5.6 k-Cousins in Digital Trees
5.6.3 k-cousins in Digital Search Trees
Var(Xn,k(T ))
−→ N(0, 1).d
Proof. Similar to the proof of Theorem 5.5.1 and Theorem 5.5.2, we only need to check that the conditions of Proposition5.1.6and Theorem5.1.7are satisfied. From5.29, we get that for given k
f˜1[k](z) = e−z∑ conditions of JS-Admissibility in Proposition 5.1.6 and Theorem 5.1.7 are satisfied. Moreover, we can easily see that Tn = 0 for all n from (5.29) and hence ∥Tn∥s = o(√
n) for all s ∈ R+. Plus Lemma 5.6.1, all the conditions are checked and the result follows.
5.6.3 k-cousins in Digital Search Trees
Similar to the Trie case, we use Xn,k(P )to denote the random variable of number of k-cousin in a symmetric DSTs built on n keys. The random variable satisfies the following distributional recurrence
Xn+1,k(P ) = the framework, we need to establish the lower bound for the variance as we have seen before.
Lemma 5.6.3.
V(Xn,k(P ))≥ cn for some positive constant c.Proof. Similar to Lemma 5.6.1. Proof. By Theorem 3.3.7, we get
G(k)(ω)≍∑
Q(1) , apply equation 2.2.6 of [5], we derive that
∑n With the help of Maple, we get the following meromorphic extension
∑n
then
G(k)(ω) ≍∑
n≥0
Ψ(k)(n)(−1)n ω + n.
By the same argument as in Example 5 of [62], we get that G(k)(ω) = Ψ(k)(−ω)Γ(ω)Γ(1 − ω).
Theorem 5.6.5. We have, as n
→ ∞,E(Xn,k(P ))∼ nϖE[k](log1/an), Var(Xn,k(P ))∼ nϖ[k]V (log1/an),
where a > 0 is a suitable constant and ϖ[k]E (z), ϖ[k]V (z) are infinitely differ-entiable, 1-periodic functions (possibly constant) for all k ∈ N. The explicit expression of ϖE[k](z) is given by
ϖ[k]E (ω) = Ψ(k)(−ω)Γ(ω)Γ(1 − ω) where Ψ(k)(ω) is defined as Proposition
5.6.4. Moreover,
Xn,k(P )− E(Xn,k(P ))
√
Var(Xn,k(P ))
−→ N(0, 1).d
Proof. Similar to Theorem 5.6.2, Theorem 5.5.1 and Theorem 5.5.2, we use the results from Lemma 5.6.3 and Proposition 5.6.4 to show that all the assumptions of Proposition5.2.2and Theorem5.2.3are satisfied. The details are emitted here.
Chapter 6 Conclusion
The main purpose of this thesis was to contribute to the analysis of additive shape parameters in random digital trees. The results in this thesis can be divided into two topic areas.
The first topic area was concerned with new applications of the recently proposed Poisson-Laplace-Mellin method. In [74], all the applications of the Poisson-Laplace-Mellin method were for shape parameters of linear order (up to a power of logarithms). In Chapter 4, we collected many examples of shape parameters which are not of linear order, including the leftmost path length, the Wiener index and the total Steiner distance. We derived asymptotic expansions of the mean and variance for these parameters. Moreover, we proved limit laws as well.
The second topic area was concerened with general framworks for central limit theorems of additive shape parameters in random digital trees. In Chapter 5, we first introduced our framework from [77] for proving central limit theorems for shape paramters in m-ary tries. Then, we extended this framework to shape parameters in symmetric digital search trees. We also gave two examples to illustrate how our frameworks work.
As for open problems, the most straightforward one is the extension of our study of the total Steiner distance to other digital trees. In fact, such a study can be performed by the methods we introduced in this thesis. However, the computations are cumbersome. Another obvious question is whether our results for symmetric DSTs can be extended to asymmetric DSTs? For pa-rameters satisfying one-sided distributional recurrences, such as the leftmost path length, we saw that the Poisson-Laplace-Mellin method still works in the asymmetric case. On the other hand, for parameters satisfying two-sided distributional recurrences, this is no longer true. Netherless, with similar tools as in our thesis, deriving asymptotic expansions of mean, variance and obtaining the limit law is still possible. However, asymptotic expressions are
not explicit. Thus, finding a general method for deriving explicit asymptotic expressions of the mean and variance of shape parameters satisfying a two-sided distributional recurrence in asymmetric DSTs is an important open question. As a final open problem, note that our frameworks in Chapter 5 are for proving central limit laws. So, a natural question is whether or not similar frameworks can be given for local limit laws and rates of convergence?
We end this thesis by placing our research in a larger context. There-fore, we point out that research of random digital trees is part of the more general study of binomial splitting processes (BSPs) in which the binomial distribution and some of its extensions play an important role. For an ex-tensive introduction into BSPs, see [75]. In this thesis, we mainly dealt with functional equations of the form
f (z) + ˜˜ f′(z) = 2 ˜f (z
2 )
+ ˜g(z) or
f (z) =˜
∑m r=1
f (p˜ rz) + ˜h(z).
Such (differential-)functional equations are special cases of the more general form
∑b j=0
(b j
)
f˜(j)(z) =
∑m r=1
arf (p˜ rz + λ) + ˜g(z)
which underlies the study of BSPs. Most of the recent research has focused on the case λ = 0. Very little is known about the case λ > 0 which is also important in applications; see [55,
56, 91]. Thus, there is still a lot of
research to be done and the we have still a long way ahead of us before having a complete understanding of the stochastic properties of BSPs.Appendices
Appendix A
We use the same notation for poissonized means, variances and covariances as in Section 2. In addition, for the node-wise Wiener index of bucket digital search trees and the internal Wiener index for tries, we denote by ˜h1(z) the Poisson generating function of E(Nn) and
H˜N(z) = ˜gN(z)− ˜h1(z)2− z˜h′1(z)2,
H˜T(z) = ˜gT(z)− ˜h1(z) ˜f1,0(z)− z˜h′1(z) ˜f1,0′ (z), H˜W(z) = ˜gW(z)− ˜h1(z) ˜f0,1(z)− z˜h′1(z) ˜f0,1′ (z),
where ˜gN(z), ˜gT(z) and ˜gW(z) denote the Poisson generating function of E(Nn2),E(NnTn) and E(NnWn), respectively.
Key-wise Wiener Index of Bucket Digital Search Trees. We have,
∑b j=0
(b j
)
f˜1,0(j)(z) = 2 ˜f1,0(z/2) + z,
∑b j=0
(b j
)
f˜0,1(j)(z) = 2 ˜f0,1(z/2) + (z + 2) ˜f1,0(z/2) + z2 2 + z and
∑b j=0
(b j
)
V˜(j)(z) = 2 ˜V (z/2) + ( b
∑
j=0
(b j
) f˜1,0(j)(z)
)2
+ z ( b
∑
j=0
(b j
)
f˜1,0(j+1)(z) )2
−
∑b j=0
(b j
) (f˜1,0(z)2+ z ˜f1,0′ (z)2 )(j)
,
∑b
Node-wise Wiener Index of Bucket Digital Search Trees. We have,
∑b
+ z
+ z
External Wiener Index of Tries. We have,
f˜1,0(z) = 2 ˜f1,0(z/2) + z− ze−z,
C(z) = 2 ˜˜ C(z/2) + z ˜V (z/2) + e−z (
z ˜f1,0(z/2) + z2 2
f˜1,0′ (z/2)−z3 2
f˜1,0′ (z/2) + 2z ˜f0,1(z/2) + z ˜f0,1′ (z/2)− z2f˜0,1′ (z/2)
) + e−z
(
z2− z3 2
) , W (z) = 2 ˜˜ W (z/2) + 2z ˜C(z/2) +
(z2 2 + z
)
V (z/2) + z˜ 2f˜1,0′ (z/2)2 + 2z2f˜1,0′ (z/2) + z2.
Internal Wiener Index of Tries.
We have,˜h1(z) =2˜h1(z/2) + 1− e−z(1 + z), f˜1,0(z) =2 ˜f1,0(z/2) + 2˜h1(z/2),
f˜0,1(z) =2 ˜f0,1(z/2) + 2 ˜f1,0(z/2)˜h1(z/2) + 2˜h1(z/2)2+ 2 ˜f1,0(z/2) + 2˜h1(z/2) and
H˜N(z) =2 ˜HN(z/2) + e−z(4˜h1(z/2) + 4z˜h1(z/2)− 2z2˜h′1(z/2)) + e−z(1 + z− e−z− 2ze−z− z2e−z− z3e−z),
H˜T(z) =2 ˜HT(z/2) + 2 ˜HN(z/2) + e−z(2˜h1(z/2) + 2z˜h1(z/2)− z2˜h′1(z/2) + 2 ˜f1,0(z/2) + 2z ˜f1,0(z/2)− z2f˜1,0′ (z/2)),
V (z) =2 ˜˜ V (z/2) + 4 ˜HT(z/2) + 2 ˜HN(z/2),
H˜W(z) =2 ˜HW(z/2) + 2 ˜HT(z/2)(˜h1(z/2) + 1) + 2 ˜HN(z/2)(2˜h1(z/2)
+ ˜f1,0(z/2) + 1) + e−z(2˜h1(z/2)2 + 2z˜h1(z/2) + 2˜h1(z/2) + 2z˜h1(z/2)
− z2˜h1(z/2)˜h′1(z/2)− z2˜h′1(z/2) + 2˜h1(z/2) ˜f1,0(z/2) + 2z˜h1(z/2) ˜f1,0(z/2)
− z2˜h1(z/2) ˜f1,0′ (z/2)− z2˜h′1(z/2) ˜f1,0(z/2) + 2 ˜f1,0(z/2) + 2z ˜f1,0(z/2)
− z2f˜1,0′ (z/2) + 2 ˜f0,1(z/2) + 2z ˜f0,1(z/2)− z2f˜0,1′ (z/2)), C(z) =2 ˜˜ C(z/2) + 2 ˜HW(z/2) + 2 ˜V (z/2)(˜h1(z/2) + 1)
+ 2 ˜HT(z/2)(3˜h1(z/2) + ˜f1,0(z/2) + 2) + 2 ˜HN(z/2)(2˜h1(z/2) + ˜f1,0(z/2) + 1),
W (z) =2 ˜˜ W (z/2) + 4 ˜C(z/2)(˜h1(z/2) + 1) + 4 ˜HW(z/2)(2˜h1(z/2) + ˜f1,0(z/2) + 1) + 2 ˜V (z/2) ˜HN(z/2) + ˜V (z/2)((2 + z)˜h1(z/2)2+ 4˜h1(z/2) + 2)
+ 2 ˜HT(z/2)2+ ˜HT(z/2)(8˜h1(z/2)2+ 16˜h1(z/2) + 4z˜h′1(z/2)2 + 4˜h1(z/2) ˜f1,0(z/2) + 2z˜h′1(z/2) ˜f1,0′ (z/2) + 4) + 4 ˜HN(z/2)2
+ 8 ˜HN(z/2)2˜h1(z/2)2+ 8 ˜HN(z/2) ˜HT(z/2) + ˜HN(z/2)(8˜h1(z/2) + 4z˜h′1(z/2)2+ 8˜h1(z/2) ˜f1,0(z/2) + 4z˜h′1(z/2) ˜f1,0′ (z/2)
+ 2 ˜f1,0(z/2)2+ 4 ˜f1,0(z/2) + z ˜f1,0′ (z/2)2+ 2) + z2˜h1(z/2)4 + 2z2˜h1(z/2)3f˜1,0(z/2) + z2˜h′1(z/2)2f˜1,0′ (z/2)2.
External Wiener Index of PATRICIA Tries. We have,
f˜1,0(z) = 2 ˜f1,0(z/2) + z− ze−z/2,
Internal Wiener Index of m-ary PATRICIA Tries.
We have,˜h(z) =
where S2 ={(r, s) : 1 ≤ r, s ≤ m, r ̸= s} and al-gebra such as maple. However, the explicit expressions of them are too complicated and hence we do not list them here. We give only the bounds since it is already enough for our purpose.
Appendix B
We will use the following notations
Ω1(n) = Q(k1)(log1/an), Ω2(n) = Q(k2)(log1/an), Ω3(n) = Q(k1,k2)(log1/an)
A(i)n (1, 2) = Bn(i)·
Ω3(In(i)) (
Ω2(n) +√ D(n)
)− Ω3(n) (
Ω2(In(i)) +
√
D(In(i)) )
2D(n) + (Ω1(n) + Ω2(n))√
D(n) ,
A(i)n (2, 1) = Bn(i)·
Ω3(In(i)) (
Ω1(n) +√ D(n)
)− Ω3(n) (
Ω1(In(i)) +
√
D(In(i)) )
2D(n) + (Ω1(n) + Ω2(n))√
D(n) ,
A(i)n (2, 2) = Bn(i)· (
Ω1(n) +√ D(n)
) (
Ω2(In(i)) +
√
D(In(i)) )
− Ω3(n)Ω3(Ir(n)) 2D(n) + (Ω1(n) + Ω2(n))√
D(n) ,
where
Bn(i) =
√ In(i)
n · vu
ut Ω1(n) + Ω2(n) + 2√ D(n) Ω1(In(i)) + Ω2(In(i)) + 2
√
D(In(i)) .
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