For the following sets of two-dimensional points, (1) provide a sketch of how they would be split into clusters by K-means for the given number of clusters and (2) indicate approximately where the resulting centroids would be. Assume that we are using the squared error objective function. If you think that there is more than one possible solution, then please indicate whether each solution is a global or local minimum. Note that the label of each diagram in Figure 8.4 matches the corresponding part of this question, e.g., Figure 8.4(a) goes with part (a).


(a) K = 2. Assuming that the points are uniformly distributed in the circle, how many possible ways are there (in theory) to partition the points into two clusters? What can you say about the positions of the two centroids? (Again, you don't need to provide exact centroid locations, just a qualitative description.)
(b) K = 3. The distance between the edges of the circles is slightly greater than the radii of the circles.
(c) K = 3. The distance between the edges of the circles is much less than the radii of the circles.
(d) K = 2.
(e) K = 3. Hint: Use the symmetry of the situation and remember that we are looking for a rough sketch of what the result would be.