If the goal is to find all points within a specified distance ? of point y, y = x, explain how you could use the triangle inequality and the al- ready calculated distances to x to potentially reduce the number of dis- tance calculations necessary? Hint: The triangle inequality, d(x, z) ? d(x, y) + d(y, x), can be rewritten as d(x, y) ? d(x, z) ? d(y, z).
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.)
Unfortunately, there is a typo and a lack of clarity in the hint. The
hint should be phrased as follows:
Hint: If z is an arbitrary point of S, then the triangle inequality,
d(x, y) ? d(x, z)+d(y, z), can be rewritten as d(y, z) ? d(x, y)?d(x, z).
Another application of the triangle inequality starting with d(x, z) ?
d(x, y) + d(y, z), shows that d(y, z) ? d(x, z) ? d(x, y). If the lower
bound of d(y, z) obtained from either of these inequalities is greater
than , then d(y, z) does not need to be calculated. Also, if the upper
bound of d(y, z) obtained from the inequality d(y, z) ? d(y, x)+d(x, z)
is less than or equal to , then d(x, z) does not need to be calculated.
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