Show that randomized quick-sort runs in O(nlogn) time with probability at least 1 – 1/n, that is,…

Show that randomized quick-sort runs in O(nlogn) time with probability at least 1 − 1/n, that is, with high probability, by answering the following:

a. For ach input element x, define Ci,j(x) to be a 0/1 random variable that is 1 if and only if element x is in j + 1sub problems that belong to size group i. Argue why we need  not define Ci,j for j > n.
b. Let Xi,j be a 0/1 random variable that is 1 with probability 1/2j, independent of any other events, and let L = …log4/3 n…. Argue why

c. Show t the expected value of

D. Show that the probability that is at most 1/n2 using the Chernoff bound that states that if X is the sum of a finite number of independent 0/1 random variables with expected value μ > 0, then Pr(X > 2……) −…,
where e = 2.71828128…. e.
Argue why the previous claim proves randomized quick-sort runs in O(nlogn) time with probability at least 1 − 1/n.

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