The.Algorithm.Design.Manual.Springer-Verlag.1998

Nearest Neighbor Search
Implementations: Ranger is a tool for visualizing and experimenting with nearest-neighbor and
orthogonal-range queries in high-dimensional data sets, using multidimensional search trees. Four
different search data structures are supported by Ranger: naive kd-trees, median kd-trees, nonorthogonal
kd-trees, and the vantage point tree.
For each of these, Ranger supports queries in up to 25 dimensions under any Minkowski metric. It
includes generators for a variety of point distributions in arbitrary dimensions. Finally, Ranger provides a
number of features to aid in visualizing multidimensional data, best illustrated by the accompanying
video [MS93]. To identify the most appropriate projection at a glance, Ranger provides a matrix of
all two-dimensional projections of the data set. Ranger is written in C, using Motif. It runs on Silicon
Graphics and HP workstations and is available in the algorithm repository http://www.cs.sunysb.edu/
algorith.
See Section for a complete collection of Voronoi diagram implementations. In particular, LEDA (see
Section ) provides an implementation of 2D Voronoi diagrams in C++, as well as planar point
location to make effective use of them for nearest-neighbor search.
A Pascal implementation of the divide-and-conquer algorithm for finding the closest pair of points in a
set of n points appears in [MS91]. See Section .
Notes: The best reference on kd-trees and other spatial data structures is two volumes by Samet
[Sam90b, Sam90a], where all major variants are developed in substantial detail. Good expositions on
finding the closest pair of points in the plane [BS76] include [CLR90, Man89]. These algorithms use a
divide-and-conquer approach instead of just selecting from the Delaunay triangulation.
A recent development in higher-dimensional nearest-neighbor search is algorithms that quickly produce a
point that if not the nearest neighbor lies provably close to the query point. A sparse weighted graph
structure is built from the data set, and the nearest neighbor is found by starting at a random point and
greedily walking in the graph towards the query point. The closest point found over several random trials
is declared the winner. Similar data structures hold promise for other problems in high-dimensional
spaces. See [AM93, AMN 94].
Related Problems: Kd-trees (see page ), Voronoi diagrams (see page ), range search (see page
).
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