Pretty: Tail bound: Pr[X>t]<e. Good Thresholding: The running time of a Las Vegas Alg. Ugly: ● Some cost (e.g.max load). ·The probability of extreme case
Tail bound: Pr[X > t] < . • The running time of a Las Vegas Alg. • Some cost (e.g. max load). • The probability of extreme case. Thresholding: Good Pretty: Ugly: Good
Tail bound: X follows distribution Pr[X>t]<e. Relate tail to some measurable characters of X character Reduce the tail bound to the analysis of the characters. Pr[X>t]<f(t,1
Tail bound: Pr[X > t] < . Relate tail to some measurable characters of X X follows distribution D character I Reduce the tail bound to the analysis of the characters. Pr[ X > t ] < f (t, I )
Markov's Inequality Markov's Inequality: For nonnegative X,for any t >0, E[X] Pr[X≥t≤ t Proof: f(x) 1 ifX≥t, →Y≤ X ≤ X E(X] Pr[X≥t=E[Y]≤E p(Xza) tight if we only know the expectation of X
Markov’s Inequality Markov’s Inequality: Pr[X ⇥ t] E[X] t . For nonnegative X , for any t > 0, Y ⇥ X t ⇥ ⇥ X t , Pr[X t] = E[Y ] E X t ⇥ = E[X] t . Proof: Y = 1 if X t, 0 otherwise. Let tight if we only know the expectation of X
Las Vegas to Monte Carlo Las Vegas:running time is B(x): random,always correct. run A(x)for 2T(n)steps; if A(x)returned ●A:Las Vegas Alg with return A(x); worst-case expected running time T(n). else return "yes" one-sided error! ● Monte Carlo:running time is fixed,correctness Pr[error] is random. ≤Pr[T(A(x)>2T(n)] ●B:Monte Carlo Alg E[T(A(x))] 1 ≤ ≤ 2T(n) -2
Las Vegas to Monte Carlo • Las Vegas: running time is random, always correct. • A: Las Vegas Alg with worst-case expected running time T(n). • Monte Carlo: running time is fixed, correctness is random. • B: Monte Carlo Alg ... B(x): run A(x) for 2T(n) steps; if A(x) returned return A(x); else return “yes”; one-sided error! Pr[error] Pr[T (A(x)) > 2T (n)] E[T (A(x))] 2T (n) 1 2
Markov's Inequality Markov's Inequality: For nonnegative X,for any t >0, E[X] Pr[X≥t≤ t
Markov’s Inequality Markov’s Inequality: Pr[X ⇥ t] E[X] t . For nonnegative X , for any t > 0