Give the asymptotic running time of each of the following functions in notation. Justify your answer (write all details). What is the asymptotic worst case running time? Whats the asymptotic expected running time? Random(n): generates a random number between 1 and n with uniform distribution. CoinFlip(): returns heads or tails with equal probability.

YOUR STEPS SHOULD LOOK LIKE THIS:

Steps 2-4 take c time. (a) In the worst case c1=c2 so statement 6 is executed. T(n)=c+T(n4)+T(n7)T(n4)+T(n7)2T(n7)22T(n14)222T(n21)n/72222T(1)2n/7c(2n/7). Since T(n)(2n/7), worst case running time T(n) has an exponential lower bound. (b) Prob(c1=c2) is 1/2. ET(n)ET(n)ET(n)=Prob(c1=c2)ET(c1=c2)+Prob(c1=c2)ET(c1=c2)=(1/2)(c+T(n4)+T(n7))+(1/2)c=c+(1/2)T(n4)+(1/2)T(n7).=c+(1/2)T(n4)+(1/2)T(n7)c+(1/2)T(n4)+(1/2)T(n4)=c+T(n4)=c+c+T(n8)=n/4c+c++T(1)=n/4c+c++c=cn/4=c+(1/2)T(n4)+n/7(1/2)T(n7)c+(1/2)T(n7)+n/7(1/2)T(n7)=c+T(n7)=c+c+T(n14)=c+c++T(1)=c+c++c=cn/7 Since cn/7ET(n)cn/4, expected running time ET(n)(n)