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Randomness in Computing L ECTURE 14 Last time Poisson distribution - - PowerPoint PPT Presentation

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Randomness in Computing L ECTURE 14 Last time Poisson distribution Poisson approximation Today Review Poisson distribution Poisson approximation Application: max load Application: Coupon Collector 3/17/2020 Sofya

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SLIDE 1

3/17/2020

Randomness in Computing

LECTURE 14

Last time

  • Poisson distribution
  • Poisson approximation

Today

  • Review Poisson distribution
  • Poisson approximation
  • Application: max load
  • Application: Coupon Collector

Sofya Raskhodnikova;Randomness in Computing

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SLIDE 2

Review: Poisson random variables

  • A Poisson random variable with parameter 𝜈 is given by

the following distribution on π‘˜ = 0,1,2, … Pr π‘Œ = π‘˜ = π‘“βˆ’πœˆπœˆπ‘˜ π‘˜!

  • 𝔽 π‘Œ = var π‘Œ = 𝜈
  • Sum of two Poisson RVs is a Poisson RV
  • A Poisson RV satisfies Chernoff-type bounds.
  • Poisson distribution is a limit of binomial distribution
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SLIDE 3

Review: Poisson approximation

  • The Balls-and-Bins model has dependences.
  • The Poisson approximation gets rid of dependencies.
  • 𝑛 balls into π‘œ bins u.i.r.

For 𝑗 ∈ π‘œ , let

(real world)

π‘Œπ‘—

(𝑛) = # of balls in bin 𝑗

(Poisson world) 𝑍

𝑗 (𝑛) ∼ π‘„π‘π‘—π‘‘π‘‘π‘π‘œ(𝜈), where 𝜈 = 𝑛 π‘œ and

𝑍

𝑗 (𝑛) are mutually independent.

  • If we condition the Poisson distribution on producing exactly 𝑙 balls, then

it’s the same as the distribution resulting from throwing 𝑙 balls into π‘œ bins.

3/17/2020

Sofya Raskhodnikova; Randomness in Computing

1 2 3 … π‘œ 1 2 3 4 5 … 𝑛-1 𝑛

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SLIDE 4

Approximating a function

  • f the loads of the bins
  • Fact π‘‡π‘’π‘—π‘ π‘šπ‘—π‘œπ‘•β€²π‘‘ π‘”π‘π‘ π‘›π‘£π‘šπ‘ : π‘œ! ∼

2πœŒπ‘œ

π‘œ 𝑓 π‘œ

  • Bounds for all π‘œ ∈ β„•:

2𝜌 π‘œ

π‘œ 𝑓 π‘œ

≀ π‘œ! ≀ 𝑓 π‘œ

π‘œ 𝑓 π‘œ

3/17/2020

Sofya Raskhodnikova; Randomness in Computing

Poisson Approximation Theorem

Let 𝑔 𝑦1, … , π‘¦π‘œ β‰₯ 0 for all 𝑦1, … , π‘¦π‘œ ∈ 0,1,2, … . Then

𝔽 𝑔 π‘Œ1

𝑛 , … , π‘Œπ‘œ 𝑛

≀ 𝒇 𝒏 β‹… 𝔽 𝑔 𝑍

1 𝑛 , … , 𝑍 π‘œ 𝑛

.

exact case Poisson case

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SLIDE 5

Approximating a function

  • f the loads of the bins

Proof: 𝔽 𝑔 𝑍

1 𝑛 , … , 𝑍 π‘œ 𝑛

= ෍

𝑙=0 ∞

𝔽 𝑔 𝑍

1 𝑛 , … , 𝑍 π‘œ 𝑛

| ෍

π‘—βˆˆ[π‘œ]

𝑍

𝑗 (𝑛) = 𝑙 β‹… Pr ෍ π‘—βˆˆ π‘œ

𝑍

𝑗 𝑛 = 𝑙

β‰₯ 𝔽 𝑔 𝑍

1 𝑛 , … , 𝑍 π‘œ 𝑛

| ෍

π‘—βˆˆ π‘œ

𝑍

𝑗 𝑛 = 𝑛 β‹… Pr ෍ π‘—βˆˆ π‘œ

𝑍

𝑗 𝑛 = 𝑛

= 𝔽 𝑔 π‘Œ1

𝑛 , … , π‘Œπ‘œ 𝑛

β‹… Pr 𝑍 = 𝑛

3/17/2020

Sofya Raskhodnikova; Randomness in Computing

Poisson Approximation Theorem Let 𝑔 𝑦1, … , π‘¦π‘œ β‰₯ 0 for all 𝑦1, … , π‘¦π‘œ ∈ 0,1,2, … . Then

𝔽 𝑔 π‘Œ1

𝑛 , … , π‘Œπ‘œ 𝑛

≀ 𝒇 𝒏 β‹… 𝔽 𝑔 𝑍

1 𝑛 , … , 𝑍 π‘œ 𝑛

.

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SLIDE 6

Approximating a function

  • f the loads of the bins
  • Poisson case: # of balls in each bin is independent Poisson

𝑛 π‘œ

  • Corollary. Any event that has probability π‘ž in the Poisson case

has probability ≀ π‘ž β‹… 𝒇 𝒏 in the exact case. Proof: Let π‘Œ be the indicator for that event. Then 𝔽[X] is the probability that event occurs.

  • Improvements to Theorem and Corollary

If 𝔽 𝑔 π‘Œ1

𝑛 , … , π‘Œπ‘œ 𝑛

is monotonically nonincreasing (or nondecreasing) in 𝑛, then 𝒇 𝒏 can be changed to 2.

3/17/2020

Sofya Raskhodnikova; Randomness in Computing

Poisson Approximation Theorem Let 𝑔 𝑦1, … , π‘¦π‘œ β‰₯ 0 for all 𝑦1, … , π‘¦π‘œ ∈ 0,1,2, … . Then

𝔽 𝑔 π‘Œ1

𝑛 , … , π‘Œπ‘œ 𝑛

≀ 𝒇 𝒏 β‹… 𝔽 𝑔 𝑍

1 𝑛 , … , 𝑍 π‘œ 𝑛

.

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SLIDE 7

𝒐 balls into 𝒐 bins

  • Before (in Discussion): Pr 𝑁𝑏𝑦𝑀𝑝𝑏𝑒 >

3 ln π‘œ ln ln π‘œ ≀ 1 π‘œ for s.l. π‘œ

  • Theorem. Pr 𝑁𝑏𝑦𝑀𝑝𝑏𝑒 <

ln π‘œ ln ln π‘œ ≀ 1 π‘œ for s.l. π‘œ

Proof: Let 𝑁 =

ln π‘œ ln ln π‘œ

3/17/2020

Sofya Raskhodnikova; Randomness in Computing

Application: Max Load

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SLIDE 8

𝒀 = # of coupons observed before obtaining 1 of each of 𝒐 types

  • Before: 𝔽[X] =

and Pr π‘Œ > π‘œ ln π‘œ + π‘‘π‘œ ≀ π‘“βˆ’π‘‘ βˆ€π‘‘ > 0 Review: Pr not obtaining coupon 𝑗 after π‘œ ln π‘œ + π‘‘π‘œ steps is

  • Theorem. Pr π‘Œ > π‘œ ln π‘œ + π‘‘π‘œ ≀ 2(1 βˆ’ π‘“βˆ’1.5β‹…π‘“βˆ’π‘‘) βˆ€π‘‘ > 0, s.l. π‘œ
  • MU: lim

π‘œβ†’βˆž Pr π‘Œ > π‘œ ln π‘œ + π‘‘π‘œ ≀ 1 βˆ’ π‘“βˆ’π‘“βˆ’π‘‘ βˆ€π‘‘ > 0

3/17/2020

Sofya Raskhodnikova; Randomness in Computing

Application: Coupon Collector

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SLIDE 9

𝒀 = # of coupons observed before obtaining 1 of each of 𝒐 types

  • Theorem. Pr π‘Œ > π‘œ ln π‘œ + π‘‘π‘œ ≀ 2(1 βˆ’ π‘“βˆ’1.5β‹…π‘“βˆ’π‘‘) βˆ€π‘‘ > 0, s.l. π‘œ

Proof: Balls and bins view π‘œ bins

  • π‘Œ = # balls thrown before all bins nonempty
  • Consider 𝑛 = π‘œ(ln π‘œ + 𝑑) balls.
  • Let 𝐢 = event that there is an empty bin.

Pr π‘Œ > π‘œ ln π‘œ + π‘‘π‘œ = Pr[B]

  • Idea: Use Poisson approximation:

# balls in each bin is Poisson(𝜈) with 𝜈 = ln π‘œ + 𝑑

  • 𝐹𝑗 = event that bin 𝑗 is empty. Then
  • Pr 𝐹𝑗 = π‘“βˆ’πœˆ = π‘“βˆ’ ln π‘œ+𝑑 =

π‘“βˆ’π‘‘ π‘œ

3/17/2020

Sofya Raskhodnikova; Randomness in Computing

Application: Coupon Collector

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SLIDE 10
  • Theorem. Pr π‘Œ > π‘œ ln π‘œ + π‘‘π‘œ ≀ 2(1 βˆ’ π‘“βˆ’1.5β‹…π‘“βˆ’π‘‘) βˆ€π‘‘ > 0, s.l. π‘œ

Proof: π‘œ bins and 𝑛 = π‘œ(ln π‘œ + 𝑑) balls.

  • Let 𝐢 = event that there is an empty bin: Pr π‘Œ > π‘œ ln π‘œ + π‘‘π‘œ = Pr[B]
  • Poisson approximation: # balls in each bin is Poisson(𝜈), 𝜈 = ln π‘œ + 𝑑
  • 𝐹𝑗 = event that bin 𝑗 is empty. Then Pr 𝐹𝑗 =

π‘“βˆ’π‘‘ π‘œ

3/17/2020

Sofya Raskhodnikova; Randomness in Computing

Application: Coupon Collector

π’‡βˆ’π’šβˆ’π’šπŸ‘ ≀ 𝟐 βˆ’ π’š ≀ π’‡βˆ’π’š for π’š ≀ 𝟐/πŸ‘ π’‡βˆ’πŸ.πŸ”π’š = π’‡βˆ’π’šβˆ’πŸ.πŸ” π’š ≀