtrees of full nodes). If them−1numbers in the root arex_j1 <· · · < x_jm−1, then the keys directed to theith subtree all have their values lying betweenx_ji−1 andx_ji, wherex_j0 := 0andx_jm :=n+ 1. All subtrees are themselvesm-ary search trees by definition. For more details, see Mahmoud [30].

While the practical usefulness ofm-ary search trees is largely overshadowed by their bal- anced counterparts such as B-trees, they have been a source of many interesting phenomena, which are to some extent universal. The study ofm-ary search trees is thus of fundamental and prototypical value. Furthermore, the close connection betweenm-ary search trees and general- ized quicksort adds an extra dimension to the richness of diverse variations and their asymptotic behaviors.

2.1 Space requirement and total path lengths

Assume that the input sequence{x1, . . . , xn}is a random permutation, where alln!permuta- tions are equally likely. The resultingm-ary search tree constructed from the given sequence is then called a randomm-ary search tree. The major shape parameters of particular algorithmic interest include the depth, the height, the space requirement, the total path length, and the pro- file; see [11,30] for more information. We are concerned in this paper with the following three random variables.

• S_n(space requirement): the total number of nodes used to store the input; the three trees in Figure2haveS10equal to10,6,6, respectively. Ifm = 2, thenSn ≡n; ifm>3, we can computeS_nrecursively byS₀ = 0, and

S_n =^d

(1, if16n < m,

S_I⁽¹⁾₁ +· · ·+S_I^(m)_m + 1, ifn >m, (5) where the S_i^(r)’s are independent copies ofS_i, 1 6 r 6 m, 0 6 i 6 n−m+ 1, and independent of(I1, . . . , Im)defined in (3).

• K_n(key path length, KPL): the sum of the distance between the root and each key; for the trees in Figure2,K₁₀ ={19,11,8}, respectively. Form>2,K_nsatisfies the recurrence

Kn

=d

(0, ifn < m,

K_I⁽¹⁾

1 +· · ·+K_I^(m)

m +n−m+ 1, ifn>m, (6)

where the K_i^(r)’s are independent copies of K_i, 1 6 r 6 m,0 6 i 6 n −m + 1, independent of(I₁, . . . , I_m).

• N_n(node path length, NPL): the sum of the distance between the root and each node; so thatN₁₀= {19,7,6}for the three trees in Figure2. Obviously,N_n =K_nwhenm = 2.

Whenm >3, Nn

=d

(0, ifn < m,

N_I⁽¹⁾

1 +· · ·+N_I^(m)

m +S_I⁽¹⁾

1 +· · ·+S_I^(m)

m , ifn >m, (7)

where the (N_i^(r), S_i^(r))’s are independent copies of(Ni, Si), 1 6 r 6 m,0 6 i 6 n− m+ 1, independent of(I₁, . . . , I_m).

While the first two random variables have been widely studied in the literature, NPL was only considered previously in [4,21] in connection with the process of cutting trees. In addition to this, our interest was to understand the extent to which the asymptotic independence for smallmbetweenS_n andK_nsubsists when the “toll function” changes from a linear function to a function that is random and may depend onS_n.

2.2 A summary of known results

LetH_m :=P

16j6mj⁻¹. Knuth [27,§6.2.4] was the first to show that E(Sn)∼φn, where φ:= 1

2(H_m−1),

(see also [1]). Hereφdenotes the “occupancy constant”, which will appear all over our analysis.

Mahmoud and Pittel [31] improved the result and derived an identity forE(S_n), which implies in particular that

E(S_n) =φ(n+ 1)− 1

m−1 +O n^α−1 ,

whereαhas the same meaning as in Introduction; see (4). They also discovered and proved the surprising result for the variance

V(S_n)∼

(C_Sn, if36m 626;

F₁(βlogn)n^2α−2, ifm>27,

where CS is a constant depending on m, F1 is a π-periodic function given in (24), α +iβ is the second largest zero (in real part) with β > 0 of the equation Λ(z) = 0(see (4)), and 2α−2>1form>27. See also [9,25,33] for a closely related fragmentation model with the same asymptotic behavior. A central limit theorem forSnwas then proved for36 m6 26in

[28,31]; see also [30] for more details. Their approach is based on an inductive approximation argument.

By the method of moments, two authors of this paper re-proved in [8] the central limit the- orem forS_nwhen36m626; the same approach was also used to establish the nonexistence of a limit law forS_ndue to inherent oscillations. Moreover, the convergence rates to the normal distribution were characterized in [22] by a refined method of moments, which undergo further change of behaviors.

Then several different approaches were developed in the literature for a deeper understand- ing of the “phase change” at m = 26; these include martingale [6], renewal theory [25], urn models [23, 32], contraction method [13, 39], method of moments [22], statistical physics [9,33], etc.

On the other hand, the KPL for general m > 2was first studied by Mahmoud [29] and he proved

E(K_n) = 2φnlogn+c₁n+o(n),

for some explicitly computable constantc1; see (21). The variance was computed in [30, §3.5]

and satisfies (Hm⁽²⁾ :=P

16j6mj⁻²)

V(K_n)∼C_Kn², where C_K = 4φ²_(m+1)H(2) m−2

m−1 − ^π₆²

. (8)

The corresponding limit law was characterized in [38] by the contraction method K_n−E(K_n)

n

−→d K, (9)

where K is given by the recursive distributional equation (44); see also [4, 34] for a general framework.

For NPLN_n, Broutin and Holmgren [4] proved that

E(N_n) = 2φ²nlogn+c₂n+o(n),

for some constant c₂ (for which no numerical value was provided); a series expression of c₂ is given in [21, p. 156]. We will give an alternative proof of this result below with tools from [8, 14]. Our approach makes the computation of c₂ feasible (although its exact value is not needed); see (27).

It should be mentioned that there is a large literature on K_n when m = 2 because it is identical to the comparison cost used by quicksort. Many fine results were obtained; see, for example, the recent papers [3,12,17,20, 37, 41] and the references therein for more informa- tion.

2.3 Covariance, correlation, dependence and phase changes

We state in this section our results for the covariance and correlation between the space require- ment and the total path lengths (KPL and NPL). The proofs and the tools needed will be given in the next sections.

Unlike the space requirementS_nwhose variance changes its asymptotic behavior form >

27, the covarianceCov(S_n, K_n)changes its asymptotic behavior atm= 14.

Theorem 2.1. The covariance betweenS_nandK_nsatisfies Cov(S_n, K_n)∼

(C_Rn, if36m613;

F₂(βlogn)n^α, ifm >14;

whereC_Ris a suitable constant andF₂(z)is a2π-periodic function given in(25)below.

This result has the following consequence.

Corollary 2.2. The correlation coefficient betweenSnandKnsatisfies

ρ(S_n, K_n)







→0, if36m626;

∼ F₂(βlogn)

pC_KF₁(βlogn), ifm>27, whereC_K >0is given in(8).

See Figure1for two different plots for the periodic functions whenm>27.

The same consideration extends easily to clarify the correlation between space requirement and NPL.

Theorem 2.3. The covariance betweenS_nandN_nsatisfies Cov(S_n, N_n)∼

(2φC_Snlogn, if36m613;

φF₂(βlogn)n^α, ifm>14, whereC_S is as in Section2.2. Moreover, the variance ofN_nsatisfies

V(N_n)∼φ²C_Kn².

Notice the appearance of an extra logn factor when 3 6 m 6 13, which reflects the additional random effect introduced by the toll function in (7). These estimates imply the following consequence.

Corollary 2.4. The correlation coefficientρ(S_n, N_n)satisfies

ρ(S_n, N_n)







→0, if36m626;

∼ρ(S_n, K_n)∼ F₂(βlogn)

pC_KF₁(βlogn), ifm>27.

The last relation suggests considering the correlation betweenK_nandN_n. Corollary 2.5. The random variableKnis asymptotically linearly correlated toNn

ρ(K_n, N_n)→1.

Indeed, we will show that

kNn−φKn−(E(Nn−φKn))k2 =o(n) which then by Slutsky’s theorem implies that

K_n−E(K_n)

n ,N_n−E(N_n) n

−→d (K, φK);

see (9), Section4.3and4.4.

These results will be proved by working out the asymptotics of the corresponding recur- rence relations, which all have the same form

a_n=m X

06j6n−m+1

π_n,ja_j+b_n, (n >m−1), where

π_n,j =

n−1−j m−2

n m−1

(06j 6n−m+ 1)

is a probability distribution, and {b_n} is a given sequence (referred to as the toll-function).

For that asymptotic purpose, our key tools will rely on theasymptotic transfer techniques(see [8, 14]), which provide a direct asymptotic translation from the asymptotic behaviors of b_n to those ofa_n. The remaining analysis will then consist of simplifying some multiple Dirichlet’s integrals.

Since Pearson’s product-moment correlation coefficientρis known to be poor in measuring nonlinear dependence between two random variables, we go further by considering the joint limit laws for(S_n, K_n)and(S_n, N_n), which exhibit a change of behavior depending on whether 36m626(convergent case) orm>27(periodic case): they are asymptotically independent in the former case but dependent in the latter.

Theorem 2.6. Assume36m626. Let(X_n)_n ∈ {(K_n)_n,(N_n)_n}andQ_n = (X_n, S_n)denote the vector of KPL or NPL and the space requirement used by a random m-ary search tree.

Then the convergence in distribution holds:

Cov(Q_n)^−1/2(Q_n−E[Q_n])−→^d (X,N ), (10) where N has the standard normal distribution and the limit law (X,N ) is described in Lemma4.2; moreover,XandN are independent.

Theorem 2.7. Assumem>27. Let(X_n)_n ∈ {(K_n)_n,(N_n)_n}and Y_n :=

X_n−E[X_n]

ι_Xn ,S_n−φn n^α−1

withι_X = 1for(X_n)_n= (N_n)_nandι_X =φ⁻¹ for(X_n)_n = (K_n)_n. Then we have

`₂(Y_n,(X,<(n^iβΛ)))→0,

whereβis as in Section2.2and(X,Λ)is a random vector whose distribution is specified as the unique fixed point solution appearing in Lemma4.1 for the choiceγ = (0, θ)(θbeing defined below in (28)).

See Section4for a more precise formulation. The proof is based on thecontraction method (see [36]) where we use the above moment asymptotics as input and combine well-known estimates within the minimal L₂-metric for the convergent case (as in [40]), and those with estimates for the periodic case (as in [13]). Similar proof techniques related to periodic distri- butional behaviors are also applied in [25, Theorem 1.3(iii)] and [26, Theorem 6.10]. If one is only interested in the asymptotic (univariate) distribution of the NPLN_n(the case of the KPL being known before), there are more direct proofs which we also discuss in Sections4.3 and 4.4.

Our study of the dependence of random variables on random m-ary search trees can be extended in at least two directions by the same methods used in this paper, namely, asymptotic transfer techniques and the contraction method.

• Extension to more general linear and nlogn shape measures: That the asymptotic co- variance undergoes a phase change afterm = 13and the asymptotic correlation under- goes a phase change afterm = 26is not restricted to the space requirement and KPL or NPL. Indeed, we can replace the space requirement by many other linear shape measures such as the number of leaves, the number of nodes of a specified type, the number of occurrences of a fixed pattern, etc. (see [8] for more examples), and KPL or NPL by other shape measures with mean of ordernlognsuch as summing over the root-node or root-key distance for certain specified nodes or patterns and weighted path length.

• Extension to other random trees of logarithmic height: the same change of asymptotic behaviors from being independent to being dependent under a varying structural pa- rameter also occurs in other classes of random log-trees; we content ourselves with the brief discussion of two classes of random trees:fringe-balanced binary search treesand quadtrees. The behaviors will be however very different for the classes of trees where the underlying distribution of the subtree sizes are dictated by a binomial distribution, which will be examined elsewhere; see a companion paper [18] for more information.

This paper is organized as follows. We prove in the next section our results for the co- variances and the correlations. These results are then used to study the bivariate distributional asymptotics in Section 4 by the multivariate contraction method (see [36]). Finally, in Sec- tion 5, we discuss the dependence and phase changes in fringe-balanced binary search trees and in quadtrees, where for the former, we study the joint behavior of the size and total path length, while for the latter (since the size is a constant) we consider the joint behavior of the number of leaves and total path length. Also we include a brief discussion for extending the study and results to other shape parameters in Section5.

3 Correlation between space requirement and path lengths

We prove in this section Theorems2.1and2.3for the covariances Cov(S_n, K_n)and Cov(S_n, N_n), respectively.

3.1 Preliminaries and recurrences

We collect here the notations to be used in the proofs. Let m > 2 be a fixed integer. For n > m, denote by I⁽ⁿ⁾ = (I₁⁽ⁿ⁾, . . . , Im⁽ⁿ⁾) the vector of the number of keys inserted in the m

ordered subtrees of the root in a randomm-ary search tree withnkeys. When the dependence on n is obvious, we write simply (I₁, . . . , I_m). Generate independently n uniform random variablesU₁, . . . , U_non[0,1]. Store the firstm−1elementsU₁, . . . , Um−1 in the root-node of the tree. Then they decompose the unit interval[0,1]into spacings of lengthsV₁, . . . , V_m, where V_j =U_(j)−U(j−1)forj = 1, . . . , mwithU₍₀₎ := 0, U_(m) := 1andU_(j)forj = 1, . . . , m−1are the order statistics ofU₁, . . . , Um−1. The uniform permutation model implies, that, conditional on U₁, . . . , U_m−1, the vector I⁽ⁿ⁾ has the multinomial distribution with success probabilities V₁, . . . , V_m, namely, we have

(I1, . . . , Im)=^d M(n−m+ 1;V1, . . . , Vm).

In particular, we have the convergence I_r

n −→V_r, (11)

for allr= 1, . . . , m, where the convergence is inL_pfor all16p < ∞. Note that we also have (3) for allm-tuplesi1, . . . , im >0withi1+· · ·+im =n−m+ 1and alln >m.

For each of the subtrees, the randomness (uniformity) is preserved; more precisely, condi- tional on the number of keys inserted in a subtree, each subtree has the same distribution as a randomm-ary search tree of that number of keys in the uniform model. Moreover, condi- tional on (I₁, . . . , I_m), the subtrees are independent. This can be seen by switching back to the ranks {1, . . . , n} of the input elements, and then by checking that a uniform random per- mutation yields independent permutations on the respective ranges. This recursive structure of the random m-ary search tree implies the recursive relations for S_n, K_n and N_n given in (5)–(7), where the summands appearing on the right-hand sides, namely, S_j⁽¹⁾, . . . , S_j^(m) and K_j⁽¹⁾, . . . , K_j^(m) andN_j⁽¹⁾, . . . , N_j^(m) have the same distributions asS_j andK_j andN_j, respec- tively. Furthermore, the triples

S_j^(r)

06j6n−m+1, K_j^(r)

06j6n−m+1, N_j^(r)

06j6n−m+1

are independent forr = 1, . . . , mand independent of(I₁, . . . , I_m). Finally, the recursive structure of them-ary search tree implies recurrences satisfied by their joint distributions. In particular, the pairQ_n:= (N_n, S_n)satisfies the recurrence

(Q_n)^t=^d X

16r6m

h 1 1 0 1

i Q^(r)_I

r

t

+ 0

1

, (n>m), (12)

where, as in (5)–(7), theQ^(r)_j ’s are distributed asQ_jfor all16r 6mand06j 6n−m+ 1, and the Q^(r)_j

06j6n−m+1 are independent for r = 1, . . . , m and independent of(I1, . . . , In).

The recurrence satisfied by the pairZ_n:= (K_n, S_n)is (Z_n)^t =^d X

16r6m

h 1 0 0 1

i Z_I^(r)_r t

+

n−m+ 1 1

, (n >m), (13) with conditions on independence and identical distributions similar to (12).

3.2 Asymptotic transfer and Dirichlet integrals

Starting from the distributional recurrences (5) and (6), we see that all centered and non- centered moments satisfy the same recurrence of the following type

a_n=m X

06j6n−m+1

π_n,ja_j +b_n, π_n,j =

n−1−j m−2

n m−1

, (14) forn > m−1, where {b_n}_n>m−1 is a given sequence. The asymptotics ofa_ncan be system- atically characterized by that of b_n through the use of the following transfer techniques; see Proposition 7 in [8] and Theorem 2.4 in [14] for details.

Proposition 3.1. Assume thatansatisfies (14) with finite initial conditionsa0, . . . , am−2. Define b_n :=a_nfor06n6m−2.

(i) Assumebn=c(n+ 1) +tn, wherec∈C. Then the conditions t_n=o(n) and

X

n>1

t_nn⁻²

<∞ are both necessary and sufficient for

a_n = 2cφnH_n+c⁰n+o(n), where

c⁰ = 2φX

j>0

t_j

(j+ 1)(j + 2) + c

2 −2cφ+ 2c(H_m⁽²⁾−1)φ²; (ii) ifb_n∼cn^v, wherev >1, then

a_n∼ c

1−^m!Γ(v+1)_Γ(v+m) n^v.

In particular, whenc= 0in(i), then we see thata_nis asymptotically linear an

n ∼2φX

j>0

bj

(j+ 1)(j + 2) iff b_n=o(n) and

X

n>1

b_nn⁻²

<∞.

We will be dealing with Dirichlet integrals of the following type I(u, v) :=

Z

x1+···+xm=1 06x1,...,xm61

X

16l6m

x^u−1_l

! X

16r6m

x^v−1_r

!

dx, (<(u),<(v)>0).

Heredxis an abbreviation fordx1· · ·dxm−1. Such integrals have a closed-form expression.

Lemma 3.2. Form>2and<(u),<(v)>0,

I(u, v) = mΓ(u+v−1) +m(m−1)Γ(u)Γ(v)

Γ(u+v+m−2) . (15)

Proof.First, the claim is easily proved form = 2. Assumem >3. Then, by symmetry, I(u, v) =

Z

x1+···+xm=1 06x1,...,xm61

mx^u+v−2₁ +m(m−1)x^u−1₁ x^v−1₂ dx

= m

(m−2)!

Z 1 0

x^u+v−2₁ (1−x₁)^m−2dx₁ +m(m−1)

(m−3)!

Z 1 0

Z 1−x1

0

x^u−1₁ x^v−1₂ (1−x₁−x₂)^m−3dx₂dx₁

= mΓ(u+v−1)

Γ(u+v+m−2)+ m(m−1)Γ(u)Γ(v) Γ(u+v +m−2) , which leads to (15).

The following two identities will be needed below.

Z

x1+···+xm=1 06x1,...,xm61

X

16l6m

x^u−1_l

! X

16r6m

x_rlogx_r

! dx

= ∂

∂vI(u, v) v=2

= mΓ(u)

Γ(m+u)(uψ(u+ 1) + (m−1)(1−γ)−(m+u−1)ψ(m+u)),

(16)

whereψ is the digamma function andγ is Euler’s constant. Similarly, Z

x1+···+x_m=1 06x1,...,xm61

X

16r6m

x_rlogx_r

!2

dx= ∂²

∂u∂vI(u, v) u=v=2

=H_m⁽²⁾+ 4

φ² − 2

m+ 1 − (m−1)π² 6(m+ 1) .

(17)

3.3 Correlation between the space requirement and KPL

We are now ready to prove Theorem2.1.

Expected values ofS_nandK_n. For convenience, letµ_n:=E(S_n)andκ_n :=E(K_n). Then, by (5) and (6), forn >m−1

µ_n=m X

06j6n−m+1

π_n,jµ_j + 1, κ_n=m X

06j6n−m+1

π_n,jκ_j+n−m+ 1,

with the initial conditionsµ₀ =κ_n= 0for06n 6m−2andµ_n= 1for16n 6m−2.

By applying Proposition3.1(i), we obtain

µ_n∼φn, and κ_n= 2φnlogn+c₁n+o(n), (18)

for some constantc₁ whose value matters less; see (21) below. The latter approximation is suf- ficient for all our purposes, but the former is not and we need the following stronger expansion (see [8,31,30])

µ_n =φ(n+ 1)− 1

m−1 + X

26k63

A_k

Γ(λ_k)n^λk−1+o(n^α−1), (19) whereλ2 =α+iβ andλ3 :=α−iβand

A_k = 1

λ_k(λ_k−1)P

06j6m−2 1 j+λk

. (20)

Note that for3 6 m 613the constant term−_m−1¹ (together withφ) is the second-order term on the right-hand side of (19), while for largerm, it is absorbed in theo-term.

On the other hand, although the explicit expression of c₁ is not needed in this paper, we provide its expression here since the known ones (see [29, 30]) are less explicit and it can be easily obtained from Proposition3.1:

c₁ =−¹₂ −4φ+ 2φ²(H_m⁽²⁾−1) +γ. (21) Variance and covariance. To compute the asymptotics of the covariance, we first derive the corresponding recurrences and then apply Proposition3.1of asymptotic transfer.

First, letS¯n =Sn−µnandK¯n =Kn−κn. We consider the moment-generating function P¯_n(u, v) := E

e^{S¯nu+ ¯Knv} . Then, using (5) and (6), we obtain forn>m−1

P¯_n(u, v) = 1

n m−1

X

j

P_j1(u, v)· · ·P_jm(u, v)e^∆ju+∇jv (22) with the initial conditionsP¯_n(u, v) = 1for06n 6m−2. Here,j= (j₁, . . . , j_m)is a vector withj1, . . . , jm >0andj1+· · ·+jm =n−m+ 1(we use this notation throughout),

∆_j = 1−µ_n+ X

16l6m

µ_jl, and ∇_j =n−m+ 1−κ_n+ X

16l6m

κ_jl. (23) Define

V_n^[S]=V(Sn), V_n^[SK]= Cov(Sn, Kn), V_n^[K] =V(Kn).

Then, by taking derivatives in (22), we obtain V_n^[X]=m X

06j6n−m+1

π_n,jV_j^[X]+b^[X_n ^], (X ∈ {S, SK, K}), where

b^[S]_n = 1

n m−1

X

j

∆²_j, b^[SK]_n = 1

n m−1

X

j

∆_j∇_j, and b^[K]_n = 1

n m−1

X

j

∇²_j. We first derive uniform asymptotic approximations for∆jand∇j.

Lemma 3.3. Uniformly inj,

∆_j = X

26k63

A_k

Γ(λ_k)n^λk−1 −1 + X

16r6m

j_r n

λk−1!

+o(n^α−1), and

∇_j =n 1 + 2φ X

16r6m

j_r n log j_r

n

!

+o(n).

Proof. This follows from substituting the asymptotic approximations (18) and (19) into (23), and standard manipulations.

Asymptotics of Vn^[S]. Although the asymptotic behaviors of the variance of Sn have been computed before, we re-derive them here by a different approach, which is easily amended for the calculation of other variances and covariances.

Consider first36m626. Thenα <3/2. Moreover, from Lemma3.3, b^[S]_n =O(n^2α−2) = O(n^1−ε),

for some0< ε <0.00171. Consequently, by applying Proposition3.1(i), V_n^[S]∼C_Sn,

for some constantC_S; see [8] for a more explicit expression and the proof thatC_S >0.

On other hand, ifm >27, sinceα >3/2, we then have, by Lemmas3.2and3.3, b^[S]_n ∼ X

26k1,k263

(m−1)!Ak1Ak2n^{λk1+λk2−2} Γ(λ_k1)Γ(λ_k2)

× Z

x1+···+xm=1 06x1,...,xm61

−1 + X

16l6m

x^λ_l^k1−1

!

−1 + X

16r6m

x^λr^k2−1

! dx

∼ X

26k1,k263

A_k1A_k2n^{λk1+λk2−2}

Γ(λk1)Γ(λk2) 1− m!Γ(λ_k1)

Γ(λk1 +m−1)− m!Γ(λ_k2) Γ(λk2 +m−1) + m!Γ(λ_k1 +λ_k2 −1)

Γ(λ_k1 +λ_k2 +m−2)+ m!(m−1)Γ(λ_k1)Γ(λ_k2) Γ(λ_k1 +λ_k2 +m−2)

! . Note that

m!Γ(λ_kj)

Γ(λ_kj +m−1) = 1, (26j 63).

Applying Proposition3.1(ii) term by term then gives V_n^[S] ∼ X

26k1,k263

A_k1A_k2n^{λk1+λk2−2} Γ(λ_k1)Γ(λ_k2)

−1 + m!(m−1)Γ(λ_k1)Γ(λ_k2)

Γ(λ_k1 +λ_k2 +m−2)−m!Γ(λ_k1 +λ_k2 −1)

=:F₁(βlogn)n^2α−2,

where

F₁(z) := 2 |A2|²

|Γ(λ₂)|²

−1 + m!(m−1)|Γ(λ2)|² Γ(2α+m−2)−m!Γ(2α−1)

+ 2<

A²₂e^2iz Γ(λ2)²

−1 + m!(m−1)Γ(λ₂)²

Γ(2λ2+m−2)−m!Γ(2λ2−1)

.

(24)

Asymptotics ofVn^[SK]. We now turn toVn^[SK]. If36m613, then, by Lemma3.3, b^[SK]_n =O(n^α),

whereα <1. Consequently, by Proposition3.1(i), V_n^[SK]∼C_Rn,

for some constantC_R. For the remaining range wherem>14, we haveα >1, and, by Lemma 3.3and (16),

b^[SK]_n ∼ X

26k63

(m−1)!A_kn^λk Γ(λ_k)

Z

x1+···+xm=1 06x1,...,xm61

−1 + X

16l6m

x^λ_l^k−1

!

1 + 2φ X

16r6m

x_rlogx_r

! dx

∼ X

26k63

A_kn^λk

Γ(λ_k) 1−2φm!Γ(λ_k+ 1) Γ(λ_k+m)

mψ(λ_k+m)−ψ(λ_k+ 1)−(m−1)(1−γ)

! . Now, we apply Proposition3.1(ii) and again after some straightforward simplifications

V_n^[SK]∼F₂(βlogn)n^α, where

F₂(z) := 2φ< (λ₂+m−1)A₂e^iz (m−1)Γ(λ₂)

1

2φ − λ₂ λ₂+m−1

mψ(λ₂+m)−ψ(λ₂+ 1)

−(m−1)(1−γ)

!!

.

(25)

Asymptotics ofVn^[K]. In a similar manner, we obtain, by Lemma3.3, b^[K]_n ∼(m−1)!n²

Z

x1+···+xm=1 06x1,...,xm61

1 + 2φ X

16l6m

x_llogx_l

!2

dx

∼4φ²n²

H_m⁽²⁾− 2

m+ 1 −π²(m−1) 6(m+ 1)

,

where the last line follows from applying (15), (16) and (17). Applying again Proposition 3.1(ii) gives

V_n^[K] ∼C_Kn², which completes the proof of Theorem2.1.

3.4 Correlation between space requirement and NPL

The calculations in this case are similar to those for ρ(S_n, K_n), so we only sketch the major steps needed. Briefly, most asymptotic estimates differ either by a factor of the occupancy constant φ or its powers. The only exception is the additional factor logn appearing in the covarianceCov(S_n, N_n)(see (2.3)).

Letν_n =E(N_n). Then

ν_n=m X

06j6n−m+1

π_n,jν_j +µ_n−1.

Consequently, by the asymptotic estimate (19) and by applying Proposition3.1(i), we obtain ν_n= 2φ²nlogn+c₂n+o(n), (26) where, by Proposition3.1,

c₂ =φc₁+ 2φ φ− 1

m−1 + X

26`6m−1

A_{`</sub> 2−λ<sub>`}

!

, (27)

c1 being given in (21) and the A`’s defined in (20). Indeed, consider the difference ξn :=

ν_n−φκ_n, which then satisfies the same recurrence (14) but with the toll function η_n:=µ_n−1−φ(n−m+ 1) =φm− m

m−1 + X

26`<m

A_`

n+λ_`−1 n

, andξ_n= 0for16n 6m−2. Then by applying Proposition3.1, we obtain

c₂−c₁φ= 2φ X

j>m−1

η_j

(j+ 1)(j + 2).

Sinceη_n =−φ(n−m+ 1)for16 n 6m−2andη₀ =φ(m−1)−1, we then derive (27) by the relation

X

j>0

λ+j−1 j

(j+ 1)(j + 2) =

Z 1 0

(1−t)^−λ+1dt= 1

2−λ (<(λ)<2).

In particular,c₂−φc₁equals

12

125,₂₁₉₇²²²,₄₅₆₅₃₃⁴⁴⁶⁷⁰,₇₅₆₉⁷¹⁰,_99806103^8990170,_1001561769^86959460 ,97908438529^8225243460,114862129381^9368632980 , form = 3, . . . ,10, and

13941168359580

175531341607271, 15364018080180

198165483844901, 36778736979244260

484907780151231137, 39706104830251860

534148059351752117, 42542306175669300 583013664848115773, 362341148683714200

5051607560589134719, 60809828396490973800

861420713064800471777, 220781849887636437400

3174476111482140491583, 1589879045909940738152200 23180880112213178399314917, 66535629228892650939112

982905224931956375768865, 69399644946307963559272

1037954891250806970920625, 72191400913204902200872 1092384284013327674677545, 911488027263952226045421464

13945777153309079949132939375, 943834826916499599456679304

14593082411910111966602252205, 3048229719576792424490262245800 47603282606571951420821994029889, 3144754504512378111611222765800

49580602253255626178697360169689, 787117453959995151898324789769400

12523181563980976087610969389067627, 809570585901011449194661971389400 12992983079952314295925927936613927, 20280854972612671613961769087339836600

328217277361176269245342166728792498003, 20806237502125190663861808383733444600 339424705221771320114642916145949390923

form= 11, . . . ,30.

Let N¯_n = N_n − ν_n. Then the moment-generating function P¯_n(u, v) := E e^{S¯nu+ ¯Nnv} satisfies forn>m−1

P¯_n(u, v) = 1

n m−1

X

j

P_j1(u+v, v)· · ·P_jm(u+v, v)e^∆ju+δjv, with the initial conditionsP¯n(u, v) = 1for06n6m−2and

δ_j :=−ν_n+ X

16l6m

(ν_jl+µ_jl). Now define

V_n^[SN]:= Cov(S_n, N_n) and V_n^[N] :=V(N_n).

Then

V_n^[X] =m X

06l6n−m+1

π_n,jV_j^[X]+b^[X_n ^], (X ∈ {SN, N}), where

b^[SN_n ^]= 1

n m−1

X

j

V_j^[S]+ ∆_jδ_j

=V_n^[S]+ 1

n m−1

X

j

∆_jδ_j−∆²_j b^[N_n ^]= 1

n m−1

X

j

V_j^[S]+ 2V_j^[SN]+δ²_j

=V_n^[S]+ 2V_n^[SN]+ 1

n m−1

X

j

δ_j²−2∆_jδ_j+ ∆²_j .

As in the case of KPL, the following uniform estimate is crucial in our analysis.

Lemma 3.4. Uniformly inj,

δ_j =φn 1 + 2φ X

16l6m

jl

n logjl

n

!

+o(n).

Proof.By the definition ofδ_j and the estimates (19) and (26).

Note that the expansion differs from that for∇_j in Lemma3.3by an additional factorφ.

If36m 613, then, by Lemmas3.3and3.4,

b^[SN_n ^]=C_Sn+O n^1−ε , for a sufficiently smallε >0. Thus, by Proposition3.1(i),

V_n^[SN]∼ C_Snlogn H_m−1 .

Assume nowm > 14. Then, again from Lemma3.3 and Lemma3.4together with the known asymptotics ofVn^[S], we see that

b^[SN_n ^]∼ 1

n m−1

X

j

∆_jδ_j ∼ φ

n m−1

X

j

∆_j∇_j ∼φb^[SK]_n .

Thus we deduce, as in the proof forVn^[SK],

V_n^[SN]∼φV_n^[SK]∼φF₂(βlogn)n^α. Similarly, we have

b^[N]_n ∼ 1

n m−1

X

j

δ_j² ∼ φ²

n m−1

X

j

∇²_j ∼φ²b^[K]_n . Consequently,

V_n^[N]∼φ²V_n^[K]∼φ²C_Kn². This completes the proof of Theorem2.3.

4 Bivariate distributional asymptotics for space requirement and path lengths

In this section, we identify the asymptotic joint distributional behaviors of the pairs (N_n, S_n) and(K_n, S_n). Although the sequences(N_n)and(K_n)converge after normalization for allm>

3with limit distributions depending on m, we split the analysis into two cases depending on 36m626orm >26due to the phase change in the limit behavior ofS_n. We discuss the pair (N_n, S_n)in detail in Sections 4.1and4.2. (the corresponding analysis for(K_n, S_n)is similar and we will not give details). Moreover, in Section4.3, we will show that the univariate limit random variables of the normalized sequences(N_n)and (K_n)do have the same distribution.

We introduce the following notation

µ(n) := µ_n =E[S_n] =φ(n+ 1) +<(θn^λ2−1) +o(1∨n^α−1), (28) where θ := 2A₂/Γ(λ₂); see (19). Similarly, write κ(n) = κ_n = E(K_n) and ν(n) = ν_n = E(N_n).

4.1 Node path length and space requirement. I. m > 27

We give in this section the precise formulation of the periodic casem>27of Theorem2.7.

Normalization. We first normalize the vectorQ_n= (N_n, S_n)as follows. LetY₀ := 0and Y_n:=

N_n−E[N_n]

n ,S_n−φn n^α−1

, (n >1).

Then the recurrence (12) implies forn>m−1 (Y_n)^t=^d X

16r6m

A⁽ⁿ⁾_r Y^(r)

Ir⁽ⁿ⁾

t

+b⁽ⁿ⁾, (29)

where

A⁽ⁿ⁾_r :=









 Ir⁽ⁿ⁾

n

Ir⁽ⁿ⁾

α−1

n 0 Ir⁽ⁿ⁾

n

!α−1











, b⁽ⁿ⁾ :=







 1 n

X

16r6m

ν I_r⁽ⁿ⁾

+φI_r⁽ⁿ⁾

−ν(n)

!

−φm−1 n^α−1







 ,

with assumptions on independence and on identical distributions as in Section3.1. The expan- sion (26) implies

1 n

X

16r6m

ν I_r⁽ⁿ⁾

+φI_r⁽ⁿ⁾

−ν(n)

!

=φ+ 2φ² X

16r6m

Ir⁽ⁿ⁾

n logIr⁽ⁿ⁾

n +o(1).

Moreover, by (11), we obtain theL₂-convergence I⁽ⁿ⁾

n

L2

−→(V₁, . . . , V_m) =:V. (30) This implies theL₂-convergences

1 n

X

16r6m

ν I_r⁽ⁿ⁾

+φI_r⁽ⁿ⁾

−ν(n)

!

→φ+ 2φ² X

16r6m

V_rlogV_r =:b_N, (31) and

b⁽ⁿ⁾→ b_N

0

, A⁽ⁿ⁾_r →

V_r 0 0 V_r^α−1

. (32)

For our limit result form>27, we first define a distribution which governs the asymptotics.

The limiting map. To describe the asymptotic behavior of Q_n, we use the following prob- ability distribution on the space R × C. Let M^R×C denote the space of all distributions L(Z, W) onR×Cand M^R₂^×C the subspace of distributions with finite second moment, i.e., k(Z, W)k₂ := (E[Z²] +E[|W|²])^1/2 <∞. Forγ = (γ1, γ2)∈R×C, let

M^R₂^×C(γ) :=n

L(Z, W)∈ M^R₂^×C

E[Z] =γ₁,E[W] =γ₂o . We define the following mapT_N onM^R₂^×C:

TN :M^R×C → M^R×C L(Z, W)7→ L X

16r6m

Vr 0 0 V_r^λ2−1

Z^(r) W^(r)

+

bN

0 !

, (33)