Fast k-nearest neighbor searching algorithms including a kd-tree, cover-tree and the algorithm implemented in class package.

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`data` |
an input data matrix. |

`query` |
a query data matrix. |

`algorithm` |
nearest neighbor searching algorithm. |

`k` |
the maximum number of nearest neighbors to search. The default value is set to 10. |

The *cover tree* is O(n) space data structure which allows us to answer queries
in the same O(log(n)) time as *kd tree* given a fixed intrinsic dimensionality.
Templated code from http://hunch.net/~jl/projects/cover_tree/cover_tree.html is used.

The *kd tree* algorithm is implemented in the Approximate Near Neighbor (ANN) C++ library (see http://www.cs.umd.edu/~mount/ANN/).
The exact nearest neighbors are searched in this package.

The *CR* algorithm is the *VR* using distance *1-x'y* assuming `x`

and `y`

are unit vectors.
The *brute* algorithm searches linearly. It is a naive method.

a list contains:

`nn.index` |
an n x k matrix for the nearest neighbor indice. |

`nn.dist` |
an n x k matrix for the nearest neighbor Euclidean distances. |

Shengqiao Li. To report any bugs or suggestions please email: shli@stat.wvu.edu.

Bentley J.L. (1975), “Multidimensional binary search trees used for associative
search,” *Communication ACM*, **18**, 309-517.

Arya S. and Mount D.M. (1993),
“Approximate nearest neighbor searching,”
*Proc. 4th Ann. ACM-SIAM Symposium on Discrete Algorithms (SODA'93)*, 271-280.

Arya S., Mount D.M., Netanyahu N.S., Silverman R. and Wu A.Y. (1998),
“An optimal algorithm for approximate nearest neighbor searching,”
*Journal of the ACM*, **45**, 891-923.

Beygelzimer A., Kakade S. and Langford J. (2006),
“Cover trees for nearest neighbor,”
*ACM Proc. 23rd international conference on Machine learning*, **148**, 97-104.

`nn2`

in RANN, `ann`

in yaImpute and `knn`

in class.

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