blog/content/posts/2024-08-10-kd-tree/index.md

title	date	draft	description	tags	categories	series	favorite	disable_feed
k-d Tree	2024-08-10T11:50:33+01:00	false	Points in spaaaaace!	algorithms data structures python	programming	Cool algorithms	false	false

The k-d Tree is a useful way to map points in space and make them efficient to query.

I ran into them during my studies in graphics, as they are one of the possible acceleration structures for ray-casting operations.

Implementation

As usual, this will be in Python, though its lack of proper discriminated enums makes it more verbose than would otherwise be necessary.

Pre-requisites

Let's first define what kind of space our k-d Tree is dealing with. In this instance k = 3 just like in the normal world.

class Point(NamedTuple):
    x: float
    y: float
    z: float

class Axis(IntEnum):
    X = 0
    Y = 1
    Z = 2

    def next(self) -> Axis:
        # Each level of the tree is split along a different axis
        return Axis((self + 1) % 3)

Representation

The tree is represented by KdTree, each of its leaf nodes is a KdLeafNode and its inner nodes are KdSplitNodes.

For each point in space, the tree can also keep track of an associated value, similar to a dictionary or other mapping data structure. Hence we will make our KdTree generic to this mapped type T.

Leaf node

A leaf node contains a number of points that were added to the tree. For each point, we also track their mapped value, hence the dict[Point, T].

class KdLeafNode[T]:
    points: dict[Point, T]

    def __init__(self):
        self.points = {}

    # Convenience constructor used when splitting a node
    @classmethod
    def from_items(cls, items: Iterable[tuple[Point, T]]) -> KdLeafNode[T]:
        res = cls()
        res.points.update(items)
        return res

Split node

An inner node must partition the space into two sub-spaces along a given axis and mid-point (thus defining a plane). All points that are "to the left" of the plane will be kept in one child, while all the points "to the right" will be in the other. Similar to a Binary Search Tree's inner nodes.

class KdSplitNode[T]:
    axis: Axis
    mid: float
    children: tuple[KdTreeNode[T], KdTreeNode[T]]

    # Convenience function to index into the child which contains `point`
    def _index(self, point: Point) -> int:
        return 0 if point[self.axis] <= self.mid else 1

Tree

The tree itself is merely a wrapper around its inner nodes.

Once annoying issue about writing this in Python is the lack of proper discriminated enum types. So we need to create a wrapper type for the nodes (KdNode) to allow for splitting when updating the tree.

class KdNode[T]:
    # Wrapper around leaf/inner nodes, the poor man's discriminated enum
    inner: KdLeafNode[T] | KdSplitNode[T]

    def __init__(self):
        self.inner = KdLeafNode()

class KdTree[T]:
    _root: KdNode[T]

    def __init__(self):
        # Tree starts out empty
        self._root = KdNode()

Inserting a point

To add a point to the tree, we simply recurse from node to node, similar to a BST's insertion algorithm. Once we've found the correct leaf node to insert our point into, we simply do so.

If that leaf node goes over the maximum number of points it can store, we must then split it along an axis, cycling between X, Y, and Z at each level of the tree (i.e: splitting along the X axis on the first level, then Y on the second, then Z after that, and then X, etc...).

# How many points should be stored in a leaf node before being split
MAX_CAPACITY = 32

def median(values: Iterable[float]) -> float:
    sorted_values = sorted(values)
    mid_point = len(sorted_values) // 2
    if len(sorted_values) % 2 == 1:
        return sorted_values[mid_point]
    a, b = sorted_values[mid_point], sorted_values[mid_point + 1]
    return a + (b - a) / 2

def partition[T](
    pred: Callable[[T], bool],
    iterable: Iterable[T]
) -> tuple[list[T], list[T]]:
    truths, falses = [], []
    for v in iterable:
        (truths if pred(v) else falses).append(v)
    return truths, falses

def split_leaf[T](node: KdLeafNode[T], axis: Axis) -> KdSplitNode[T]:
    # Find the median value for the given axis
    mid = median(p[axis] for p in node.points)
    # Split into left/right children according to the mid-point and axis
    left, right = partition(lambda kv: kv[0][axis] <= mid, node.points.items())
    return KdSplitNode(
        split_axis,
        mid,
        (KdLeafNode.from_items(left), KdLeafNode.from_items(right)),
    )

class KdTree[T]:
    def insert(self, point: Point, val: T) -> bool:
        # Forward to the root node, choose `X` as the first split axis
        return self._root.insert(point, val, Axis.X)

class KdLeafNode[T]:
    def insert(self, point: Point, val: T, split_axis: Axis) -> bool:
        # Check whether we're overwriting a previous value
        was_mapped = point in self.points
        # Store the corresponding value
        self.points[point] = val
        # Return whether we've performed an overwrite
        return was_mapped

class KdSplitNode[T]:
    def insert(self, point: Point, val: T, split_axis: Axis) -> bool:
        # Find the child which contains the point
        child = self.children[self._index(point)]
        # Recurse into it, choosing the next split axis
        return child.insert(point, val, split_axis.next())

class KdNode[T]:
    def insert(self, point: Point, val: T, split_axis: Axis) -> bool:
        # Add the point to the wrapped node...
        res = self.inner.insert(point, val, split_axis)
        # ... And take care of splitting leaf nodes when necessary
        if (
            isinstance(self.inner, KdLeafNode)
            and len(self.inner.points) > MAX_CAPACITY
        ):
            self.inner = split_leaf(self.inner, split_axis)
        return res