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 ? of at least one of the points 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 ? of a specified point from the data set.
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 does not matter if x ? S.)
Let x and y be the two points and let x? and y? be the points in S
that are closest to the two points, respectively. If d(x?, y?)+2 ? ?,
then we can safely conclude d(x, y) ? ?. Likewise, if d(x?, y?)?2 ? ?,
then we can safely conclude d(x, y) ? ?. These formulas are derived
by considering the cases where x and y are as far from x? and y? as
possible and as far or close to each other as possible.
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