Closest pair of points problem

From Wikipedia, the free encyclopedia
Closest pair of points shown in red

The closest pair of points problem or closest pair problem is a problem of computational geometry: given n points in metric space, find a pair of points with the smallest distance between them. The closest pair problem for points in the Euclidean plane[1] was among the first geometric problems that were treated at the origins of the systematic study of the computational complexity of geometric algorithms.

A naive algorithm of finding distances between all pairs of points in a space of dimension d and selecting the minimum requires O(n2) time. It turns out that the problem may be solved in O(n log n) time in a Euclidean space or Lp space of fixed dimension d.[2] In the algebraic decision tree model of computation, the O(n log n) algorithm is optimal, by a reduction from the element uniqueness problem. In the computational model that assumes that the floor function is computable in constant time the problem can be solved in O(n log log n) time.[3] If we allow randomization to be used together with the floor function, the problem can be solved in O(n) time.[4][5]

Dynamic closest-pair problem[]

The dynamic version for the closest-pair problem is stated as follows:

  • Given a dynamic set of objects, find algorithms and data structures for efficient recalculation of the closest pair of objects each time the objects are inserted or deleted.

If the bounding box for all points is known in advance and the constant-time floor function is available, then the expected O(n) space data structure was suggested that supports expected-time O(log n) insertions and deletions and constant query time. When modified for the algebraic decision tree model, insertions and deletions would require O(log2 n) expected time.[6] It is worth noting, though, that the complexity of the dynamic closest pair algorithm cited above is exponential in the dimension d, and therefore such an algorithm becomes less suitable for high-dimensional problems.

An algorithm for the dynamic closest-pair problem in d dimensional space was developed by Sergey Bespamyatnikh in 1998.[7] Points can be inserted and deleted in O(log n) time per point (in the worst case).

See also[]

Notes[]

  1. ^ M. I. Shamos and . "Closest-point problems." In Proc. 16th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 151—162, 1975 (DOI 10.1109/SFCS.1975.8)
  2. ^ K. L. Clarkson, "Fast algorithms for the all nearest neighbors problem", FOCS 1983.
  3. ^ S. Fortune and J.E. Hopcroft. "A note on Rabin's nearest-neighbor algorithm." Information Processing Letters, 8(1), pp. 20—23, 1979
  4. ^ S. Khuller and Y. Matias. A simple randomized sieve algorithm for the closest-pair problem. Inf. Comput., 118(1):34—37,1995
  5. ^ Richard Lipton (24 September 2011). "Rabin Flips a Coin".
  6. ^ Mordecai Golin, Rajeev Raman, Christian Schwarz, Michiel Smid, "Randomized Data Structures For The Dynamic Closest-Pair Problem", SIAM J. Comput., vo. 27, no. 4, 1998, preliminary version reported at the 4th Annu. ACM-SIAM Symp. on Discrete Algorithms, pp. 301–310 (1993)
  7. ^ Sergey Bespamyatnikh, An Optimal Algorithm for Closest-Pair Maintenance. Discrete Comput. Geom., 19:175-195, 1998.

References[]

Retrieved from ""