All Algorithms questions

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$\Theta(m + n)$
• B$\Theta(m + n\log(n))$
• C$\Theta(nm)$
• D$\Theta(n^3)$

Single correct

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

Algorithm B1:  For i = 1, 2, ..., n
                 For j = 1, 2, ..., n
                   T[i][j] = 2T[i-1][j] + 3T[i][j-1]

Algorithm B2:  For s = 2, 3, ..., 2n
                 For i = 1, 2, ..., n
                   For j = 1, 2, ..., n
                     If (i + j == s)
                       T[i][j] = 2T[i-1][j] + 3T[i][j-1]

Single correct

• ABoth algorithms $B_1$ and $B_2$ are correct
• BAlgorithm $B_1$ is correct, but algorithm $B_2$ is incorrect
• CAlgorithm $B_2$ is correct, but algorithm $B_1$ is incorrect
• DBoth algorithms $B_1$ and $B_2$ are incorrect

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

• AEvery MST remains an MST, and every SP remains an SP.
• BMSTs need not remain MSTs, and every SP remains an SP.
• CEvery MST remains an MST, and SPs need not remain SPs.
• DMSTs need not remain MSTs, and SPs need not remain SPs.

Single correct

• A$\Theta(n)$
• B$\Theta(1)$
• C$\Theta(\log^2 n)$
• D$\Theta(\log n)$

someAlgo(G)
  1. Let v be any vertex in G. Run BFS on G starting at v.
     Let u be a vertex in G at maximum distance from v as given by the BFS.
  2. Run BFS on G again with u as the starting vertex.
     Let z be the vertex at maximum distance from u as given by the BFS.
  3. Output the distance between u and z in G.

Numerical answer

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

• AAll edges in $G$ have even weight
• BAll edges in $G$ have even weight OR all edges in $G$ have odd weight
• CIn each cycle $C$ in $G$, all edges in $C$ have even weight
• DIn each cycle $C$ in $G$, either all edges in $C$ have even weight OR all edges in $C$ have odd weight

Numerical answer

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

Single correct

• A21
• B23
• C25
• D30

Answer key coming soon

Single correct

• AIf $f(n)$ is $n\log_2(n)$, then $T(n)$ is $\Theta(n\log_2(n))$.
• BIf $f(n)$ is $\dfrac{n}{\log_2(n)}$, then $T(n)$ is $\Theta(\log_2(n))$.
• CIf $f(n)$ is $O(n^{\log_b(a) - \epsilon})$ for some $\epsilon > 0$, then $T(n)$ is $\Theta(n^{\log_b(a)})$.
• DIf $f(n)$ is $\Theta(n^{\log_b(a)})$, then $T(n)$ is $\Theta(n^{\log_b(a)})$.

Answer key coming soon

One or more correct

• AThe graph does not have a topological order.
• BA depth-first traversal starting at vertex S classifies three directed edges as back edges.
• CThe graph does not have a strongly connected component.
• DFor each pair of vertices $u$ and $v$, there is a directed path from $u$ to $v$.

Answer key coming soon