Proximity problems

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Proximity problems is a class of problems in computational geometry which involve estimation of distances between geometric objects.

A subset of these problems stated in terms of points only are sometimes referred to as closest point problems,[1] although the term "closest point problem" is also used synonymously to the nearest neighbor search.

A common trait for many of these problems is the possibility to establish the Θ(n log n) lower bound on their computational complexity by reduction from the element uniqueness problem basing on an observation that if there is an efficient algorithm to compute some kind of minimal distance for a set of objects, it is trivial to check whether this distance equals to 0.

Atomic problems[edit]

While these problems pose no computational complexity challenge, some of them are notable because of their ubiquity in computer applications of geometry.

Problems on points[edit]



  1. ^ J. R. Sack and J. Urrutia (eds.) (2000). Handbook of Computational Geometry. North Holland. ISBN 0-444-82537-1. 
  2. ^ V. J. Lumelsky (1985). "On fast computation of distance between line segments". Inform. Process. Lett. 21 (2): 55–61. doi:10.1016/0020-0190(85)90032-8.