$4.$ Use the similarity matrix in the table below to perform single and complete link hierarchical clustering. Show your results by drawing a dendrogram. The dendrogram should clearly show the order in which the points are merged.

$5.$ Hierarchical clustering is sometimes used to generate $\backslash ($ K $\backslash )$ clusters, $\backslash ($ K$>1\backslash )$ by taking the clusters at the $\backslash ($ K$^\{\backslash$text $\{$th $\}\}\backslash )$ level of the dendrogram. $($Root is at level $1.)$ By looking at the clusters produced in this way, we can evaluate the behavior of hierarchical clustering on different types of data and clusters, and also compare hierarchical approaches to K$-$means.

The following is a set of one$-$dimensional points: $\backslash (\backslash \{6,12,18,24,30,42,48\backslash \}\backslash ).$

$($a$)$ For each of the following sets of initial centroids, create two clusters by assigning each point to the nearest centroid, and then calculate the total squared error for each set of two clusters. Show both the clusters and the total squared error for each set of centroids.

i$.\backslash (\backslash \{18,45\backslash \}\backslash )$

ii$.\backslash (\backslash$quad$\backslash \{15,40\backslash \}\backslash )$

$($b$)$ Do both sets of centroids represent stable solutions; i$.$e$.$ if the K$-$means algorithm was run on this set of points using the given centroids as the starting centroids, would there be any change in the clusters generated?

$($c$)$ What are the two clusters produced by single link?

$($d$)$ Which technique, K$-$means or single link, seems to produce the "most natural" clustering in this situation? $($For K$-$means, take the clustering with the lowest squared error.$)($e$)$ What definition$($s$)$ of clustering does this natural clustering correspond to$?($Wellseparated$,$ center$-$based, contiguous, or density.$)$

$($f$)$ What well$-$known characteristic of the K$-$means algorithm explains the previous behavior?