All Algorithms questions

Single correct

• A$T_1(n) = \Theta(n^2)$
• B$T_1(n) = \Theta(n^2 \log_2 n)$
• C$T_1(n) = \Theta(n^{\log_4 5})$
• D$T_1(n) = \Theta(n^{\log_4 5} \log_2 n)$

Single correct

• A$\log(n),\ n^{1/3},\ 2^{\log(n)},\ \log(n!)$
• B$n^{1/3},\ \log(n),\ \log(n!),\ 2^{\log(n)}$
• C$\log(n),\ n^{1/3},\ \log(n!),\ 2^{\log(n)}$
• D$2^{\log(n)},\ n^{1/3},\ \log(n),\ \log(n!)$

One or more correct

• A$T(n) = T(n-1) + 1, \quad T(1) = 1$
• B$T(n) = 2T\left(\frac{n}{2}\right) + 1, \quad T(1) = 1$
• C$T(n) = 2T\left(\frac{n}{2}\right) + n, \quad T(1) = 1$
• D$T(n) = T(n-1) + n, \quad T(1) = 1$

Numerical answer

Single correct

• A$d_1(u, v) = d_2(u, v)$
• B$d_1(u, v) \le d_2(u, v)$
• C$d_1(u, v) \ge d_2(u, v)$
• D$d_1(u, v) \ne d_2(u, v)$

Single correct

• A$T(n) = \Theta(n^2 2^n)$
• B$T(n) = \Theta(n 2^n)$
• C$T(n) = \Theta((\log n)^2 2^n)$
• D$T(n) = \Theta(4^n)$

fun(int A[0,…,n-1]){
    for i=0 to n-2
        for j=0 to n-i-2
            if (A[j]>A[j+1])
                then swap A[j] and A[j+1]
}

Numerical answer

Single correct

• A1
• B$N - 1$
• C$N$
• D$2N - 1$

One or more correct

• AA DFS tree of $G$ is a Shortest Path tree of $G$.
• BEvery non-tree edge of $G$ with respect to a DFS tree is a forward/back edge.
• CIf $(u, v)$ is a non-tree edge of $G$ with respect to a BFS tree, then the distances from the source vertex $s$ to $u$ and $v$ in the BFS tree are within $\pm 1$ of each other.
• DBoth BFS and DFS can be used to find the connected components of $G$.

Single correct

• Aboth $O(N)$ and $\Omega(N)$
• B$O(N)$ but not $\Omega(N)$
• C$\Omega(N)$ but not $O(N)$
• Dneither $O(N)$ nor $\Omega(N)$

Single correct

• A$T(n) = \Theta(2^n)$
• B$T(n) = \Theta(n2^n)$
• C$T(n) = \Theta(3^n)$
• D$T(n) = \Theta(n3^n)$

Numerical answer

One or more correct

• A$f \in O(g)$
• B$f \in \Omega(g)$
• C$f \in o(g)$
• D$f \in \Theta(g)$

Answer key coming soon

Single correct

• A$f(n^2) = \theta(f(n)^2)$, when $f(n)$ is a polynomial
• B$f(n^2) = o(f(n)^2)$
• C$f(n^2) = O(f(n)^2)$, when $f(n)$ is an exponential function
• D$f(n^2) = \Omega(f(n)^2)$

Answer key coming soon

Single correct

• A$t > 2n - 2$
• B$t > 3\left\lceil \frac{n}{2} \right\rceil$ and $t \le 2n - 2$
• C$t > n$ and $t \le 3\left\lceil \frac{n}{2} \right\rceil$
• D$t > \lceil \log_2(n) \rceil$ and $t \le n$

Answer key coming soon

Single correct

• A$f_3, f_2, f_1$
• B$f_2, f_1, f_3$
• C$f_1, f_2, f_3$
• D$f_2, f_3, f_1$

Answer key coming soon

Single correct

• ASelection sort
• BMergesort
• CInsertion sort
• DQuicksort using the last element as pivot

Answer key coming soon

Numerical answer

Answer key coming soon

Single correct

• ABoth $S_1$ and $S_2$ are true.
• B$S_1$ is true and $S_2$ is false.
• C$S_1$ is false and $S_2$ is true.
• DBoth $S_1$ and $S_2$ are false.

Answer key coming soon

Single correct

• A$\Theta(\sqrt{n})$
• B$\Theta(\log_2(n))$
• C$\Theta(n^2)$
• D$\Theta(n)$

Answer key coming soon