Algorithm::KDimensionalTree

Raku package with implementations of the K-Dimensional Tree (K-D Tree) algorithm.

Remark: This package should not be confused with "Algorithm::KdTree", [ITp1], which provides Raku bindings to a C-implementation of the K-D Tree algorithm. (A primary motivation for making this package, "Algorithm::KDimensionalTree", was to have a pure-Raku implementation.)

Remark: The versions "ver<0.1.0+>" with the API "api<1>" of this package are better utilized via the package "Math::Nearest".

Compared to the simple scanning algorithm this implementation of the K-D Tree algorithm is often 2÷15 times faster for randomly generated low dimensional data. (Say. less than 6D.) It can be 200+ times faster for "real life" data, like, Geo-locations of USA cities.

The implementation is tested for correctness against Mathematica's Nearest. (See the resource files.)

Features

• Finds both top-k Nearest Neighbors (NNs).
  • See the method k-nearest.
• Finds NNs within a ball with a given radius.
  • See the method nearest-within-ball.
• Can return indexes and distances for the found NNs.
  • The result shape is controlled with the adverb :values.
• Works with arrays of numbers, arrays of arrays of numbers, and arrays of pairs.
• Utilizes distance functions from "Math::DistanceFunctions".
  • Which can be specified by their string names, like, "bray-curtis" or "cosine-distance".
• Allows custom distance functions to be used.

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Installation

From Zef ecosystem:

zef install Algorithm::KDimensionalTree

From GitHub:

zef install https://github.com/antononcube/Raku-Algorithm-KDimensionalTree.git

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Usage examples

Setup

use Algorithm::KDimensionalTree;
use Data::TypeSystem;
use Text::Plot;

# (Any)

Set of points

Make a random set of points:

my @points = ([(^100).rand, (^100).rand] xx 30).unique;
deduce-type(@points);

# Vector(Vector(Atom((Numeric)), 2), 30)

Create the K-dimensional tree object

my $kdTree = Algorithm::KDimensionalTree.new(@points);

say $kdTree;

# Algorithm::KDimensionalTree(points => 30, distance-function => &euclidean-distance)

Nearest k-neighbors

Use as a search point one from the points set:

my @searchPoint = |@points.head;

# [89.27640360563157 68.36845738910054]

Find 6 nearest neighbors:

my @res = $kdTree.k-nearest(@searchPoint, 6);
.say for @res;

# [89.27640360563157 68.36845738910054]
# [91.96352893754062 73.56174634865347]
# [80.0160216888336 69.80400458389097]
# [99.2564424012451 76.5797786385005]
# [79.5081497932748 52.564329376808196]
# [76.82139591710131 47.8616016873568]

Plot

Plot the points, the found nearest neighbors, and the search point:

my @point-char =  <* ⏺ ▲>;
say <data nns search> Z=> @point-char;
say text-list-plot(
[@points, @res, [@searchPoint,]],
:@point-char,
x-limit => (0, 100),
y-limit => (0, 100),
width => 60,
height => 20);

# (data => * nns => ⏺ search => ▲)
# ++----------+-----------+----------+-----------+----------++       
# +                                                          + 100.00
# |             *                                            |       
# |           *                          *                   |       
# +                                                          +  80.00
# |                                                    ⏺    ⏺|       
# |               *                    *         ⏺    ▲      |       
# |                     *           *                        |       
# +                                                          +  60.00
# |      *      *                   *           ⏺            |       
# |                                            ⏺            *|       
# +    *                                 **       *          +  40.00
# |                                                          |       
# |                            *       *           *         |       
# |                   *                                      |       
# +              *                                           +  20.00
# |               *                                          |       
# | *                              *           *             |       
# +                                                          +   0.00
# ++----------+-----------+----------+-----------+----------++       
#  0.00       20.00       40.00      60.00       80.00      100.00

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TODO

• TODO Implementation
  • DONE Using distance functions from an "universal" package
    • E.g. "Math::DistanceFunctions"
  • DONE Using distance functions other than Euclidean distance
  • DONE Returning properties
    • DONE Points
    • DONE Indexes
    • DONE Distances
    • DONE Labels
    • DONE Combinations of those
    • This is implemented by should be removed.
      • There is another package -- "Math::Nearest" -- that handles all nearest neighbors finders.
      • Version "0.1.0 with "api<1>" is without the .nearest method.
  • DONE Having an umbrella function nearest
    • Instead of creating a KDTree object etc.
    • This might require making a functor nearest-function
    • This is better done in a different package
      • See "Math::Nearest"
  • TODO Make the nearest methods work with strings
    • For example, using Hamming distance over a collection of words.
    • Requires using the distance function as a comparator for the splitting hyperplanes.
    • This means, any objects can be used as long as they provide a distance function.
• DONE Extensive correctness tests
  • Derived with Mathematica / WL (see the resources)
• TODO Documentation
  • DONE Basic usage examples with text plots
  • TODO More extensive documentation with a Jupyter notebook
    • Using "JavaScript::D3".
  • TODO Corresponding blog post
  • MAYBE Corresponding video

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References

[AAp1] Anton Antonov, Math::DistanceFunctions Raku package, (2024), GitHub/antononcube.

[AAp2] Anton Antonov, Math::Nearest Raku package, (2024), GitHub/antononcube.

[AAp3] Anton Antonov, Data::TypeSystem Raku package, (2023), GitHub/antononcube.

[AAp4] Anton Antonov, Text::Plot Raku package, (2022), GitHub/antononcube.

[ITp1] Itsuki Toyota, Algorithm::KdTree Raku package, (2016-2024), GitHub/titsuki.