$x_1=(0,16),x_2=(0,9),x_3=(-4,0),x_4=(4,0).$

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 Kmeans? Explicitly write out the steps that the Kmeans algorithm will take to find the final cluster assignments. In this case, does Kmeans find the assignment of points to clusters that minimizes the following Equation?

${\displaystyle \underset{i=1}{\overset{m}{}}}d(x^{(i)},{}_{l_i}),$

where ${}_c$ 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

$d(x,y)=({\displaystyle \underset{i=1}{\overset{n}{}}}(x_i-y_i{)}^2{)}^{\frac{1}{2}}$