Chapter 13, Problem 4P

a.

Program Plan Intro

To show that the treap associated with the given nodes is unique.

b.

Program Plan Intro

To show that the expected height of treap is $\Theta \left(\lg n\right)$ and therefore the expected time to search a value is $\Theta \left(\lg n\right)$ .

c.

Program Plan Intro

To explain the procedure of TREAP-INSERT in English and provide its pseudo code.

d.

Program Plan Intro

To show that expected running time of TREAP-INSERT is $\Theta \left(\lg n\right)$ .

e.

Program Plan Intro

To prove that the total number of rotations performed during insertion of x is equal to C + D.

f.

Program Plan Intro

To prove that $X_{ik}=1$ if and only if y.priority >x.priority, y.key < x.key and for every z such that y.key < z.key < x.key there is a condition that y.priority < z.priority.

g.

Program Plan Intro

To prove that $\mathrm{Pr}\left\{X_{ik}=1\right\}=\frac{\left(k-i-1\right)!}{\left(k-i+1\right)!}=\frac{1}{\left(k-i+1\right)\left(k-i\right)}$

h.

Program Plan Intro

To prove that $E\left[C\right]={\displaystyle \sum _{j=1}^{k-1}\frac{1}{j\left(j+1\right)}}=1-\frac{1}{k}$

i.

Program Plan Intro

To prove that $E\left[D\right]=1-\frac{1}{n-k+1}$ , using symmetry argument.

j.

Program Plan Intro

To conclude that expected number of rotations performed during insertion of node into a treap is less than 2.