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+---
+title: "k-d Tree"
+date: 2024-08-10T11:50:33+01:00
+draft: false # I don't care for draft mode, git has branches for that
+description: "Points in spaaaaace!"
+tags:
+  - algorithms
+  - data structures
+  - python
+categories:
+  - programming
+series:
+  - Cool algorithms
+favorite: false
+disable_feed: false
+---
+
+The [_k-d Tree_][wiki] 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.
+
+[wiki]: https://en.wikipedia.org/wiki/K-d_tree
+[ray-casting]: https://en.wikipedia.org/wiki/Ray_casting
+
+<!--more-->
+
+## 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.
+
+```python
+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 `KdSplitNode`s.
+
+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]`.
+
+```python
+class KdLeafNode[T]:
+    points: dict[Point, T]
+
+    def __init__(self):
+        self.points = {}
+```
+
+#### 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_][bst]'s inner nodes.
+
+[bst]: https://en.wikipedia.org/wiki/Binary_search_tree
+
+```python
+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.
+
+```python
+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()
+
+    # Convenience constructor used when splitting a node
+    @classmethod
+    def from_items(cls, items: Iterable[tuple[Point, T]]) -> KdNode[T]:
+        res = cls()
+        res.inner.points.update(items)
+        return res
+
+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...).
+
+```python
+# 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,
+        (KdNode.from_items(left), KdNode.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
+```
+
+### Searching for a point
+
+Looking for a given point in the tree look very similar to a _BST_'s search,
+each leaf node dividing the space into two sub-spaces, only one of which
+contains the point.
+
+```python
+class KdTree[T]:
+    def lookup(self, point: Point) -> T | None:
+        # Forward to the root node
+        return self._root.lookup(point)
+
+class KdNode[T]:
+    def lookup(self, point: Point) -> T | None:
+        # Forward to the wrapped node
+        return self.inner.lookup(point)
+
+class KdLeafNode[T]:
+    def lookup(self, point: Point) -> T | None:
+        # Simply check whether we've stored the point in this leaf
+        return self.points.get(point)
+
+class KdSplitNode[T]:
+    def lookup(self, point: Point) -> T | None:
+        # Recurse into the child which contains the point
+        return self.children[self._index(point)].lookup(point)
+```
+
+### Closest points
+
+Now to look at the most interesting operation one can do on a _k-d Tree_:
+querying for the objects which are closest to a given point (i.e: the [Nearest
+neighbour search][nns].
+
+This is a more complicated algorithm, which will also need some modifications to
+current _k-d Tree_ implementation in order to track just a bit more information
+about the points it contains.
+
+[nns]: https://en.wikipedia.org/wiki/Nearest_neighbor_search
+
+#### A notion of distance
+
+To search for the closest points to a given origin, we first need to define
+which [distance](https://en.wikipedia.org/wiki/Distance) we are using in our
+space.
+
+For this example, we'll simply be using the usual definition of [(Euclidean)
+distance][euclidean-distance].
+
+[euclidean-distance]: https://en.wikipedia.org/wiki/Euclidean_distance
+
+```python
+def dist(point: Point, other: Point) -> float:
+    return sqrt(sum((a - b) ** 2 for a, b in zip(self, other)))
+```
+
+#### Tracking the tree's boundaries
+
+To make the query efficient, we'll need to track the tree's boundaries: the
+bounding box of all points contained therein. This will allow us to stop the
+search early once we've found enough points and can be sure that the rest of the
+tree is too far away to qualify.
+
+For this, let's define the `AABB` (Axis-Aligned Bounding Box) class.
+
+```python
+class Point(NamedTuple):
+    # Convenience function to replace the coordinate along a given dimension
+    def replace(self, axis: Axis, new_coord: float) -> Point:
+        coords = list(self)
+        coords[axis] = new_coord
+        return Point(coords)
+
+class AABB(NamedTuple):
+    # Lowest coordinates in the box
+    low: Point
+    # Highest coordinates in the box
+    high: Point
+
+    # An empty box
+    @classmethod
+    def empty(cls) -> AABB:
+        return cls(
+            Point(*(float("inf"),) * 3),
+            Point(*(float("-inf"),) * 3),
+        )
+
+    # Split the box into two along a given axis for a given mid-point
+    def split(axis: Axis, mid: float) -> tuple[AABB, AABB]:
+        assert self.low[axis] <= mid <= self.high[axis]
+        return (
+            AABB(self.low, self.high.replace(axis, mid)),
+            AABB(self.low.replace(axis, mid), self.high),
+        )
+
+    # Extend a box to contain a given point
+    def extend(self, point: Point) -> None:
+        low = NamedTuple(*(map(min, zip(self.low, point))))
+        high = NamedTuple(*(map(max, zip(self.high, point))))
+        return AABB(low, high)
+
+    # Return the shortest between a given point and the box
+    def dist_to_point(self, point: Point) -> float:
+        deltas = (
+            max(self.low[axis] - point[axis], 0, point[axis] - self.high[axis])
+            for axis in Axis
+        )
+        return dist(Point(0, 0, 0), Point(*deltas))
+```
+
+And do the necessary modifications to the `KdTree` to store the bounding box and
+update it as we add new points.
+
+```python
+class KdTree[T]:
+    _root: KdNode[T]
+    # New field: to keep track of the tree's boundaries
+    _aabb: AABB
+
+    def __init__(self):
+        self._root = KdNode()
+        # Initialize the empty tree with an empty bounding box
+        self._aabb = AABB.empty()
+
+    def insert(self, point: Point, val: T) -> bool:
+        # Extend the AABB for our k-d Tree when adding a point to it
+        self._aabb = self._aabb.extend(point)
+        return self._root.insert(point, val, Axis.X)
+```
+
+#### `MaxHeap`
+
+Python's builtin [`heapq`][heapq] module provides the necessary functions to
+create and interact with a [_Priority Queue_][priority-queue], in the form of a
+[_Binary Heap_][binary-heap].
+
+Unfortunately, Python's library maintains a _min-heap_, which keeps the minimum
+element at the root. For this algorithm, we're interested in having a
+_max-heap_, with the maximum at the root.
+
+Thankfully, one can just reverse the comparison function for each element to
+convert between the two. Let's write a `MaxHeap` class making use of this
+library, with a `Reverse` wrapper class to reverse the order of elements
+contained within it (similar to [Rust's `Reverse`][reverse]).
+
+[binary-heap]: https://en.wikipedia.org/wiki/Binary_heap
+[heapq]: https://docs.python.org/3/library/heapq.html
+[priority-queue]: https://en.wikipedia.org/wiki/Priority_queue
+[reverse]: https://doc.rust-lang.org/std/cmp/struct.Reverse.html
+
+```python
+# Reverses the wrapped value's ordering
+@functools.total_ordering
+class Reverse[T]:
+    value: T
+
+    def __init__(self, value: T):
+        self.value = value
+
+    def __lt__(self, other: Reverse[T]) -> bool:
+        return self.value > other.value
+
+    def __eq__(self, other: Reverse[T]) -> bool:
+        return self.value == other.value
+
+class MaxHeap[T]:
+    _heap: list[Reverse[T]]
+
+    def __init__(self):
+        self._heap = []
+
+    def __len__(self) -> int:
+        return len(self._heap)
+
+    def __iter__(self) -> Iterator[T]:
+        yield from (item.value for item in self._heap)
+
+    # Push a value on the heap
+    def push(self, value: T) -> None:
+        heapq.heappush(self._heap, Reverse(value))
+
+    # Peek at the current maximum value
+    def peek(self) -> T:
+        return self._heap[0].value
+
+    # Pop and return the highest value
+    def pop(self) -> T:
+        return heapq.heappop(self._heap).value
+
+    # Pushes a value onto the heap, pops and returns the highest value
+    def pushpop(self, value: T) -> None:
+        return heapq.heappushpop(self._heap, Reverse(value)).value
+```
+
+#### The actual Implementation
+
+Now that we have written the necessary building blocks, let's tackle the
+Implementation of `closest` for our _k-d Tree_.
+
+```python
+# Wrapper type for closest points, ordered by `distance`
+@dataclasses.dataclass(order=True)
+class ClosestPoint[T](NamedTuple):
+    point: Point = field(compare=False)
+    value: T = field(compare=False)
+    distance: float
+
+class KdTree[T]:
+    def closest(self, point: Point, n: int = 1) -> list[ClosestPoint[T]]:
+        assert n > 0
+        # Create the output heap
+        res = MaxHeap()
+        # Recurse onto the root node
+        self._root.closest(point, res, n, self._aabb)
+        # Return the resulting list, from closest to farthest
+        return sorted(res)
+
+class KdNode[T]:
+    def closest(
+        self,
+        point: Point,
+        out: MaxHeap[ClosestPoint[T]],
+        n: int,
+        bounds: AABB,
+    ) -> None:
+        # Forward to the wrapped node
+        self.inner.closest(point, out, n, bounds)
+
+class KdLeafNode[T]:
+    def closest(
+        self,
+        point: Point,
+        out: MaxHeap[ClosestPoint[T]],
+        n: int,
+        bounds: AABB,
+    ) -> None:
+        # At the leaf, simply iterate over all points and add them to the heap
+        for p, val in self.points.items():
+            item = ClosestPoint(p, val, dist(p, point))
+            if len(out) < n:
+                # If the heap isn't full, just push
+                out.push(item)
+            elif out.peek().distance > item.distance:
+                # Otherwise, push and pop to keep the heap at `n` elements
+                out.pushpop(item)
+
+class KdSplitNode[T]:
+    def closest(
+        self,
+        point: Point,
+        out: list[ClosestPoint[T]],
+        n: int,
+        bounds: AABB,
+    ) -> None:
+        index = self._index(point)
+        children_bounds = bounds.split(self.axis, self.mid)
+        # Iterate over the child which contains the point, then its neighbour
+        for i in (index, 1 - index):
+            child, bounds = self.children[i], children_bounds[i]
+            # `min_dist` is 0 for the first child, and the minimum distance of
+            # all points contained in the second child
+            min_dist = bounds.dist_to_point(point)
+            # If the heap is at capacity and the child to inspect too far, stop
+            if len(out) == n and min_dist > out.peek().distance:
+                return
+            # Otherwise, recurse
+            child.closest(point, out, n, bounds)
+```