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

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 Raskhodnikova;Randomness in Computing

Review: Poisson random variables • A Poisson random variable with parameter 𝜈 is given by the following distribution on 𝑘 = 0,1,2, … Pr 𝑌 = 𝑘 = 𝑓 −𝜈 𝜈 𝑘 𝑘! • 𝔽 𝑌 = var 𝑌 = 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

Review: Poisson approximation • The Balls-and-Bins model has dependences. • The Poisson approximation gets rid of dependencies. 1 2 3 4 5 … 𝑛 -1 𝑛 𝑛 balls into 𝑜 bins u.i.r. • For 𝑗 ∈ 𝑜 , let 1 2 3 … 𝑜 (𝑛) = # of balls in bin 𝑗 𝑌 𝑗 (real world) (𝑛) ∼ 𝑄𝑝𝑗𝑡𝑡𝑝𝑜(𝜈) , where 𝜈 = 𝑛 (Poisson world) 𝑍 𝑜 and 𝑗 (𝑛) are mutually independent. 𝑍 𝑗 • 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 𝑜 bins. Sofya Raskhodnikova; Randomness in Computing 3/17/2020

Approximating a function of the loads of the bins Poisson Approximation Theorem Let 𝑔 𝑦 1 , … , 𝑦 𝑜 ≥ 0 for all 𝑦 1 , … , 𝑦 𝑜 ∈ 0,1,2, … . Then Poisson case exact case 𝑛 , … , 𝑌 𝑜 𝑛 , … , 𝑍 𝑛 𝑛 𝔽 𝑔 𝑌 1 ≤ 𝒇 𝒏 ⋅ 𝔽 𝑔 𝑍 . 𝑜 1 𝑜 𝑜 • F act 𝑇𝑢𝑗𝑠𝑚𝑗𝑜𝑕 ′ 𝑡 𝑔𝑝𝑠𝑛𝑣𝑚𝑏 : 𝑜! ∼ 2𝜌𝑜 𝑓 𝑜 𝑜 𝑜 𝑜 • Bounds for all 𝑜 ∈ ℕ: 2𝜌 𝑜 ≤ 𝑜! ≤ 𝑓 𝑜 𝑓 𝑓 Sofya Raskhodnikova; Randomness in Computing 3/17/2020

Approximating a function of the loads of the bins Poisson Approximation Theorem Let 𝑔 𝑦 1 , … , 𝑦 𝑜 ≥ 0 for all 𝑦 1 , … , 𝑦 𝑜 ∈ 0,1,2, … . Then 𝑛 , … , 𝑌 𝑜 𝑛 , … , 𝑍 𝑛 𝑛 𝔽 𝑔 𝑌 1 ≤ 𝒇 𝒏 ⋅ 𝔽 𝑔 𝑍 . 𝑜 1 𝑛 , … , 𝑍 𝑛 Proof: 𝔽 𝑔 𝑍 𝑜 1 ∞ (𝑛) = 𝑙 ⋅ Pr ෍ 𝑛 , … , 𝑍 𝑛 = 𝑙 𝑛 = ෍ 𝔽 𝑔 𝑍 | ෍ 𝑍 𝑍 𝑜 1 𝑗 𝑗 𝑙=0 𝑗∈[𝑜] 𝑗∈ 𝑜 𝑛 , … , 𝑍 𝑛 = 𝑛 ⋅ Pr ෍ 𝑛 = 𝑛 𝑛 ≥ 𝔽 𝑔 𝑍 | ෍ 𝑍 𝑍 𝑜 1 𝑗 𝑗 𝑗∈ 𝑜 𝑗∈ 𝑜 𝑛 , … , 𝑌 𝑜 𝑛 = 𝔽 𝑔 𝑌 1 ⋅ Pr 𝑍 = 𝑛 Sofya Raskhodnikova; Randomness in Computing 3/17/2020

Approximating a function of the loads of the bins Poisson Approximation Theorem Let 𝑔 𝑦 1 , … , 𝑦 𝑜 ≥ 0 for all 𝑦 1 , … , 𝑦 𝑜 ∈ 0,1,2, … . Then 𝑛 , … , 𝑌 𝑜 𝑛 , … , 𝑍 𝑛 𝑛 𝔽 𝑔 𝑌 1 ≤ 𝒇 𝒏 ⋅ 𝔽 𝑔 𝑍 . 𝑜 1 𝑛 • Poisson case: # of balls in each bin is independent Poisson 𝑜 • Corollary. Any event that has probability 𝑞 in the Poisson case has probability ≤ 𝑞 ⋅ 𝒇 𝒏 in the exact case. Proof: Let 𝑌 be the indicator for that event. Then 𝔽 [X] is the probability that event occurs. • Improvements to Theorem and Corollary 𝑛 , … , 𝑌 𝑜 𝑛 If 𝔽 𝑔 𝑌 1 is monotonically nonincreasing (or nondecreasing) in 𝑛, then 𝒇 𝒏 can be changed to 2. Sofya Raskhodnikova; Randomness in Computing 3/17/2020

Application: Max Load 𝒐 balls into 𝒐 bins 3 ln 𝑜 1 • Before (in Discussion): Pr 𝑁𝑏𝑦𝑀𝑝𝑏𝑒 > ln ln 𝑜 ≤ 𝑜 for s.l. 𝑜 ln 𝑜 1 • Theorem. Pr 𝑁𝑏𝑦𝑀𝑝𝑏𝑒 < ln ln 𝑜 ≤ 𝑜 for s.l. 𝑜 ln 𝑜 Proof: Let 𝑁 = ln ln 𝑜 Sofya Raskhodnikova; Randomness in Computing 3/17/2020

Application: Coupon Collector 𝒀 = # of coupons observed before obtaining 1 of each of 𝒐 types and Pr 𝑌 > 𝑜 ln 𝑜 + 𝑑𝑜 ≤ 𝑓 −𝑑 ∀𝑑 > 0 • Before: 𝔽 [X] = Review: Pr not obtaining coupon 𝑗 after 𝑜 ln 𝑜 + 𝑑𝑜 steps is • Theorem. Pr 𝑌 > 𝑜 ln 𝑜 + 𝑑𝑜 ≤ 2(1 − 𝑓 −1.5⋅𝑓 −𝑑 ) ∀𝑑 > 0, s.l. 𝑜 𝑜→∞ Pr 𝑌 > 𝑜 ln 𝑜 + 𝑑𝑜 ≤ 1 − 𝑓 −𝑓 −𝑑 ∀𝑑 > 0 • MU: lim Sofya Raskhodnikova; Randomness in Computing 3/17/2020

Application: Coupon Collector 𝒀 = # of coupons observed before obtaining 1 of each of 𝒐 types • Theorem. Pr 𝑌 > 𝑜 ln 𝑜 + 𝑑𝑜 ≤ 2(1 − 𝑓 −1.5⋅𝑓 −𝑑 ) ∀𝑑 > 0, s.l. 𝑜 Proof: Balls and bins view 𝑜 bins • 𝑌 = # balls thrown before all bins nonempty • Consider 𝑛 = 𝑜(ln 𝑜 + 𝑑) balls. • Let 𝐶 = event that there is an empty bin. Pr 𝑌 > 𝑜 ln 𝑜 + 𝑑𝑜 = Pr[B] • Idea: Use Poisson approximation: # balls in each bin is Poisson( 𝜈 ) with 𝜈 = ln 𝑜 + 𝑑 • 𝐹 𝑗 = event that bin 𝑗 is empty. Then 𝑓 −𝑑 • Pr 𝐹 𝑗 = 𝑓 −𝜈 = 𝑓 − ln 𝑜+𝑑 = 𝑜 Sofya Raskhodnikova; Randomness in Computing 3/17/2020

Application: Coupon Collector Theorem. Pr 𝑌 > 𝑜 ln 𝑜 + 𝑑𝑜 ≤ 2(1 − 𝑓 −1.5⋅𝑓 −𝑑 ) ∀𝑑 > 0, s.l. 𝑜 Proof: 𝑜 bins and 𝑛 = 𝑜(ln 𝑜 + 𝑑) balls. • Let 𝐶 = event that there is an empty bin: Pr 𝑌 > 𝑜 ln 𝑜 + 𝑑𝑜 = Pr[B] • Poisson approximation: # balls in each bin is Poisson( 𝜈 ), 𝜈 = ln 𝑜 + 𝑑 𝑓 −𝑑 • 𝐹 𝑗 = event that bin 𝑗 is empty. Then Pr 𝐹 𝑗 = for 𝒚 ≤ 𝟐/𝟑 𝑜 𝒇 −𝒚−𝒚 𝟑 ≤ 𝟐 − 𝒚 ≤ 𝒇 −𝒚 𝒇 −𝟐.𝟔𝒚 = 𝒇 −𝒚−𝟏.𝟔 𝒚 ≤ Sofya Raskhodnikova; Randomness in Computing 3/17/2020