5. This problem from Tan-Steinbach-Kumar, Ch. 2. You are given a set of points S in Euclidean space, as well
as the distance of each point in S to a point $x$. (It doesn't matter if $x \in S$).
(a) If the goal is to find all points within a specified distance, $\epsilon$, of point $y$, $y \neq x$, explain how you could
use the triangle inequality and the already calculated distances to $x$ to potentially reduce the number of
distance calculations necessary? Hint The triangle inequality, $d(x, z) \leq d(x, y) + d(y, z)$ can be rewritten
as $d(x, y) \geq d(x, z) - d(y, z)$.
(b) In general, how would the distance between $x$ and $y$ affect the number of distance calculations?
(c) Suppose that you can find a small subset of points $S'$ from the original data set, such that every point in
the data set is within a specified distance $\epsilon$ of at least one point in $S'$, and that you also have the pairwise
distance matrix for $S'$. Describe a technique that uses this information to compute, with a minimum of
distance calculations, the set of all points within a distance of $\beta$ of a specified point from the data set.
