$1.$ Which of the following statements about Breadth$-$First Search $($BFS$)$ is NOT true?

$($A$.)$ It takes time O$($n long n$)$ to visit the site.

$($B$.)$ It can be used to test whether a graph is connected or not.

$($C$.)$ It can calculate the shortest path from the starting point to each vertex.

$($D$.)$ can calculate whether there are loops in the graph.

$2.$ Which of the following descriptions of Depth$-$First Search $($DFS$)$ is NOT true?

$($a$.)$ It costs O$($n$+$m$)$ to visit the entire graph.

$($b$.)$ The spanning tree of a graph can be computed if the graph is connected.

$($C$.)$ Finding loops in a graph.

$($d$.)$ find the shortest path between two vertices.

$3.$ Is the quick sort algorithm a stable algorithm, depending on whether the input data is sorted or randomized? What is the average complexity of the time required for sorting?

a$.$ Unstable, with expectation O$($n log n$)$

b$.$ Unstable, O$($n log n$)$

c$.$ Stable, O$($n$^2)$

d$.$ Stable, O$($n log n$)$

$4.$ Consider a tree T storing n data items. If T is a binary search tree, what is the worst$-$case height of T$?$

$($a$.)$ n $/2$

$($b$.)$ n log n

$($C$)$ log n

$($D$)$ n

$5.$

Consider a tree T storing n data items. If T is a splay$-$tree, what is the worst$-$case height of T$?$

$($a$.)$ None of the following

$($b$.)$ n

$($C$.)$ log n

$($d$.)$ n log n