$($a$)$ Write a recursive procedure for the Quick Sort algorithm including a separate sode for

the partition process.

$($b$)$ Apply the algorithm in $($a$)$ to sort the dataset $25,57,48,37,12,92,86$ in the ascending

order. Analyze the time$-$complexity of your solution. $a.s.$

$($c$)$ Consider the dataset in $($b$).$ How can you find the key $=86$ in the above dataset?

Analyze the time$-$complexity of your solution. $($a$)$ Write a procedure for the Bubble Sort algorithm.

$($b$)$ Apply the algorithm in $($a$)$ to sort the dataset $25,57,48,37,12,92,86$ in the ascending

order. Analyze the time$-$complexity of your solution.

$($c$)$ Consider the dataset in $($b$)$ after sorting has taken place. How can you find the key $=86$

iff the above sorted dataset? Analyze the time$-$complexity of your solution$\mathrm{.}4$ a$)$ Define bal:nce factor and a height balanced binary search tree.

b$)$ Show that the binary search tree constructed with nodes as $8,6,10,4,7,9,11,3,5,2$ is

not height$-$balanced. You should first show the step$-$by$-$step construction of the tree taking

the first data as the root.

c$)$ Define order and size.of a graph with examples.

od$)$ Show that the number of edges in a complete bipartite graph of order $2n$ is $n^2.$

e$)$ Prove or disprove: The graph $K_5$ is planar.