Consider the following program.

for (\(i = 0\); \(i < n\,/\,2\); \(i++\))

for (\(j = 0\); \(j < n\,/\,2\); \(j++\))

print(\(j\))A. (5 points) Assume the print statement is \(O(b)\). Write an expression \(f(n)\) that gives the run time of this code in the form of summations (\(\sum\)).

B. (5 points) Simplify your answer \(f(n)\) from Part A as much as possible. Show your work.

C. (5 points) Justify the Big-\(O\) order of your answer \(f(n)\) from Part B by finding \(c\) and \(n_0\) such that the definition of \(O\) holds.

Consider the following very slightly different program.

for (\(i = 0\); \(i < n\,/\,2\); \(i++\))

for (\(j = 0\); \(j < i\,/\,2\); \(j++\))

print(\(j\))A. (5 points) Assume the print statement is \(O(b)\). Write an expression \(f(n)\) that gives the run time of this code in the form of summations (\(\sum\)).

B. (5 points) Simplify your answer \(f(n)\) from Part A as much as possible. Show your work.

C. (5 points) Justify the Big-\(O\) order of your answer \(f(n)\) from Part B by finding \(c\) and \(n_0\) such that the definition of \(O\) holds.

(10 points) Simplify the following recursion to a non-recursive form. \[f(n)=\begin{cases} a & \text{if}\ n = 1\\b\,n + 3\,f(n\,/\,3) & \text{otherwise}\end{cases}\]

Consider the following recursive definition of the binary search. It seeks a value \(x\) in array \(A\) and would be invoked as search(\(A\), 0, \(n\), \(x\)). Assume \(A\) is indexed beginning with 0.

search(\(A\), \(f\), \(n\), \(x\))

if (\(n = 1\))

return \(f\)

else

\(m = n\,/\,2\)

if (\(x < A[f + m]\))

return search(\(A\), \(f\), \(m\), \(x\))

else

return search(\(A\), \(f + m\), \(n - m\), \(x\))A. (5 points) Show an example execution given the input \[A = [ 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25 ]\] and \(x = 13\). Specifically, list all calls to the search function in the order in which they occur, with arguments.

B. (5 points) Write an expression \(f(n)\) that gives the run time of this code in the form of a recursion. \[f(n)=\begin{cases}\text{base case} \\ \text{recursive case}\end{cases}\] Assume the base case is at least \(O(a)\) and the recursive case is at least \(O(b)\).

C. (10 points) Simplify your answer \(f(n)\) from Part B as much as possible, eliminating the recursion. Show your work.

D. (5 points) Justify the Big-\(O\) order of your answer \(f(n)\) from Part C by finding \(c\) and \(n_0\) such that the definition of \(O\) holds.

(10 points) Assume a stack \(S = [ 1, 2, 3, 4, 5, 6, 7, 8 ]\) (where 1 is at the top) with push and pop operations, where pop removes and returns the top value. Also assume an empty queue \(Q = [\,]\) with enqueue and dequeue operations, where dequeue removes and returns the front value. Show a sequence of stack and queue operations that result in \(S = [ 2, 3, 4, 1, 5, 6, 7, 8 ]\) and \(Q = [\,]\) using no other variables or containers.