Q$1.(20$ pts$)$ Consider the points:

x$\_(1)=(0,16),$x$\_(2)=(0,9),$x$\_(3)=(-4,0),$x$\_(4)=(4,0)$

Suppose we wish to separate these points into two clusters, and we initialize the K$-$means algorithm by

randomly assigning points to clusters. This random initialization assigns x$\_(1)$ to cluster $1$ and all other

points to cluster $2.$

With these points and these initial assignments of points to clusters, what will be the final assignment of

points to clusters computed by K$-$means? Explicitly write out the steps the K$-$means algorithm will take to

find the final cluster assignments. In this case, does K$-$means find the assignment of points to clusters that

minimize the following Equation?

$\backslash$sum$\_($i$=1)^$m d$($x$^(($i$)),\backslash$mu $\_($l$\_($i$)))$

where $\backslash$mu $\_($c$)$ is the centroid of all points x$^(($i$))$ with label l$\_($i$)=$c and d$($x$,$y$)$ is the distance between points x

and y$.$ Usually, d is the Euclidean distance, given by

d$($x$,$y$)=(\backslash$sum$\_($i$=1)^$n $($x$\_($i$)-$y$\_($i$))^(2))^((1)/(2))$