Consider the following recurrence relations:

For all $n>1$,

$$ \begin{aligned} & T_1(n)=4 T_1\left(\frac{n}{2}\right)+T_2(n) \\ & T_2(n)=5 T_2\left(\frac{n}{4}\right)+\theta\left(\log _2 n\right) \end{aligned} $$

Assume that for all $n \leq 1, T_1(n)=1$ and $T_2(n)=1$. Which one of the following options is correct?

Let $G(V, E)$ be an undirected, edge-weighted graph with integer weights. The weight of a path is the sum of the weights of the edges in that path. The length of a path is the number of edges in that path.

Let $s \in V$ be a vertex in $G$. For every $u \in V$ and for every $k \geq 0$, let $d_k(u)$ denote the weight of a shortest path (in terms of weight) from $s$ to $u$ of length at most $k$. If there is no path from $s$ to $u$ of length at most $k$, then $d_k(u)=\infty$.

Consider the statements:

S1: For every $k \geq 0$ and $u \in V, d_{k+1}(u) \leq d_k(u)$.

S2: For every $(u, v) \in E$, if $(u, v)$ is part of a shortest path (in terms of weight) from $s$ to $v$, then for every $k \geq 0, d_k(u) \leq d_k(v)$.

Which one of the following options is correct?

Let $G(V, E)$ be a simple, undirected, edge-weighted graph with unique edge weights. Which of the following statements about the minimum spanning trees (MST) of $G$ is/are true?

Consider the following C statements:

char str1 = "Hello; / Statement S1 */

char str2 = "Hello;"; / Statement S2 */

int str3 = "Hello"; / Statement S3 */

Which of the following options is/are correct?