Chapter 7, Problem 3P

(a)

Program Plan Intro

To argue the statement that probability of any particular element is chosen as the pivot is $1/n$ and also defined $E[Xi]$ .

(b)

Program Plan Intro

To argue that running time of quick-sort represented by $T[n]$ is- $E[T(n)]=E[{\displaystyle \sum _{q=1}^{n}Xq(T(q-1)+T(n-q)+\theta (n))}]$ .

(c)

Program Plan Intro

To show that equation $E[T(n)]=E[{\displaystyle \sum _{}^{}Xq(T(q-1)+T(n-q)+\theta (n))}]$ can be written as- $E[T(n)]=2/n{\displaystyle \sum _{q=2}^{n-1}E[T(q)]+\theta (n)}$ .

(d)

Program Plan Intro

To shows that ${\displaystyle \sum _{k=2}^{n-1}k\lg k}\le 1/2n^2\lg n-1/8n^2$ .

(e)

Program Plan Intro

To prove the recurrence relation ${\displaystyle \sum _{k=2}^{n-1}k\lg k}\le 1/2n^2\lg n-1/8n^2$ has the solution $E[T(n)]=\theta (n\lg n)$ .