Randomness in Computing L ECTURE 13 Last time Finished routing on - - PowerPoint PPT Presentation

▶

Oct 16, 2022 263 likes •350 views

Randomness in Computing L ECTURE 13 Last time Finished routing on hypercube Balls into bins model Today Poisson distribution Poisson approximation 3/5/2020 Sofya Raskhodnikova;Randomness in Computing The number of empty bins

SLIDE 1

3/5/2020

Randomness in Computing

LECTURE 13

Last time

Finished routing on hypercube
Balls into bins model

Today

Poisson distribution
Poisson approximation

Sofya Raskhodnikova;Randomness in Computing

SLIDE 2

𝒏 balls into 𝒐 bins

The probability that bin 1 is empty is
Expected number of empty bins

𝑌 = the number of empty bins 𝑌𝑗 = 𝔽 𝑌 =

3/5/2020

Sofya Raskhodnikova; Randomness in Computing

The number of empty bins

𝒇−𝒚−𝒚𝟑 ≤ 𝟐 − 𝒚 ≤ 𝒚−𝒚 for 𝒚 ≤ 𝟐/𝟑

SLIDE 3

𝒏 balls into 𝒐 bins, 𝒔 is a small constant

The probability 𝑞𝑠 that bin 1 has 𝑠 balls is

𝑞𝑠=

3/5/2020

Sofya Raskhodnikova; Randomness in Computing

The number of bins with 𝒔 balls

SLIDE 4

Poisson random variables

A Poisson random variable with parameters 𝜈 is given

by the following distribution on 𝑘 = 0,1,2, … Pr 𝑌 = 𝑘 = 𝑓−𝜈𝜈𝑘 𝑘!

Check that probabilities sum to 1:

𝑘=0

∞

Pr 𝑌 = 𝑘 = 𝑘=0

∞

𝑓−𝜈𝜈𝑘 𝑘!

=

The expectation of a Poisson R.V. 𝑌 is

𝔽 𝑌 = var 𝑌 = 𝜈 (See Ex. 5.5)

Taylor expansion: 𝒇𝒚 = 𝒌=𝟏

∞ 𝒚𝒌 𝒌!

SLIDE 5

Independent Poisson RVs

3/5/2020

Sofya Raskhodnikova; Randomness in Computing

Theorem

Let 𝑌 and 𝑍 be independent Poisson RVs with means 𝜈𝑌 and 𝜈𝑍. Then 𝑌 + 𝑍 is a Poisson RV with mean 𝜈𝑌 + 𝜈𝑍.

SLIDE 6

Chernoff Bounds for Poisson RVs

Theorem. Let 𝑌 be a Poisson RV with mean 𝜈.
(upper tail, additive) If 𝑦 > 0, then

Pr 𝑌 ≥ 𝜈 + 𝑦 ≤ 𝑓−𝜈 𝑓𝜈 𝑦 𝑦𝑦 .

(lower tail, additive) If 𝑦 < 𝜈, then

Pr 𝑌 ≤ 𝑦 ≤ 𝑓−𝜈 𝑓𝜈 𝑦 𝑦𝑦 .

(upper tail, multiplicative) For any 𝜀 > 0,

Pr 𝑌 ≥ 1 + 𝜀 𝜈 ≤ 𝑓𝜀 1 + 𝜀 1+𝜀

𝜈

.

(lower tail, multiplicative) For any 𝜀 ∈ (0,1),

Pr 𝑌 ≤ 1 − 𝜀 𝜈 ≤ 𝑓𝜀 1 − 𝜀 1−𝜀

𝜈

.

3/5/2020

Sofya Raskhodnikova; Randomness in Computing

SLIDE 7

Poisson Distribution is Limit of Binomial Distribution

Applies to balls-and-bins fi 𝑛 = 𝑜𝑑.

3/5/2020

Sofya Raskhodnikova; Randomness in Computing

Theorem

Let 𝑌𝑜 ∼ Bin 𝑜, 𝑞 , where 𝑞 is a function of 𝑜 and lim

𝑜→∞ 𝑜𝑞 = 𝜈,

a constant independent of 𝑜. Then, for all fixed 𝑙,

lim

𝑜→∞ Pr[𝑌𝑜 = 𝑙] = 𝑓−𝜈𝜈𝑙

𝑙! .

SLIDE 8

The Poisson Approximation

The Balls-and-Bins model has dependences.
E.g. if Bin 1 is empty, then Bin 2 is less likely to

be empty.

The Poisson Approximation gets rid of

dependencies.

(on the board).

3/5/2020

Sofya Raskhodnikova; Randomness in Computing