Dependence and phase changes in random $m$-ary search trees

0.65 MB·English

by  Hua-Huai Chern

Dependence and phase changes in random m-ary
search trees
Hua-Huai Chern Michael Fuchs∗
Department of Computer Science Department of Applied Mathematics
National Taiwan Ocean University National Chiao Tung University
Keelung 202 Hsinchu 300
6
Taiwan Taiwan
1
0
2 Hsien-Kuei Hwang† Ralph Neininger‡

b Institute of Statistical Science Institute for Mathematics
e
F Academia Sinica Goethe University

5 Taipei 115 60054 Frankfurt a.M.
2
Taiwan Germany

]
R
P February 26, 2016
.
h
t
a
m
Abstract
[

We study the joint asymptotic behavior of the space requirement and the total path
3
v length(eithersummingoverallroot-keydistancesoroverallroot-nodedistances)inran-
5 dom m-ary search trees. The covariance turns out to exhibit a change of asymptotic be-
3
havior: it is essentially linear when 3 (cid:54) m (cid:54) 13 but becomes of higher order when
1
5 m (cid:62) 14. Surprisingly, the corresponding asymptotic correlation coefficient tends to zero
0 when 3 (cid:54) m (cid:54) 26 but is periodically oscillating for larger m, and we also prove asymp-
.
1 toticindependencewhen3 (cid:54) m (cid:54) 26. Suchalessanticipatedphenomenonisnotexcep-
0
tionalandweextendtheresultsintwodirections: oneformoregeneralshapeparameters,
5
1 and the other for other classes of random log-trees such as fringe-balanced binary search
: treesandquadtrees. Themethodsofproofcombineasymptotictransferfortheunderlying
v
i recurrencerelationswiththecontractionmethod.
X
r AMS2010subjectclassifications. Primary60F05,68Q25;secondary68P05,60C05,05A16.
a
Keywords. m-arysearchtree,correlation,dependence,recurrencerelations,fringe-balanced
binary search tree, quadtree, asymptotic analysis, limit law, asymptotic transfer, contraction
method.
∗PartiallysupportedbytheMinistryofScienceandTechnology,TaiwanunderthegrantMOST-103-2115-M-
009-007-MY2.
†Thisauthor’sresearchstayatJ.W.Goethe-Universita¨twaspartiallysupportedbytheSimonsFoundationand
bytheMathematischesForschungsinstitutOberwolfach.
‡SupportedbyDFGgrantNE828/2-1.
1
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
sectionformoredetails. Whenconstructedfromarandompermutationofnelements,thespace
requirement (total number of nodes to store the input) S of such random m-ary search trees
n
(m (cid:62) 3) is known to exhibit a phase change phenomenon: its distribution is asymptotically
Gaussian for large n when the branching factor m satisfies 3 (cid:54) m (cid:54) 26 but does not approach
a limit law when m (cid:62) 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 (the sum over all distances from the root to
n
any key) does not change its limiting behavior when m varies, and tends asymptotically, after
properlycenteredandnormalized,toalimitlawforeachm (cid:62) 3. Anothercloselyrelatedshape
measure, the total node pathlength N (summing over all distances from the root to anynode)
n
alsofollowsasymptoticallyaverysimilarbehavior.
Our motivating question was “how does K or N depend on S ?” Surprisingly, despite
n n n
the strong dependence of the definition of N on S (see (2)), we show that the correlation
n n
coefficientρ(S ,N )satisfies
n n
(cid:40)
0, if3 (cid:54) m (cid:54) 26;
ρ(S ,N ) ∼ (1)
n n F (βlogn), ifm (cid:62) 27,
ρ
where F (t) is a 2π-periodic function and β = β is a structural constant depending on m.
ρ m
The same type of results also holds for ρ(S ,K ). In words, N and S are asymptotically
n n n n
uncorrelatedfor3 (cid:54) m (cid:54) 26andtheircorrelationfluctuates(between−1and1)form (cid:62) 27;
seeFigure1foranillustration.
Figure 1: The periodic functions F (2πt) for m = 27,...,100 (left) and F (βlogn) for m =
ρ ρ
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 on S in the recursive equations satisfied by both
n n
randomvariables
(cid:40)
S =d S(1) +···+S(m) +1,
n I1 Im (2)
N =d N(1) +···+N(m) +S(1) +···+S(m),
n I1 Im I1 Im
where the (S(r),N(r))’s are independent copies of (S ,N ), respectively, also independent of
i i i i
(I ,...,I ),and
1 m
1
P(I = i ,...,I = i ) = , (3)
1 1 m m (cid:0) n (cid:1)
m−1
2
when i ,...,i (cid:62) 0 and i + ··· + i = n − m + 1. Intuitively, we expect, from the above
1 m 1 m
relations,thatthenodepathlengthN wouldhaveastrongcorrelationwithS .
n n
While one might ascribe this seemingly less intuitive result to the possibly nonlinear de-
pendence between N and S , we enhance such an uncorrelation by a stronger joint limit law
n n
for (S ,N ) for 3 (cid:54) m (cid:54) 26, which further accents the asymptotic independence between N
n n n
and S ; for m (cid:62) 27, they are asymptotically dependent and we will derive a precise character-
n
ization of their joint asymptotic distributions. See Section 4 for a more precise description of
thejointasymptoticbehaviorsof(S ,N )and(S ,K ).
n n n n
Let α 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 < 14 and 1 < α < 3 for 14 (cid:54) m (cid:54) 26; see Table 1. Also α → 2 as
2
m → ∞; see [30, Sec. 3.3] for more properties of α. The main reason that ρ(S ,N ) → 0 for
n n
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
Table1: Approximatenumericalvaluesofα = α for3 (cid:54) m (cid:54) 26.
m
3 (cid:54) m (cid:54) 26 is roughly that their covariance is of order max{nlogn,nα} (see Theorem 2.3
√
below), while the standard deviations for S and N are of orders n and n, respectively. So
n n
that
 (cid:16) (cid:17)
O n−12 logn , if3 (cid:54) m (cid:54) 13;
ρ(Sn,Nn) = O(cid:16)n−32+α(cid:17), if14 (cid:54) m (cid:54) 26,
which tends to zero in both cases. Briefly, the large quadratic variance of N is the major
n
causeoftheasymptoticindependencebetweenS andN for3 (cid:54) m (cid:54) 26.
n n
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-
balancedbinarysearchtreesandquadtrees;atypicalrelatedinstancestatesthat: thenumberof
comparisons (or exchanges) used by the median-of-(2t + 1) quicksort is asymptotically inde-
pendentofthenumberofpartitioningstageswhen0 (cid:54) t (cid:54) 58,butisasymptoticallydependent
fort (cid:62) 59.
2 M-ary search trees
We briefly introduce m-ary search trees in this section and then describe the random variables
wearestudyinginthispaper.
Anm-arytreeiseitheremptyorcomprisesofasinglenodecalledtheroot,togetherwithan
ordered m-tuple of subtrees, each of which is, by definition, an m-ary tree. Given a sequence
3
6
2,6 2,4,6
2 8
1 4,5 7,8 1 3 5 7,8,9
1 4 7 10
3 9,10 10
3 5 9
Figure 2: Three m-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 {x ,...,x }, we construct an m-ary search tree by the following procedure,
1 n
m (cid:62) 2. If 1 (cid:54) n < m, then all keys are stored in the root. If n (cid:62) m the first m − 1
keys are sorted and stored in the root, the remaining keys are directed to the m subtrees, each
corresponding to one of the m intervals formed by the m−1 sorted keys in the root node; see
Figure 2 for an illustration (the rectangular nodes denote yet empty subtrees of full nodes). If
the m−1 numbers in the root are x < ··· < x , then the keys directed to the ith subtree
j1 jm−1
allhavetheirvalueslyingbetweenx andx ,wherex := 0andx := n+1. Allsubtrees
ji−1 ji j0 jm
arethemselvesm-arysearchtreesbydefinition. Formoredetails,seeMahmoud[30].
While the practical usefulness of m-ary search trees is largely overshadowed by their bal-
anced counterparts such as B-trees, they have been a source of many interesting phenomena,
whicharetosomeextentuniversal. Thestudyofm-arysearchtreesisthusoffundamentaland
prototypicalvalue. Furthermore,thecloseconnectionbetweenm-arysearchtreesandgeneral-
izedquicksortaddsanextradimensiontotherichnessofdiversevariationsandtheirasymptotic
behaviors.
2.1 Space requirement and total path lengths
Assume that the input sequence {x ,...,x } is a random permutation, where all n! permuta-
1 n
tionsareequallylikely. Theresultingm-arysearchtreeconstructedfromthegivensequenceis
then called a random m-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]formoreinformation. Weareconcernedinthispaperwiththefollowingthree
randomvariables.
• S (spacerequirement): thetotalnumberofnodesusedtostoretheinput;thethreetrees
n
in Figure 2 have S equal to 10,6,6, respectively. If m = 2, then S ≡ n; if m (cid:62) 3, we
10 n
cancomputeS recursivelybyS = 0,and
n 0
(cid:40)
1, if1 (cid:54) n < m,
d
S = (5)
n S(1) +···+S(m) +1, ifn (cid:62) m,
I1 Im
where the S(r)’s are independent copies of S , 1 (cid:54) r (cid:54) m, 0 (cid:54) i (cid:54) n − m + 1, and
i i
independentof(I ,...,I )definedin(3).
1 m
4
• K (keypathlength,KPL):thesumofthedistancebetweentherootandeachkey;forthe
n
treesinFigure2,K = {19,11,8},respectively. Form (cid:62) 2,K satisfiestherecurrence
10 n
(cid:40)
0, ifn < m,
d
K = (6)
n K(1) +···+K(m) +n−m+1, ifn (cid:62) m,
I1 Im
where the K(r)’s are independent copies of K , 1 (cid:54) r (cid:54) m,0 (cid:54) i (cid:54) n − m + 1,
i i
independentof(I ,...,I ).
1 m
• N (node path length, NPL): the sum of the distance between the root and each node; so
n
that N = {19,7,6} for the three trees in Figure 2. Obviously, N = K when m = 2.
10 n n
Whenm (cid:62) 3,
(cid:40)
0, ifn < m,
d
N = (7)
n N(1) +···+N(m) +S(1) +···+S(m), ifn (cid:62) m,
I1 Im I1 Im
where the (N(r),S(r))’s are independent copies of (N ,S ), 1 (cid:54) r (cid:54) m,0 (cid:54) i (cid:54) n −
i i i i
m+1,independentof(I ,...,I ).
1 m
While the first two random variables have been widely studied in the literature, NPL was
onlyconsideredpreviouslyin[4,21]inconnectionwiththeprocessofcuttingtrees. Inaddition
to this, our interest was to understand the extent to which the asymptotic independence for
small m between S and K subsists when the “toll function” changes from a linear function
n n
toafunctionthatisrandomandmaydependonS .
n
2.2 A summary of known results
LetH := (cid:80) j−1. Knuth[27,§6.2.4]wasthefirsttoshowthat
m 1(cid:54)j(cid:54)m
1
E(S ) ∼ φn, where φ := ,
n
2(H −1)
m
(seealso[1]). Hereφdenotesthe“occupancyconstant”,whichwillappearalloverouranalysis.
MahmoudandPittel[31]improvedtheresultandderivedanidentityforE(S ),whichimplies
n
inparticularthat
1
E(S ) = φ(n+1)− +O(cid:0)nα−1(cid:1),
n
m−1
whereαhasthesamemeaningasinIntroduction;see(4). Theyalsodiscoveredandprovedthe
surprisingresultforthevariance
(cid:40)
C n, if3 (cid:54) m (cid:54) 26;
V(S ) ∼ S
n F (βlogn)n2α−2, ifm (cid:62) 27,
1
where C is a constant depending on m, F is a π-periodic function given in (24), α + iβ
S 1
is the second largest zero (in real part) with β > 0 of the equation Λ(z) = 0 (see (4)), and
2α−2 > 1form (cid:62) 27. Seealso[9,25,33]foracloselyrelatedfragmentationmodelwiththe
same asymptotic behavior. A central limit theorem for S was then proved for 3 (cid:54) m (cid:54) 26 in
n
5
[28,31];seealso[30]formoredetails. Theirapproachisbasedonaninductiveapproximation
argument.
By the method of moments, two authors of this paper re-proved in [8] the central limit the-
oremforS when3 (cid:54) m (cid:54) 26;thesameapproachwasalsousedtoestablishthenonexistence
n
ofalimitlawforS duetoinherentoscillations. Moreover,theconvergenceratestothenormal
n
distributionwerecharacterizedin[22]byarefinedmethodofmoments,whichundergofurther
changeofbehaviors.
Thenseveraldifferentapproachesweredevelopedintheliteratureforadeeperunderstand-
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 (cid:62) 2 was first studied by Mahmoud [29] and he
proved
E(K ) = 2φnlogn+c n+o(n),
n 1
for some explicitly computable constant c ; see (21). The variance was computed in [30, §3.5]
1
andsatisfies(H(2) := (cid:80) j−2)
m 1(cid:54)j(cid:54)m
(cid:16) (cid:17)
V(K ) ∼ C n2, where C = 4φ2 (m+1)Hm(2)−2 − π2 . (8)
n K K m−1 6
Thecorrespondinglimitlawwascharacterizedin[38]bythecontractionmethod
K −E(K )
n n d
−→ K, (9)
n
where K is given by the recursive distributional equation (44); see also [4, 34] for a general
framework.
ForNPLN ,BroutinandHolmgren[4]provedthat
n
E(N ) = 2φ2nlogn+c n+o(n),
n 2
for some constant c (for which no numerical value was provided); a series expression of c
2 2
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 c feasible (although its exact value is not
2
needed);see(27).
It should be mentioned that there is a large literature on K when m = 2 because it is
n
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
Westateinthissectionourresultsforthecovarianceandcorrelationbetweenthespacerequire-
ment and the total path lengths (KPL and NPL). The proofs and the tools needed will be given
inthenextsections.
Unlike the space requirement S whose variance changes its asymptotic behavior for m (cid:62)
n
27,thecovarianceCov(S ,K )changesitsasymptoticbehavioratm = 14.
n n
6
Theorem2.1. ThecovariancebetweenS andK satisfies
n n
(cid:40)
C n, if3 (cid:54) m (cid:54) 13;
R
Cov(S ,K ) ∼
n n F (βlogn)nα, ifm (cid:62) 14;
2
whereC isasuitableconstantandF (z)isa2π-periodicfunctiongivenin(25)below.
R 2
Thisresulthasthefollowingconsequence.
Corollary2.2. ThecorrelationcoefficientbetweenS andK satisfies
n n

→ 0, if3 (cid:54) m (cid:54) 26;

ρ(S ,K ) F (βlogn)
n n ∼ 2 , ifm (cid:62) 27,
 (cid:112)
 C F (βlogn)
K 1
whereC > 0isgivenin(8).
K
SeeFigure1fortwodifferentplotsfortheperiodicfunctionswhenm (cid:62) 27.
Thesameconsiderationextendseasilytoclarifythecorrelationbetweenspacerequirement
andNPL.
Theorem2.3. ThecovariancebetweenS andN satisfies
n n
(cid:40)
2φC nlogn, if3 (cid:54) m (cid:54) 13;
S
Cov(S ,N ) ∼
n n φF (βlogn)nα, ifm (cid:62) 14,
2
whereC isasinSection2.2. Moreover,thevarianceofN satisfies
S n
V(N ) ∼ φ2C n2.
n K
Notice the appearance of an extra logn factor when 3 (cid:54) m (cid:54) 13, which reflects the
additional random effect introduced by the toll function in (7). These estimates imply the
followingconsequence.
Corollary2.4. Thecorrelationcoefficientρ(S ,N )satisfies
n n

→ 0, if3 (cid:54) m (cid:54) 26;

ρ(S ,N ) F (βlogn)
n n ∼ ρ(S ,K ) ∼ 2 , ifm (cid:62) 27.
 n n (cid:112)
 C F (βlogn)
K 1
ThelastrelationsuggestsconsideringthecorrelationbetweenK andN .
n n
Corollary2.5. TherandomvariableK isasymptoticallylinearlycorrelatedtoN
n n
ρ(K ,N ) → 1.
n n
7
Indeed,wewillshowthat
(cid:107)N −φK −(E(N −φK ))(cid:107) = o(n)
n n n n 2
whichthenbySlutsky’stheoremimpliesthat
(cid:18)K −E(K ) N −E(N )(cid:19)
n n n n d
, −→ (K,φK);
n n
see(9),Section4.3and4.4.
These results will be proved by working out the asymptotics of the corresponding recur-
rencerelations,whichallhavethesameform
(cid:88)
a = m π a +b , (n (cid:62) m−1),
n n,j j n
0(cid:54)j(cid:54)n−m+1
where
(cid:0)n−1−j(cid:1)
π = m−2 (0 (cid:54) j (cid:54) n−m+1)
n,j (cid:0) n (cid:1)
m−1
is a probability distribution, and {b } is a given sequence (referred to as the toll-function).
n
For that asymptotic purpose, our key tools will rely on the asymptotic transfer techniques (see
[8, 14]), which provide a direct asymptotic translation from the asymptotic behaviors of b to
n
those of a . The remaining analysis will then consist of simplifying some multiple Dirichlet’s
n
integrals.
SincePearson’sproduct-momentcorrelationcoefficientρisknowntobepoorinmeasuring
nonlinear dependence between two random variables, we go further by considering the joint
limitlawsfor(S ,K )and(S ,N ),whichexhibitachangeofbehaviordependingonwhether
n n n n
3 (cid:54) m (cid:54) 26(convergentcase)orm (cid:62) 27(periodiccase): theyareasymptoticallyindependent
intheformercasebutdependentinthelatter.
Theorem 2.6. Assume 3 (cid:54) m (cid:54) 26. Let (X ) ∈ {(K ) ,(N ) } and Q = (X ,S ) denote
n n n n n n n n n
the vector of KPL or NPL and the space requirement used by a random m-ary search tree.
Thentheconvergenceindistributionholds:
Cov(Q )−1/2(Q −E[Q ]) −→d (X,N ), (10)
n n n
where N has the standard normal distribution and the limit law (X,N ) is described in
Lemma4.2;moreover,X andN areindependent.
Theorem2.7. Assumem (cid:62) 27. Let(X ) ∈ {(K ) ,(N ) }and
n n n n n n
(cid:18)X −E[X ] S −φn(cid:19)
n n n
Y := ,
n ι n nα−1
X
withι = 1for(X ) = (N ) andι = φ−1 for(X ) = (K ) . Thenwehave
X n n n n X n n n n
(cid:96) (Y ,(X,(cid:60)(niβΛ))) → 0,
2 n
whereβ isasinSection2.2and(X,Λ)isarandomvectorwhosedistributionisspecifiedasthe
unique fixed point solution appearing in Lemma 4.1 for the choice γ = (0,θ) (θ being defined
belowin(28)).
8
SeeSection4foramorepreciseformulation. Theproofisbasedonthecontractionmethod
(see [36]) where we use the above moment asymptotics as input and combine well-known
estimates within the minimal L -metric for the convergent case (as in [40]), and those with
2
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 NPL N (the case of the KPL
n
being known before), there are more direct proofs which we also discuss in Sections 4.3 and
4.4.
Our study of the dependence of random variables on random m-ary search trees can be
extendedinatleasttwodirectionsbythesamemethodsusedinthispaper,namely,asymptotic
transfertechniquesandthecontractionmethod.
• Extension to more general linear and nlogn shape measures: That the asymptotic co-
variance undergoes a phase change after m = 13 and the asymptotic correlation under-
goes a phase change after m = 26 is not restricted to the space requirement and KPL or
NPL.Indeed,wecanreplacethespacerequirementbymanyotherlinearshapemeasures
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 ordernlogn such as summing over the root-node or
root-keydistanceforcertainspecifiednodesorpatternsandweightedpathlength.
• 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 trees and
quadtrees. Thebehaviorswillbehoweververydifferentfortheclassesoftreeswherethe
underlyingdistributionofthesubtreesizesaredictatedbyabinomialdistribution,which
willbeexaminedelsewhere;seeacompanionpaper[18]formoreinformation.
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
studyandresultstoothershapeparametersinSection5.
3 Correlation between space requirement and path lengths
WeproveinthissectionTheorems2.1and2.3forthecovariancesCov(S ,K )andCov(S ,N ),
n n n n
respectively.
3.1 Preliminaries and recurrences
We collect here the notations to be used in the proofs. Let m (cid:62) 2 be a fixed integer. For
n (cid:62) m, denote by I(n) = (I(n),...,I(n)) the vector of the number of keys inserted in the m
1 m
9
ordered subtrees of the root in a random m-ary search tree with n keys. When the dependence
on n is obvious, we write simply (I ,...,I ). Generate independently n uniform random
1 m
variables U ,...,U on [0,1]. Store the first m−1 elements U ,...,U in the root-node of
1 n 1 m−1
thetree. Thentheydecomposetheunitinterval[0,1]intospacingsoflengthsV ,...,V ,where
1 m
V = U −U forj = 1,...,mwithU := 0,U := 1andU forj = 1,...,m−1are
j (j) (j−1) (0) (m) (j)
the order statistics of U ,...,U . The uniform permutation model implies, that, conditional
1 m−1
on U ,...,U , the vector I(n) has the multinomial distribution with success probabilities
1 m−1
V ,...,V ,namely,wehave
1 m
d
(I ,...,I ) = M(n−m+1;V ,...,V ).
1 m 1 m
Inparticular,wehavetheconvergence
I
r
−→ V , (11)
r
n
forallr = 1,...,m,wheretheconvergenceisinL forall1 (cid:54) p < ∞. Notethatwealsohave
p
(3)forallm-tuplesi ,...,i (cid:62) 0withi +···+i = n−m+1andalln (cid:62) m.
1 m 1 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 random m-ary search tree of that number of keys in the uniform model. Moreover, condi-
tional on (I ,...,I ), the subtrees are independent. This can be seen by switching back to
1 m
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 ,K and N given in
n n n
(5)–(7), where the summands appearing on the right-hand sides, namely, S(1),...,S(m) and
j j
K(1),...,K(m) and N(1),...,N(m) have the same distributions as S and K and N , respec-
j j j j j j j
(cid:16) (cid:17)
tively. Furthermore, the triples (cid:0)S(r)(cid:1) ,(cid:0)K(r)(cid:1) ,(cid:0)N(r)(cid:1) are
j 0(cid:54)j(cid:54)n−m+1 j 0(cid:54)j(cid:54)n−m+1 j 0(cid:54)j(cid:54)n−m+1
independent for r = 1,...,m and independent of (I ,...,I ). Finally, the recursive structure
1 m
of the m-ary search tree implies recurrences satisfied by their joint distributions. In particular,
thepairQ := (N ,S )satisfiestherecurrence
n n n
(cid:18) (cid:19)
(cid:88) (cid:104) 1 1 (cid:105)(cid:16) (cid:17)t 0
(Q )t =d Q(r) + , (n (cid:62) m), (12)
n 0 1 Ir 1
1(cid:54)r(cid:54)m
where,asin(5)–(7),theQ(r)’saredistributedasQ forall1 (cid:54) r (cid:54) mand0 (cid:54) j (cid:54) n−m+1,
j j
and the (cid:0)Q(r)(cid:1) are independent for r = 1,...,m and independent of (I ,...,I ).
j 0(cid:54)j(cid:54)n−m+1 1 n
TherecurrencesatisfiedbythepairZ := (K ,S )is
n n n
(cid:18) (cid:19)
(cid:88) (cid:104) 1 0 (cid:105)(cid:16) (cid:17)t n−m+1
(Z )t =d Z(r) + , (n (cid:62) m), (13)
n 0 1 Ir 1
1(cid:54)r(cid:54)m
withconditionsonindependenceandidenticaldistributionssimilarto(12).
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