kd tree data structure implementation (TheAlgorithms#11532)

* Implemented KD-Tree Data Structure

* Implemented KD-Tree Data Structure. updated DIRECTORY.md.

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* Create __init__.py

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* Replaced legacy `np.random.rand` call with `np.random.Generator` in kd_tree/example_usage.py

* Replaced legacy `np.random.rand` call with `np.random.Generator` in kd_tree/hypercube_points.py

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* docstring for search()

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* Added tests. Updated docstrings/typehints

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* updated tests and used | for type annotations

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* E501 for build_kdtree.py, hypercube_points.py, nearest_neighbour_search.py

* I001 for example_usage.py and test_kdtree.py

* I001 for example_usage.py and test_kdtree.py

* Update data_structures/kd_tree/build_kdtree.py

Co-authored-by: Christian Clauss <cclauss@me.com>

* Update data_structures/kd_tree/example/hypercube_points.py

Co-authored-by: Christian Clauss <cclauss@me.com>

* Update data_structures/kd_tree/example/hypercube_points.py

Co-authored-by: Christian Clauss <cclauss@me.com>

* Added new test cases requested in Review. Refactored the test_build_kdtree() to include various checks.

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* Considered ruff errors

* Considered ruff errors

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* Update kd_node.py

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---------

Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>
Co-authored-by: Christian Clauss <cclauss@me.com>

Author: 3 people

DIRECTORY.md

@@ -285,6 +285,12 @@
   * Trie
     * [Radix Tree](data_structures/trie/radix_tree.py)
     * [Trie](data_structures/trie/trie.py)
+  * KD Tree
+    * [KD Tree Node](data_structures/kd_tree/kd_node.py)
+    * [Build KD Tree](data_structures/kd_tree/build_kdtree.py)
+    * [Nearest Neighbour Search](data_structures/kd_tree/nearest_neighbour_search.py)
+    * [Hypercibe Points](data_structures/kd_tree/example/hypercube_points.py)
+    * [Example Usage](data_structures/kd_tree/example/example_usage.py)
 
 ## Digital Image Processing
   * [Change Brightness](digital_image_processing/change_brightness.py)

data_structures/kd_tree/__init__.py

@@ -0,0 +1,35 @@
+from data_structures.kd_tree.kd_node import KDNode
+
+
+def build_kdtree(points: list[list[float]], depth: int = 0) -> KDNode | None:
+    """
+    Builds a KD-Tree from a list of points.
+
+    Args:
+        points: The list of points to build the KD-Tree from.
+        depth: The current depth in the tree
+                     (used to determine axis for splitting).
+
+    Returns:
+        The root node of the KD-Tree,
+                       or None if no points are provided.
+    """
+    if not points:
+        return None
+
+    k = len(points[0])  # Dimensionality of the points
+    axis = depth % k
+
+    # Sort point list and choose median as pivot element
+    points.sort(key=lambda point: point[axis])
+    median_idx = len(points) // 2
+
+    # Create node and construct subtrees
+    left_points = points[:median_idx]
+    right_points = points[median_idx + 1 :]
+
+    return KDNode(
+        point=points[median_idx],
+        left=build_kdtree(left_points, depth + 1),
+        right=build_kdtree(right_points, depth + 1),
+    )

data_structures/kd_tree/build_kdtree.py

@@ -0,0 +1,38 @@
+import numpy as np
+
+from data_structures.kd_tree.build_kdtree import build_kdtree
+from data_structures.kd_tree.example.hypercube_points import hypercube_points
+from data_structures.kd_tree.nearest_neighbour_search import nearest_neighbour_search
+
+
+def main() -> None:
+    """
+    Demonstrates the use of KD-Tree by building it from random points
+    in a 10-dimensional hypercube and performing a nearest neighbor search.
+    """
+    num_points: int = 5000
+    cube_size: float = 10.0  # Size of the hypercube (edge length)
+    num_dimensions: int = 10
+
+    # Generate random points within the hypercube
+    points: np.ndarray = hypercube_points(num_points, cube_size, num_dimensions)
+    hypercube_kdtree = build_kdtree(points.tolist())
+
+    # Generate a random query point within the same space
+    rng = np.random.default_rng()
+    query_point: list[float] = rng.random(num_dimensions).tolist()
+
+    # Perform nearest neighbor search
+    nearest_point, nearest_dist, nodes_visited = nearest_neighbour_search(
+        hypercube_kdtree, query_point
+    )
+
+    # Print the results
+    print(f"Query point: {query_point}")
+    print(f"Nearest point: {nearest_point}")
+    print(f"Distance: {nearest_dist:.4f}")
+    print(f"Nodes visited: {nodes_visited}")
+
+
+if __name__ == "__main__":
+    main()

data_structures/kd_tree/example/__init__.py

@@ -0,0 +1,21 @@
+import numpy as np
+
+
+def hypercube_points(
+    num_points: int, hypercube_size: float, num_dimensions: int
+) -> np.ndarray:
+    """
+    Generates random points uniformly distributed within an n-dimensional hypercube.
+
+    Args:
+        num_points: Number of points to generate.
+        hypercube_size: Size of the hypercube.
+        num_dimensions: Number of dimensions of the hypercube.
+
+    Returns:
+        An array of shape (num_points, num_dimensions)
+                    with generated points.
+    """
+    rng = np.random.default_rng()
+    shape = (num_points, num_dimensions)
+    return hypercube_size * rng.random(shape)

data_structures/kd_tree/example/example_usage.py

@@ -0,0 +1,30 @@
+from __future__ import annotations
+
+
+class KDNode:
+    """
+    Represents a node in a KD-Tree.
+
+    Attributes:
+        point: The point stored in this node.
+        left: The left child node.
+        right: The right child node.
+    """
+
+    def __init__(
+        self,
+        point: list[float],
+        left: KDNode | None = None,
+        right: KDNode | None = None,
+    ) -> None:
+        """
+        Initializes a KDNode with the given point and child nodes.
+
+        Args:
+            point (list[float]): The point stored in this node.
+            left (Optional[KDNode]): The left child node.
+            right (Optional[KDNode]): The right child node.
+        """
+        self.point = point
+        self.left = left
+        self.right = right

data_structures/kd_tree/example/hypercube_points.py

@@ -0,0 +1,71 @@
+from data_structures.kd_tree.kd_node import KDNode
+
+
+def nearest_neighbour_search(
+    root: KDNode | None, query_point: list[float]
+) -> tuple[list[float] | None, float, int]:
+    """
+    Performs a nearest neighbor search in a KD-Tree for a given query point.
+
+    Args:
+        root (KDNode | None): The root node of the KD-Tree.
+        query_point (list[float]): The point for which the nearest neighbor
+                                    is being searched.
+
+    Returns:
+        tuple[list[float] | None, float, int]:
+            - The nearest point found in the KD-Tree to the query point,
+              or None if no point is found.
+            - The squared distance to the nearest point.
+            - The number of nodes visited during the search.
+    """
+    nearest_point: list[float] | None = None
+    nearest_dist: float = float("inf")
+    nodes_visited: int = 0
+
+    def search(node: KDNode | None, depth: int = 0) -> None:
+        """
+        Recursively searches for the nearest neighbor in the KD-Tree.
+
+        Args:
+            node: The current node in the KD-Tree.
+            depth: The current depth in the KD-Tree.
+        """
+        nonlocal nearest_point, nearest_dist, nodes_visited
+        if node is None:
+            return
+
+        nodes_visited += 1
+
+        # Calculate the current distance (squared distance)
+        current_point = node.point
+        current_dist = sum(
+            (query_coord - point_coord) ** 2
+            for query_coord, point_coord in zip(query_point, current_point)
+        )
+
+        # Update nearest point if the current node is closer
+        if nearest_point is None or current_dist < nearest_dist:
+            nearest_point = current_point
+            nearest_dist = current_dist
+
+        # Determine which subtree to search first (based on axis and query point)
+        k = len(query_point)  # Dimensionality of points
+        axis = depth % k
+
+        if query_point[axis] <= current_point[axis]:
+            nearer_subtree = node.left
+            further_subtree = node.right
+        else:
+            nearer_subtree = node.right
+            further_subtree = node.left
+
+        # Search the nearer subtree first
+        search(nearer_subtree, depth + 1)
+
+        # If the further subtree has a closer point
+        if (query_point[axis] - current_point[axis]) ** 2 < nearest_dist:
+            search(further_subtree, depth + 1)
+
+    search(root, 0)
+    return nearest_point, nearest_dist, nodes_visited

data_structures/kd_tree/kd_node.py

@@ -0,0 +1,100 @@
+import numpy as np
+import pytest
+
+from data_structures.kd_tree.build_kdtree import build_kdtree
+from data_structures.kd_tree.example.hypercube_points import hypercube_points
+from data_structures.kd_tree.kd_node import KDNode
+from data_structures.kd_tree.nearest_neighbour_search import nearest_neighbour_search
+
+
+@pytest.mark.parametrize(
+    ("num_points", "cube_size", "num_dimensions", "depth", "expected_result"),
+    [
+        (0, 10.0, 2, 0, None),  # Empty points list
+        (10, 10.0, 2, 2, KDNode),  # Depth = 2, 2D points
+        (10, 10.0, 3, -2, KDNode),  # Depth = -2, 3D points
+    ],
+)
+def test_build_kdtree(num_points, cube_size, num_dimensions, depth, expected_result):
+    """
+    Test that KD-Tree is built correctly.
+
+    Cases:
+        - Empty points list.
+        - Positive depth value.
+        - Negative depth value.
+    """
+    points = (
+        hypercube_points(num_points, cube_size, num_dimensions).tolist()
+        if num_points > 0
+        else []
+    )
+
+    kdtree = build_kdtree(points, depth=depth)
+
+    if expected_result is None:
+        # Empty points list case
+        assert kdtree is None, f"Expected None for empty points list, got {kdtree}"
+    else:
+        # Check if root node is not None
+        assert kdtree is not None, "Expected a KDNode, got None"
+
+        # Check if root has correct dimensions
+        assert (
+            len(kdtree.point) == num_dimensions
+        ), f"Expected point dimension {num_dimensions}, got {len(kdtree.point)}"
+
+        # Check that the tree is balanced to some extent (simplistic check)
+        assert isinstance(
+            kdtree, KDNode
+        ), f"Expected KDNode instance, got {type(kdtree)}"
+
+
+def test_nearest_neighbour_search():
+    """
+    Test the nearest neighbor search function.
+    """
+    num_points = 10
+    cube_size = 10.0
+    num_dimensions = 2
+    points = hypercube_points(num_points, cube_size, num_dimensions)
+    kdtree = build_kdtree(points.tolist())
+
+    rng = np.random.default_rng()
+    query_point = rng.random(num_dimensions).tolist()
+
+    nearest_point, nearest_dist, nodes_visited = nearest_neighbour_search(
+        kdtree, query_point
+    )
+
+    # Check that nearest point is not None
+    assert nearest_point is not None
+
+    # Check that distance is a non-negative number
+    assert nearest_dist >= 0
+
+    # Check that nodes visited is a non-negative integer
+    assert nodes_visited >= 0
+
+
+def test_edge_cases():
+    """
+    Test edge cases such as an empty KD-Tree.
+    """
+    empty_kdtree = build_kdtree([])
+    query_point = [0.0] * 2  # Using a default 2D query point
+
+    nearest_point, nearest_dist, nodes_visited = nearest_neighbour_search(
+        empty_kdtree, query_point
+    )
+
+    # With an empty KD-Tree, nearest_point should be None
+    assert nearest_point is None
+    assert nearest_dist == float("inf")
+    assert nodes_visited == 0
+
+
+if __name__ == "__main__":
+    import pytest
+
+    pytest.main()