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The Node Profile of Symmetric Digital Search Trees (joint with M. - - PowerPoint PPT Presentation

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The Node Profile of Symmetric Digital Search Trees (joint with M. Drmota, H.-K. Hwang and R. Neininger) Michael Fuchs Department of Applied Mathematics National Chiao Tung University June 8th, 2015 Michael Fuchs (NCTU) Node Profile of DSTs

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SLIDE 1

The Node Profile of Symmetric Digital Search Trees

(joint with M. Drmota, H.-K. Hwang and R. Neininger) Michael Fuchs

Department of Applied Mathematics National Chiao Tung University

June 8th, 2015

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 1 / 28

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SLIDE 2

Node Profile of (Rooted) Trees

Bn,k = number of external nodes at level k; In,k = number of internal nodes at level k.

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 2 / 28

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SLIDE 3

Node Profile of (Rooted) Trees

Bn,k = number of external nodes at level k; In,k = number of internal nodes at level k. Example:

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 2 / 28

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SLIDE 4

Node Profile of (Rooted) Trees

Bn,k = number of external nodes at level k; In,k = number of internal nodes at level k. Example: B5,0 = 0, B5,1 = 0, B5,2 = 2, B5,3 = 4,

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 2 / 28

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SLIDE 5

Node Profile of (Rooted) Trees

Bn,k = number of external nodes at level k; In,k = number of internal nodes at level k. Example: B5,0 = 0, I5,0 = 1; B5,1 = 0, I5,1 = 2; B5,2 = 2, I5,2 = 2; B5,3 = 4, I5,3 = 0.

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 2 / 28

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Relations to Other Shape Parameters

Many shape parameters can by analyzed through the profile.

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 3 / 28

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SLIDE 7

Relations to Other Shape Parameters

Many shape parameters can by analyzed through the profile. Depth: P(Dn = k) = Bn,k/(n + 1); Width: max{Bn,k : k ≥ 0}; Total Path Length:

k kBn,k;

Height: max{k : Bn,k > 0}; Shortest Path: min{k : Bn,k > 0}; Fill-up Level: max{k : In,k = 2k}; Etc.

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 3 / 28

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SLIDE 8

Profile of Random Trees

√n-Trees: Aldous (1991); Drmota and Gittenberger (1997); Kersting (1998); Pitman (1999); etc.

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 4 / 28

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SLIDE 9

Profile of Random Trees

√n-Trees: Aldous (1991); Drmota and Gittenberger (1997); Kersting (1998); Pitman (1999); etc. log n-Trees:

Binary Search Trees: Chauvin, Drmota, Jabbour-Hattab (2001); Drmota and Hwang (2005); F., Hwang, Neininger (2006). Recursive Trees: Drmota and Hwang (2005); F., Hwang, Neininger (2006). Plane-oriented Recursive Trees: Hwang (2007). m-ary Seach Trees: Drmota, Janson, Neininger (2008).

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 4 / 28

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Tries

Ren´ e de la Briandais (1959) Name from data retrieval (suggested by Fredkin).

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 5 / 28

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Tries

Ren´ e de la Briandais (1959) Name from data retrieval (suggested by Fredkin). Example: 011011 010101 101110 010000 101010 001100

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 5 / 28

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SLIDE 12

Tries

Ren´ e de la Briandais (1959) Name from data retrieval (suggested by Fredkin). Example: 011011 010101 101110 010000 101010 001100

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 5 / 28

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Tries

Ren´ e de la Briandais (1959) Name from data retrieval (suggested by Fredkin). Example:

1 1

011011 010101 101110 010000 101010 001100

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 5 / 28

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Tries

Ren´ e de la Briandais (1959) Name from data retrieval (suggested by Fredkin). Example:

1 1 1

011011 010101 101110 010000 101010 001100

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 5 / 28

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Tries

Ren´ e de la Briandais (1959) Name from data retrieval (suggested by Fredkin). Example:

1 1 1 1

011011 010101 101110 010000 101010 001100

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 5 / 28

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SLIDE 16

Tries

Ren´ e de la Briandais (1959) Name from data retrieval (suggested by Fredkin). Example:

1 1 1 1 1 1

011011 010101 101110 010000 101010 001100

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 5 / 28

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SLIDE 17

Tries

Ren´ e de la Briandais (1959) Name from data retrieval (suggested by Fredkin). Example:

1 1 1 1 1 1

011011 010101 101110 010000 101010 001100

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 5 / 28

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SLIDE 18

Digital Search Trees (DSTs)

Edward G. Coffman & James Eve (1970) Closely related to Lempel-Ziv compression scheme.

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 6 / 28

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Digital Search Trees (DSTs)

Edward G. Coffman & James Eve (1970) Closely related to Lempel-Ziv compression scheme. Example: 011011 010101 101110 010000 101010 001100

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 6 / 28

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SLIDE 20

Digital Search Trees (DSTs)

Edward G. Coffman & James Eve (1970) Closely related to Lempel-Ziv compression scheme. Example:

1

011011 010101 101110 010000 101010 001100

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 6 / 28

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SLIDE 21

Digital Search Trees (DSTs)

Edward G. Coffman & James Eve (1970) Closely related to Lempel-Ziv compression scheme. Example:

1 1

011011 010101 101110 010000 101010 001100

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 6 / 28

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SLIDE 22

Digital Search Trees (DSTs)

Edward G. Coffman & James Eve (1970) Closely related to Lempel-Ziv compression scheme. Example:

1 1 1

011011 010101 101110 010000 101010 001100

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 6 / 28

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SLIDE 23

Digital Search Trees (DSTs)

Edward G. Coffman & James Eve (1970) Closely related to Lempel-Ziv compression scheme. Example:

1 1 1 1

011011 010101 101110 010000 101010 001100

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 6 / 28

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SLIDE 24

Digital Search Trees (DSTs)

Edward G. Coffman & James Eve (1970) Closely related to Lempel-Ziv compression scheme. Example:

1 1 1 1 1

011011 010101 101110 010000 101010 001100

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 6 / 28

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SLIDE 25

Digital Search Trees (DSTs)

Edward G. Coffman & James Eve (1970) Closely related to Lempel-Ziv compression scheme. Example:

1 1 1 1 1 1

011011 010101 101110 010000 101010 001100

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 6 / 28

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SLIDE 26

Random Model

Bits generated by iid Bernoulli random variables with mean p − → Bernoulli model

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 7 / 28

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SLIDE 27

Random Model

Bits generated by iid Bernoulli random variables with mean p − → Bernoulli model Two types: p = 1/2: symmetric digital trees; p = 1/2: asymmetric digital trees.

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 7 / 28

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SLIDE 28

Random Model

Bits generated by iid Bernoulli random variables with mean p − → Bernoulli model Two types: p = 1/2: symmetric digital trees; p = 1/2: asymmetric digital trees. Question: What can be said about the profile?

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 7 / 28

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SLIDE 29

Random Model

Bits generated by iid Bernoulli random variables with mean p − → Bernoulli model Two types: p = 1/2: symmetric digital trees; p = 1/2: asymmetric digital trees. Question: What can be said about the profile? In this talk, we are interested in mean, variance and limit laws of the profile for symmetric DSTs.

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 7 / 28

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SLIDE 30

Profile of Digital Trees

Tries: Mean, variance, limit laws: Hwang, Nicod´ eme, Park and Szpankowski (2009).

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 8 / 28

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SLIDE 31

Profile of Digital Trees

Tries: Mean, variance, limit laws: Hwang, Nicod´ eme, Park and Szpankowski (2009). PATRICIA tries: Mean: Magner, Knessl, Szpankowski (2014); Variance & limit laws: Szpankowkski & Magner (→ Thursday).

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 8 / 28

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SLIDE 32

Profile of Digital Trees

Tries: Mean, variance, limit laws: Hwang, Nicod´ eme, Park and Szpankowski (2009). PATRICIA tries: Mean: Magner, Knessl, Szpankowski (2014); Variance & limit laws: Szpankowkski & Magner (→ Thursday). Asymmetric DSTs: Mean: Drmota and Szpankowski (2011); Variance: Kazemi and Vahidi-Asl (2011); so far no limit laws.

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 8 / 28

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SLIDE 33

Profile of Digital Trees

Tries: Mean, variance, limit laws: Hwang, Nicod´ eme, Park and Szpankowski (2009). PATRICIA tries: Mean: Magner, Knessl, Szpankowski (2014); Variance & limit laws: Szpankowkski & Magner (→ Thursday). Asymmetric DSTs: Mean: Drmota and Szpankowski (2011); Variance: Kazemi and Vahidi-Asl (2011); so far no limit laws. Symmetric DSTs: Variance & limit laws: Drmota, F., Hwang, Neininger (→ this talk).

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 8 / 28

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SLIDE 34

Profile of Tries

Hwang, Nicod´ eme, Park, Szpankowski (2009)

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 9 / 28

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SLIDE 35

Plot of Mean Profile of Symmetric Tries

Hwang, Nicod´ eme, Park, Szpankowski (2009):

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 10 / 28

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SLIDE 36

Symmetric Tries: Mean

We have, µn,k := E(Bn,k) ∼

n(1 − 2−k)n−1,

if 2−kn → ∞; ˜ Mk,1(n), if 4−kn → 0, where ˜ Mk,1(z) = z(e−z/2k − e−z/2k−1).

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 11 / 28

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SLIDE 37

Symmetric Tries: Mean

We have, µn,k := E(Bn,k) ∼

n(1 − 2−k)n−1,

if 2−kn → ∞; ˜ Mk,1(n), if 4−kn → 0, where ˜ Mk,1(z) = z(e−z/2k − e−z/2k−1). In particular, ˜ Mk,1(n) ∼      ne−n/2k, if 2−kn → ∞; Θ(n), if 2−kn = Θ(1); 2−kn2, if 2−kn → 0.

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 11 / 28

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SLIDE 38

Symmetric Tries: Mean

We have, µn,k := E(Bn,k) ∼

n(1 − 2−k)n−1,

if 2−kn → ∞; ˜ Mk,1(n), if 4−kn → 0, where ˜ Mk,1(z) = z(e−z/2k − e−z/2k−1). In particular, ˜ Mk,1(n) ∼      ne−n/2k, if 2−kn → ∞; Θ(n), if 2−kn = Θ(1); 2−kn2, if 2−kn → 0. Thus, the profile has maximum of order n (asymmetric tries: n/√log n)

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 11 / 28

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SLIDE 39

Symmetric Tries: Variance

We have, σ2

n,k := Var(Bn,k) ∼

n(1 − 2−k)n−1,

if 2−kn → ∞; ˜ Vk(n), if 4−kn → 0, where ˜ Vk(z) = z(e−z/2k − e−z/2k−1) + 2−kz2e−z/2k−1 − 21−kz2(e−z/2k − e−z/2k−1)2.

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 12 / 28

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SLIDE 40

Symmetric Tries: Variance

We have, σ2

n,k := Var(Bn,k) ∼

n(1 − 2−k)n−1,

if 2−kn → ∞; ˜ Vk(n), if 4−kn → 0, where ˜ Vk(z) = z(e−z/2k − e−z/2k−1) + 2−kz2e−z/2k−1 − 21−kz2(e−z/2k − e−z/2k−1)2. In particular, ˜ Vk(n) ∼      ne−n/2k ∼ ˜ Mk(n), if 2−kn → ∞; Θ(n), if 2−kn = Θ(1); 21−kn2 ∼ 2 ˜ Mk(n), if 2−kn → 0.

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 12 / 28

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SLIDE 41

Poissonization and Depoissonization

Poisson Model: Build digital tree from Poisson-distributed number of records.

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 13 / 28

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SLIDE 42

Poissonization and Depoissonization

Poisson Model: Build digital tree from Poisson-distributed number of records. Poisson moments: ˜ Mk,ℓ(z) = E(Bℓ

Pois(z),k) = e−z n≥0

E(Bℓ

n,k)zn

n! .

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 13 / 28

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SLIDE 43

Poissonization and Depoissonization

Poisson Model: Build digital tree from Poisson-distributed number of records. Poisson moments: ˜ Mk,ℓ(z) = E(Bℓ

Pois(z),k) = e−z n≥0

E(Bℓ

n,k)zn

n! . Poisson Heuristic: ˜ Mk,ℓ(z) sufficiently smooth = ⇒ E(Bℓ

n,k) ≈ ˜

Mk,ℓ(n).

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 13 / 28

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SLIDE 44

Poissonization and Depoissonization

Poisson Model: Build digital tree from Poisson-distributed number of records. Poisson moments: ˜ Mk,ℓ(z) = E(Bℓ

Pois(z),k) = e−z n≥0

E(Bℓ

n,k)zn

n! . Poisson Heuristic: ˜ Mk,ℓ(z) sufficiently smooth = ⇒ E(Bℓ

n,k) ≈ ˜

Mk,ℓ(n). Poisson heuristic made precise by the Theory of Analytic Depoissonization (Jacquet & Szpankowski; 1998).

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 13 / 28

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SLIDE 45

Poisson Variance

Correct choice is crucial!

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 14 / 28

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SLIDE 46

Poisson Variance

Correct choice is crucial! Asymmetric Digital Trees: ˜ Vk(z) = ˜ Mk,2(z) − ˜ Mk,1(z)2.

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 14 / 28

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SLIDE 47

Poisson Variance

Correct choice is crucial! Asymmetric Digital Trees: ˜ Vk(z) = ˜ Mk,2(z) − ˜ Mk,1(z)2. Symmetric Digital Trees: ˜ Vk(z) = ˜ Mk,2(z) − ˜ Mk,1(z)2 − z ˜ M′

k,1(z)2.

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 14 / 28

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SLIDE 48

Poisson Variance

Correct choice is crucial! Asymmetric Digital Trees: ˜ Vk(z) = ˜ Mk,2(z) − ˜ Mk,1(z)2. Symmetric Digital Trees: ˜ Vk(z) = ˜ Mk,2(z) − ˜ Mk,1(z)2 − z ˜ M′

k,1(z)2.

With this choice: Var(Bn,k) ∼ ˜ Vk(n) when 4−kn → 0.

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 14 / 28

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SLIDE 49

Symmetric DSTs: Mean

Let Q(z) =

∞

ℓ=1
1 − z2−ℓ

, Qn =

n

ℓ=1
1 − 2−ℓ

= Q(2−n) Q(1) .

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 15 / 28

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SLIDE 50

Symmetric DSTs: Mean

Let Q(z) =

∞

ℓ=1
1 − z2−ℓ

, Qn =

n

ℓ=1
1 − 2−ℓ

= Q(2−n) Q(1) . Theorem We have, µn,k    ∼ 2k Qk

1 − 2−kn

, if 2−kn → ∞; = 2kF(n/2k) + O(1), if 4−kn → 0, where F(x) is the positive function F(x) =

j≥0

(−1)j2−(j

2)

QjQ(1) e−2jx.

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 15 / 28

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SLIDE 51

F(x) (i)

As x → ∞, F(x) = e−x Q(1) + O(e−2x)

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 16 / 28

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SLIDE 52

F(x) (i)

As x → ∞, F(x) = e−x Q(1) + O(e−2x) and as x → 0, F(x) ∼ X1/ log 2 √ 2πx exp  −(log X log X)2 log 2 −

j∈Z

cj(X log X)−χj   , where X = 1/(x log 2), χj = 2jπi/ log 2, c0 = log 2 12 + π2 6 log 2 and cj = 1 2j sinh(2jπ/ log 2), (j = 0).

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 16 / 28

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SLIDE 53

F(x) (ii)

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 17 / 28

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SLIDE 54

Some Details of the Proof (i)

We have, ˜ Mk,1(z) + ˜ M′

k,1(z) = 2 ˜

Mk−1,1(z/2).

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 18 / 28

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SLIDE 55

Some Details of the Proof (i)

We have, ˜ Mk,1(z) + ˜ M′

k,1(z) = 2 ˜

Mk−1,1(z/2). By Laplace transform and its inverse, ˜ Mk,1(z) = 2k

0≤j≤k

(−1)j2−(j

2)

QjQk−j e−z/2k−j.

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 18 / 28

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SLIDE 56

Some Details of the Proof (i)

We have, ˜ Mk,1(z) + ˜ M′

k,1(z) = 2 ˜

Mk−1,1(z/2). By Laplace transform and its inverse, ˜ Mk,1(z) = 2k

0≤j≤k

(−1)j2−(j

2)

QjQk−j e−z/2k−j. From this, µn,k = 2k

0≤j≤k

(−1)j2−(j

2)

QjQk−j

1 − 2j−kn

. This formula was first derived by Louchard (1987).

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 18 / 28

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SLIDE 57

Some Details of the Proof (i)

We have, ˜ Mk,1(z) + ˜ M′

k,1(z) = 2 ˜

Mk−1,1(z/2). By Laplace transform and its inverse, ˜ Mk,1(z) = 2k

0≤j≤k

(−1)j2−(j

2)

QjQk−j e−z/2k−j. From this, µn,k = 2k

0≤j≤k

(−1)j2−(j

2)

QjQk−j

1 − 2j−kn

. This formula was first derived by Louchard (1987). This is useful if n2−k → ∞.

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 18 / 28

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SLIDE 58

Some Details of the Proof (ii)

If 4−kn → 0, Poisson heuristic holds.

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 19 / 28

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SLIDE 59

Some Details of the Proof (ii)

If 4−kn → 0, Poisson heuristic holds. Lemma We have, ˜ Mk,1(z) = 2k

r≥0

2−(r+1

2 )−kr

Qr F (r) z 2k

.

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 19 / 28

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SLIDE 60

Some Details of the Proof (ii)

If 4−kn → 0, Poisson heuristic holds. Lemma We have, ˜ Mk,1(z) = 2k

r≥0

2−(r+1

2 )−kr

Qr F (r) z 2k

.

This gives, ˜ Mk,1(z) = 2kF z 2k

+ O(1).

Result follows from depoissonization.

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 19 / 28

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SLIDE 61

Symmetric DSTs: Variance

Theorem (Drmota, F., Hwang, Neininger) We have, σ2

n,k

   ∼ 2k Qk

1 − 2−kn

, if 2−kn → ∞; = 2kH(n/2k) + O(1), if 4−kn → 0, where H(x) is a function with H(x) = e−x Q(1) + O(xe−2x), (x → ∞) and H(x) ∼ 2F(x), (x → 0).

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 20 / 28

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SLIDE 62

H(x) (i)

We have, H(x) =

∞

j,r=0
0≤h,ℓ≤j

2−j(−1)r+h+ℓ2−(r

2)−(h 2)−(ℓ 2)+2h+2ℓ

QrQ(1)QhQj−hQℓQj−ℓ ϕ(2r+j, 2h+2ℓ; x), where ϕ(u, v; x) =    e−ux − ((v − u)x + 1)e−vx (v − u)2 , if u = v; x2e−ux/2, if u = v.

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 21 / 28

slide-63

SLIDE 63

H(x) (i)

We have, H(x) =

∞

j,r=0
0≤h,ℓ≤j

2−j(−1)r+h+ℓ2−(r

2)−(h 2)−(ℓ 2)+2h+2ℓ

QrQ(1)QhQj−hQℓQj−ℓ ϕ(2r+j, 2h+2ℓ; x), where ϕ(u, v; x) =    e−ux − ((v − u)x + 1)e−vx (v − u)2 , if u = v; x2e−ux/2, if u = v. Proposition (Drmota, F., Hwang, Neininger) H(x) is a positive function on (0, ∞).

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 21 / 28

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SLIDE 64

H(x) (ii)

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 22 / 28

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SLIDE 65

Some Details of the Proof (i)

We have, ˜ Vk(z) + ˜ V ′

k(z) = 2 ˜

Vk−1(z/2) + z ˜ M′′

k,2(z)2.

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 23 / 28

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SLIDE 66

Some Details of the Proof (i)

We have, ˜ Vk(z) + ˜ V ′

k(z) = 2 ˜

Vk−1(z/2) + z ˜ M′′

k,2(z)2.

By Laplace transform and its inverse, ˜ Vk(z) =

(j,r,h,ℓ)∈V

2k−j(−1)r+h+ℓ2−(r

2)−(h 2)−(ℓ 2)+2h+2ℓ

QrQk−j−rQhQj−hQℓQj−ℓ ϕ

2r+j, 2h + 2ℓ, z

2k

with

V = {(j, r, h, ℓ) : 0 ≤ j ≤ k, 0 ≤ r ≤ k − j, 0 ≤ h, ℓ ≤ j} and ϕ(u, v; x) =    e−ux − ((v − u)x + 1)e−vx (v − u)2 , if u = v; x2e−ux/2, if u = v.

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 23 / 28

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SLIDE 67

Some Details of the Proof (ii)

Lemma We have, ˜ Vk(z) = 2k

m≥0

2−(m+1

2 )−km

Qm H(m) z 2k

.

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 24 / 28

slide-68

SLIDE 68

Some Details of the Proof (ii)

Lemma We have, ˜ Vk(z) = 2k

m≥0

2−(m+1

2 )−km

Qm H(m) z 2k

.

The Laplace transform of H(z): L [H(z); s] =

j≥0

4−j ˜ g∗

j (2−js)

Q(−21−js) where ˜ g∗

j (s) =

0≤k,ℓ≤j

(−1)h+ℓ2−(h

2)−(ℓ 2)+2h+2ℓ

QkQj−kQℓQj−ℓ 1 (2js + 2h + 2ℓ)2 .

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 24 / 28

slide-69

SLIDE 69

Some Details of the Proof (iii)

Lemma We have, as s → ∞, ˜ g∗

0(s)

Q(−2s) ∼ 1 s2Q(−2s), 4−1 ˜ g∗

1(2−1s)

Q(−s) ∼ 9 sQ(−2s) and, for j ≥ 2, 4−j ˜ g∗

j (2−js)

Q(−21−js) ∼ (2j − 3)! ((j − 2)!)2 2(j

2)

sj−2Q(−2s). Thus, L [H(z); s] ∼ 2 Q(−2s) and hence, H(x) ∼ 2F(x) as x → 0.

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 25 / 28

slide-70

SLIDE 70

Symmetric DSTs: Limit Laws

Corollary (Drmota, F., Hwang, Neininger) We have, µn,k − → ∞ iff σ2

n,k −

→ ∞. Theorem (Drmota, F., Hwang, Neininger) Assume that µn,k − → ∞. Then, Bn,k − µn,k σn,k

d

− → N(0, 1), where N(0, 1) denotes a standard normal distribution.

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 26 / 28

slide-71

SLIDE 71

Application to the Height

Hn =height of a symmetric DST of size n.

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 27 / 28

slide-72

SLIDE 72

Application to the Height

Hn =height of a symmetric DST of size n. Theorem (Drmota, F., Hwang, Neininger) Set kn = min{k ≥ log2 n : 2kF(n/2k) ≤ 1}. Then, kn = log2 n +

2 log2 n − log2
log2 n
+ O(1).

Moreover, P(Hn = kn − 2 or Hn = kn − 1) → 1.

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 27 / 28

slide-73

SLIDE 73

Application to the Height

Hn =height of a symmetric DST of size n. Theorem (Drmota, F., Hwang, Neininger) Set kn = min{k ≥ log2 n : 2kF(n/2k) ≤ 1}. Then, kn = log2 n +

2 log2 n − log2
log2 n
+ O(1).

Moreover, P(Hn = kn − 2 or Hn = kn − 1) → 1. This solves an open problem of Aldous & Shields.

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 27 / 28

slide-74

SLIDE 74

Summary of Results for Symmetric DSTs

Mean profile tends to infinity when k is roughly in the range log2 n − log2 log n ≤ k ≤ log2 n +

2 log2 n;
therwise it is bounded.

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 28 / 28

slide-75

SLIDE 75

Summary of Results for Symmetric DSTs

Mean profile tends to infinity when k is roughly in the range log2 n − log2 log n ≤ k ≤ log2 n +

2 log2 n;
therwise it is bounded.

Maximum of mean profile is of linear order.

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 28 / 28

slide-76

SLIDE 76

Summary of Results for Symmetric DSTs

Mean profile tends to infinity when k is roughly in the range log2 n − log2 log n ≤ k ≤ log2 n +

2 log2 n;
therwise it is bounded.

Maximum of mean profile is of linear order. Variance has same order as the mean. Thus, it tends to infinity iff mean tends to infinity.

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 28 / 28

slide-77

SLIDE 77

Summary of Results for Symmetric DSTs

Mean profile tends to infinity when k is roughly in the range log2 n − log2 log n ≤ k ≤ log2 n +

2 log2 n;
therwise it is bounded.

Maximum of mean profile is of linear order. Variance has same order as the mean. Thus, it tends to infinity iff mean tends to infinity. If mean tends to infinity, a central limit theorem holds.

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 28 / 28

slide-78

SLIDE 78

Summary of Results for Symmetric DSTs

Mean profile tends to infinity when k is roughly in the range log2 n − log2 log n ≤ k ≤ log2 n +

2 log2 n;
therwise it is bounded.

Maximum of mean profile is of linear order. Variance has same order as the mean. Thus, it tends to infinity iff mean tends to infinity. If mean tends to infinity, a central limit theorem holds. Our results have many applications, e.g., they allow us to solve a problem of Aldous & Shields.

Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 28 / 28