Balls-into-Bins Model and Chernoff Bounds Advanced Algorithms - PowerPoint PPT Presentation

Chernoff Bounds in Action: Load Balancing “ m balls are thrown into n bins uniformly and independently at random” Question: maximum number of balls in a bin? For 𝑛 ≥ 𝑜 ln 𝑜 , 𝜈 ≥ ln 𝑜

Chernoff Bounds in Action: Load Balancing “ m balls are thrown into n bins uniformly and independently at random” Question: maximum number of balls in a bin? For 𝑛 ≥ 𝑜 ln 𝑜 , 𝜈 ≥ ln 𝑜 𝑛 when 𝑛 = Ω(𝑜 log 𝑜) max load is 𝑃 w.h.p. 𝑜

Chernoff Bounds in Action: Load Balancing “ m balls are thrown into n bins uniformly and independently at random” Question: maximum number of balls in a bin? For 𝑛 ≥ 𝑜 ln 𝑜 , 𝜈 ≥ ln 𝑜 𝑛 when 𝑛 = Ω(𝑜 log 𝑜) max load is 𝑃 w.h.p. 𝑜

Chernoff Bounds in Action: Load Balancing “ m balls are thrown into n bins uniformly and independently at random” Question: maximum number of balls in a bin? For 𝑛 ≥ 𝑜 ln 𝑜 , 𝜈 ≥ ln 𝑜 𝑛 when 𝑛 = Ω(𝑜 log 𝑜) max load is 𝑃 w.h.p. 𝑜 𝑛 when 𝑛 = Ω(𝑜 log 𝑜) min load is Ω w.h.p. 𝑜

Load Balancing “ m balls are thrown into n bins uniformly and independently at random” Question: maximum number of balls in a bin?

Load Balancing “ m balls are thrown into n bins uniformly and independently at random” Question: maximum number of balls in a bin?

Chernoff Bounds

Generalization of Markov’s Inequality

Moment Generating Functions

Moment Generating Functions by Taylor’s expansion:

𝜇 > 0 , and generalized Markov’s inequality

𝜇 > 0 , and generalized Markov’s inequality

𝜇 > 0 , and generalized Markov’s inequality independence of X i not linearity of expectation

𝜇 > 0 , and generalized Markov’s inequality independence of X i not linearity of expectation

𝜇 > 0 , and generalized Markov’s inequality independence of X i not linearity of expectation

𝜇 > 0 , and generalized Markov’s inequality independence of X i not linearity of expectation

𝜇 > 0 , and generalized Markov’s inequality independence of X i not linearity of expectation 1 + 𝑧 ≤ 𝑓 𝑧

𝜇 > 0 , and generalized Markov’s inequality independence of X i not linearity of expectation 1 + 𝑧 ≤ 𝑓 𝑧

𝜇 > 0 , and generalized Markov’s inequality independence of X i not linearity of expectation 1 + 𝑧 ≤ 𝑓 𝑧

𝜇 > 0 , and generalized Markov’s inequality minimized when 𝜇 = ln(1 + 𝜀) independence of X i not linearity of expectation 1 + 𝑧 ≤ 𝑓 𝑧

𝜇 > 0 , and generalized Markov’s inequality (a) apply Markov’s inequality to moment generating function minimized when 𝜇 = ln(1 + 𝜀) independence of X i not linearity of expectation 1 + 𝑧 ≤ 𝑓 𝑧

𝜇 > 0 , and generalized Markov’s inequality (a) apply Markov’s inequality to moment generating function minimized when 𝜇 = ln(1 + 𝜀) independence of X i not linearity of expectation (b) bound the value of the moment generating function 1 + 𝑧 ≤ 𝑓 𝑧

𝜇 > 0 , and generalized Markov’s inequality (a) apply Markov’s inequality to moment generating function minimized when 𝜇 = ln(1 + 𝜀) (c) optimize the bound of the moment generating function independence of X i not linearity of expectation (b) bound the value of the moment generating function 1 + 𝑧 ≤ 𝑓 𝑧

Chernoff Bounds ???

Chernoff Bounds ???

Hoeffding’s Inequality

(Convenient) Hoeffding’s Inequality

Hoeffding’s Lemma

𝜇 > 0 , and generalized Markov’s inequality