Consider a directed graph $G=(V, E)$, where $V=\{0,1,2, \ldots, 100\}$ and $E=\{(i$, $j): 0 < j-i \leq 2$, for all $i, j \in V\}$. Suppose the adjacency list of each vertex is in decreasing order of vertex number, and depth-first search (DFS) is performed at vertex 0 . The number of vertices that will be discovered after vertex 50 is___________

Consider designing a linear classifier

$$ y=\operatorname{sign}(f(x ; w ; b)), f(x ; w, b)=w^{\mathrm{T}} x+b $$

on a dataset

$$ \begin{aligned} & D=\left\{\left(x_1, y_1\right),\left(x_2, y_2\right) \ldots \ldots\left(x_N, y_N\right)\right\} \\ & x_i \in \mathbb{R}^d, y_i \in\{+1,-1\}, i=1,2, \ldots \ldots, N \end{aligned} $$

Recall that the sign function outputs +1 if the argument is positive, and -1 if the argument is non-positive. The parameters $w$ and $b$ are updated as per the following training algorithm:

$$ w_{\text {new }}=w_{\text {old }}+y_n x_n, b_{\text {new }}=b_{\text {old }}+y_n $$

Whenever sign $\left(f\left(x_n ; w_{\text {old }}, b_{\text {old }}\right)\right) \neq y_n$ In other words, whenever the classifier wrongly predicts a sample $\left(x_n, y_n\right)$ from the dataset, $w_{\text {old }}$ gets updated to $w_{\text {new }}$, and likewise $b_{\text {old }}$ gets updated to $b_{\text {new }}$. Consider the case

$$ \left(x_n,+1\right), f\left(x_n ; w_{\text {old }}, b_{\text {old }}\right)<0 \text {. Then } $$

Let $C_1$ and $C_2$ be two sets of objects. Let $D(x, y)$ be a measure of dissimilarity between two objects $x$ and $y$. Consider the following definitions of dissimilarity between $C_1$ and $C_2$.

DIS-1 $\left(C_1, C_2\right)=\max _{x \in C_1, y \in C_2} D(x, y)$ DIS-2 $\left(C_1, C_2\right)=\min _{x \in C_1, y \in C_2} D(x, y)$

Which of the following statements Which of the following statements is/are correct?

Given data $\{(-1,1),(2,-5),(3,5)\}$ of the form $(x, y)$, we fit a model $y=w x$ using linear least-squares regression. The optimal value of $w$ is _________