$(1)(12$pt$)$ Consider the following set of one$-$dimensional points:

$\backslash [$

$\backslash \{0\mathrm{.}1,0\mathrm{.}2,0\mathrm{.}45,0\mathrm{.}55,0\mathrm{.}8,0\mathrm{.}9\backslash \}.$

$\backslash ]$

All the points are located in the range between $[0,1].$ Suppose we apply k$-$means clustering to obtain three clusters, $\backslash (\backslash$mathrm$\{$A$\},\backslash$mathrm$\{$B$\}\backslash ),$ and C $.$ If the initial centroids are located at $\backslash (\backslash \{0\backslash ),\backslash (0\mathrm{.}4,1\backslash \}\backslash ),$ respectively, please find:

$($a$)(8$ pt $)$ the cluster assignments $($fill in either $\backslash (\backslash$mathrm$\{$A$\},\backslash$mathrm$\{$B$\}\backslash ),$ or C for each data point$)$;

$($b$)(4$ pt$\backslash ()\backslash )$ locations of the centroids $($coordinate$)$ after the first three iterations by filling out the following table.

$\backslash$begin$\{$tabular$\}\{|$c$|$c$|$c$|$c$|$c$|$c$|$c$|$c$|$c$|$c$|\}$

$\backslash$hline $\backslash$multirow$\{2\}\{*\}\{$ Iter $\}$ & $\backslash$multicolumn$\{4\}\{|$c$|\}\{$ Cluster assignment of data points $\}$ & $\backslash$multicolumn$\{3\}\{$c$|\}\{$ Centroid Locations $\}\backslash \backslash$

$\backslash$cline $\{2-10\}$ & $0\mathrm{.}10$ & $0\mathrm{.}20$ & $0\mathrm{.}45$ & $0\mathrm{.}55$ & $0\mathrm{.}80$ & $0\mathrm{.}90$ & A & B & C $\backslash \backslash$

$\backslash$hline $0$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ & $0\mathrm{.}00$ & $0\mathrm{.}40$ & $1\mathrm{.}00\backslash \backslash$

$\backslash$hline $1$ & & & & & & & & & $\backslash \backslash$

$\backslash$hline $2$ & & & & & & & & & $\backslash \backslash$

$\backslash$hline $3$ & & & & & & & & & $\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$tabular$\}(2)\backslash ((6\backslash$mathrm$\{$pt$\})\backslash )$ The following table shows the clustering results in a land cover classification dataset that consists of many pieces of land. The number provided in the table is the number of objects $($pieces of land$)$ that are clustered into each cluster that belongs to each category. For example, the number in the forest column and cluster $1$ row means that $10$ forest items are clustered into cluster $1.$

Table: Clustering results for land cover classification dataset.

Which cluster has the smallest entropy? Which cluster has the largest entropy?