$1.$ Consider the following clustering method called Leader Clustering. It receives two parameters: an integer k and a real number t$.$ Similar to k$-$means, it starts by selecting k instances

$($which will be called leaders$)$ and assigns each training instances to the cluster of the closest leader. During the assignment step, however, if the distance of a training instance to its closest leader is greater than the input threshold t$,$ then this training instance becomes a

new leader. During the same assignment step, remaining points can be assigned to these new leaders. After all the training instances have been assigned to a leader$$s cluster, new leaders are calculated by averaging each cluster. The process is then repeated until the cluster assignments do not change. $[$Use diagram to illustrate following answers$]$

a$.$ Given a dataset and a value k$,$ let t vary from $0$ to a very large value. When does Leader Clustering produce more, the same number, or fewer clusters than k$-$means, assuming that the k initial centers are the same for both? When will the clusterings produced by Leader Clustering and k$-$means be identical?

b$.$ Which of the two methods, k$-$means or Leader Clustering, will be best at dealing with outliers $($data instances that are $$far away$$ or very different to the other instances in the dataset$)?$ Explain.