FIND REVISITED Uwe Roesler and Diether Knof Stochastic fixed points for processes. Setting: On D X

FIND REVISITED

Uwe Roesler and Diether Knof Stochastic fixed points for processes.

Setting: On D

X =^{D X}

i A_iX_i ◦ B_i + C.

((A_i, B_i)_i, C), X_j, j ∈ IN independent, X_j ∼ X. Why another talk on Find?

– Toy example

– Introduction of FIND.

– Runtime analysis of Gruebel-Roesler.

– Problem of 2- and 3-version.

– FIND-process as fixed point.

– Knof’s approach via finite marginals.

– Solution via Weighted Branching Process.

FIND or QUICKSELECT Algorithm: FIND or QUICKSELECT and Algorithm 63, PARTITION, Hoare 61,

– Input: Set S of n different reals.

– Output: l-th smallest in S

– Procedure: Divide and conquer FIND(S, l):

Choose random pivot within S Split S into S_<,{p}, S_>

Continue with S_< or {p} or S_>.

r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r

l

u

u u u

u u

l

internal versus external randomness FIND(S, l)=FIND(|S|, l)

ANALYSIS Running time is proportional to

X_n(l) = number of comparisons for FIND(n, l) Key equation:

X_n(l) = n − 1 + 11_Un>lX_Un−1(l) + 11_Un<lX_n−Un(l − U_n) X, X, U_n independent,

U_n uniformly on {1, 2, . . . , n}, X ∼ X

Notice U_n is rank of pivot, (= position after compari- sons).

DIVERSES best case: X_n(l) = n − 1

worst case: X_n(l) = ⁿ⁽ⁿ₂⁻¹⁾

average: H_n harmonic numbers

EX_n(l) = 2((n+1)H_n−(n+3−k)H_n+1−l−(l+2)H_l+n+3) liml

n→t

EX_n(l)

n = 2 − 2tlnt − 2(1 − t) ln(1 − t) Knuth 71

variance: explicit Kirschenhofer, Prodinger 98, tail: ^Xn_n^(l) = C, Devroye 84,

E(C) ≤ 4 E(C^k) = O(1) P(C ≥ c) ≤







3 4







(1+o(1))c

RECALLS Recalls:

R_n(l) = number of recalls for FIND(n, l).

R_n(l) = 11^D _{U >l}R_U−1(l) + 11_{U <l}R_n−V(l − V ) + 1 ER_n(l) = H_l + H_n−l − 1

variance R_n(l) = ....

MAPLE, Kirschenhofer, Prodinger 1998 Gr¨ubel 2003

JOINT AVERAGES Joint averages:

X_n(J) = number for FIND(n, J)

Grand averages: Mahmoud et al 95 X_n^(p) := 1

n p

! X

|J|=pX_n(J) Analog R_n(J) and R^(p)_n .

First and second moment, Prodinger 95

DISTRIBUTION Normalization:

Y_n( l

n) = X_n(l)

n Y_n : [0,1] → IR, linear Theorem Gr¨ubel-R¨osler 96

For FIND 2-version exists versions Y_n →ⁿ Z

pointwise in D[0, 1] and Skorodhod metric.

Not true for 3 version!

3-version ⇔ split into S_<, {p}, S_>

2-version ⇔ split into S_<, S_≥

Skorodhod metric d on cadlad functions

d(f, g) = inf{ǫ | ∃λ kλ − idk∞, kf − g ◦ λk∞ < ǫ} where exists λ : [0,1] → [0, 1] one-one and increasing.

Intuition: Always Y_n →n Z in distribution.

But: (D, d) not complete.

FIND PROCESS Characterize Find process (Z(t))_t∈[0,1]

p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p

t

r r r r r

r r

r r r r

Z(t) = sum over the length of intervals containing t

r r r r r r r r

6

t

Z(t) = height at t

RANDOM BINARY NUMBERS

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u u

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00 01 10 11

011

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u

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u u u u

t

0 1

00 01 10 11

000 001 100

[0, 1) ∋ value t ↔ (t₁, t₂, . . .) path ∈ {0, 1}^IN Z_n(t) := 1 + ^Xn

i=1L(t₁, ..., t_i)

FIXED POINT EQUATION

Forward view and backward view Forward view

Zⁿ₊₁(t) = Zⁿ(t) +L(t₁, . . . , tⁿ₊₁).

Zⁿ(t) րⁿ Z(t) Backward view

Zⁿ+1(t) = 1 + 11^{D U >t}U Zⁿ( t

U) + 11^U_≤^t(1−U)Zⁿ(t−U 1−U)

= with suitable choosen versions (Zⁿ+1(t))t = 1 + 11^{U >t}U Zn⁽⁰⁾( t

U) + 11^U_≤^t(1−U)Z⁽¹⁾n (t−U 1−U)

!

t

Fixed point equation

(Z(t))^t =^D 1 + 11^t<UU Z( t

U) + 11^U_≤^t(1−U)Z(t−U 1−U)

!

t

Z, Z, U independent, U uniform distributed.

Compare to key equation for Yⁿ

Yⁿ(_n^l) =^D 11^Un

n >^l

n

Un

n Y^Un(_U^k_n) + 11^Un

n <^l

n

n−Un

n Yⁿ₋^Un(_n^l−₋^U_Uⁿ_n) + ⁿ⁻_n¹

↓ ↓ ↓ ↓

Z(t) =^D 11^{U >t}U Z(_U^t) + 11^U_≤^t(1−U)Z(₁^t−₋^U_U) + 1

FIND MOMENTS Expectation: a(t) = EZ(t)

a(t) = ^Z_t¹ua(t

u)du + ^Z₀^t(1 − u)a(t − u

1 − u)du + 1 Fixed point equation

Rate l₁











X_n,l^k n , Z









l − 1/2 n

















 ∼ ln l⁵(n − l + 1)⁵ n²

Kodaj and Mori 1997 in Wasserstein metric

l₁(µ, ν) = inf kX − Y k1 = ^Z |F_µ(x) − F_ν(x)|dx (Orlicz distance, limiting process)

HIGHER MOMENTS Higher moments:

nlim→∞EX_n^k(l_n)

n = E(Z^k(t)) =: m_k(t)

l_n

n →ⁿ t ∈ [0, 1]

m_k as fixed point, Paulsen 1997 K_k(f)(t) := ^Z_t¹u^kf(t

u)du+^Z₀^t(1−u)^kf(t − u

1 − u)du+b_k(t) b_k(t) := ^Xk

i=1(−1)ⁱ⁻¹







k i





m_k−i(t).

m_k is unique fixed point of K_k

K_k(m_k) = m_k D^k+2m_k(t) =







1

1 − t − 1 t





D^k+1m_k(t) + D^k+2b_k(t) D differentiation t.

Explicit solutions for m₁, m₂, More?

numerical approximations

Exponential moments exists, Devroye 84, Gr¨ubel- R¨osler 96

supt P(Z(t) ≥ z) ≤ e^{z+zln 4−zlnz} Gr¨ubel 99, also discrete

SUPREMUM of FIND Supremum: sup_l X_n(l)

M := sup_t Z(t) satisfies fixed point equation M = 1 + (U M^D ) ∨ ((1 − U)M) EM² < ∞, unique solution Gr¨ubel-R¨osler 96

Devroye 01

2.3862 < 1+2 ln 2 ≤ E(M−1) ≤ 5

√2π+1212e

5 < 8.5185 E(M − 1)^k ≤ 3^k−1k!E(M − 1).

P(M ≥ t) ≤ Ce^−λt.

∀λ > 0∃C∀t.

Lemma M = sup_tZ(t) or M = sup_l ^Xn_n^(l). Then P(M ≥ t) ≤ Ce^−λt. (1) Proof: Define

K(µ) = L(1 + (U X) ∨ ((1 − U)X) If µ satisfies (1) then K(µ) satisfies (1)

K a strict contraction in l₂−metric µ, K(µ), K²(µ), . . .

µ satisfies the condition, all do, limit does.

Discrete analog K_n, yes q.e.d.

FIND as FIXED POINT FIND process satisfies

(Z(t))_t = (11^D _{U >t}U Z₁( t

U) + 11_U≤t(1− U)Z₂(t − U

1 − U) + 1)_t Distribution is fixed point of map K

Contraction method:

– K contraction for suitable metric – iterate K

Knof in thesis 2007 – On D

K(µ) ∼ ^X

i A_iX_i ◦ B_i + C

((A_i, B_i)_i, C), X_j, j ∈ IN independent, X_j ∼ µ.

– Suitable metric on finite dimensional marginals – Finite dimensional marginals converge

(Kⁿ(µ))_J →ⁿ ϕ(J) weakly

– (ϕ(J))_J is consistent family of pm

– projective limit ν (on product space IR^[0,1]) – outer ν^∗(D) = 1

Via Dubins-Hahn and recursive structure.

– Restrict ν to D. Done Next aim FIND: Y_n → Z.

WEIGHTED BRANCHING PROCESS

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Weighted Branching Process is tupel (V, T,(G, ∗)).

– Tree V = ∪^∞_n=0INⁿ as graph

– (G,∗) semigroup with grave and neutral – edge (v, vi) carries random weight T_i(v) Assumption: T(v) = (T_i(v))_i, v ∈ V iid.

L_v path length to v L_vi = L_v ∗ T_i(v).

Weighted Branching Process with utility is tupel (V, T, U,(G, ∗, R)).

– (V, T,(G,∗)) is WBP.

– (G,∗) operates transitiv on R via ∗ : G × R → R – Every knot v carries rv U_v with values in R

Assumption: (T(v), C_v), v ∈ V iid.

Object of interest L_v ∗ U_v or functions of it.

SETTING Setting, on D

X =^{D X}

i A_iX_i ◦ B_i + C A_i, C rv D₊-valued, B_i rvs D_↑-valued

σ-field on D via Skorodhod metric, Billingsley

(isomorph to product space (IR^I,B^I) where 1 ∈ I dense in [0, 1].)

G = D₊ × D_↑ with semi group operation

∗((a, b),(a^′, b^′)) = (aa^′ ◦ b, b^′ ◦ b).

grave is function identically 0,

neutral element function identically 1 R = D₊ with pointwise multiplication edge weight (T(v), U_v) ∼ ((A_i, B_i)_i, C) object of interest

R_n = ^X

|v|≤nL_v ∗ U_v. Backward equation

R_n+1 = ^X

i∈IN L_i∗R_n(i) +L_∅∗U_∅ = ^X

i∈IN A_iR_n(i)◦B_i+C

FIXED POINT K(µ) =^{D X}

i∈IN A_iX_i ◦ B_i + C.

Weighted Branching Process R_n = ^X

|v|≤nL_v ∗ U_v.

• Kⁿ(δ₀) ∼ R_n.

Kⁿ(δ₀) →ⁿ fixed point ⇔ R_n →ⁿ R.

• R_n ր R = ^P_v∈V L_v ∗ U_v.

A(s) := ^P_i A_i(s) f = sup_s |f(s)|

Theorem Assume EA < 1 and EC < ∞. Then ER_n+1 − R_n ≤ Eⁿ(A)E(C)

and R takes values in D.

The Supremum via fixed point of supremum-type R ≤ Asup

i R(i) + C.

TheoremAssume Y < 1 a.s., ^P_i Y_i < ∞ and some exponential moments. Then R has exponential mo- ments.

MEDIAN-VERSION of FIND

r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r

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– pivot is median of m randomly choosen – costs n − 1 ≤ C_n ≤ n + m².

– Rank of median normalized ^I_nⁿ →n U weak U median of m iid uniformly distributed – Limit

(Z(t))_t = 11^D _t<UU Z₁( t

U) + 11_t≥U(1 − U)Z₂(t − U

1 − U) + 1 Corollary Y_n weakly to Z.

ADAPTED VERSION of FIND – m = m_n → ∞ slowly

– choose as pivot the kⁿ(l) smallest out of m_n – costs n − 1 ≤ C_n ≤ n + (m_n)².

– ^k_m^n(l)

n close to _n^l Corollary R¨osler Y_n → Z weakly,

Z(s) = 1 + min{s,1 − s} The very best possible!

Martinez, Panario and Viola ’03

- 6

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1 1

CONCLUSION

– (weaker) measure versus (stronger) process conver- gence

– Stochastic fixed points by contraction of X =^{D X}

i A_iX_i ◦ B_i + C

Study of Weighted Branching Processes provi- des

– Fixed points of

X = ^X

i A_iX_i ◦ B_i + C – complete analysis of FIND

– Identified the very best version of FIND