  The nearest-neighbor rule is a well-known classification technique that,
given a training set P of labeled points, classifies any unlabeled query point
with the label of its closest point in P. The nearest-neighbor condensation
problem aims to reduce the training set without harming the accuracy of the
nearest-neighbor rule.
  FCNN is the most popular algorithm for condensation. It is heuristic in
nature, and theoretical results for it are scarce. In this paper, we settle the
question of whether reasonable upper-bounds can be proven for the size of the
subset selected by FCNN. First, we show that the algorithm can behave poorly
when points are too close to each other, forcing it to select many more points
than necessary. We then successfully modify the algorithm to avoid such cases,
thus imposing that selected points should "keep some distance". This
modification is sufficient to prove useful upper-bounds, along with
approximation guarantees for the algorithm.
