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use std::cmp::Ordering;
use std::collections::BinaryHeap;
/// A `Point` is something that exists in some sort of metric space, and
/// can thus calculate its distance to another `Point`, and can be moved
/// a certain distance towards another `Point`.
pub trait Point: Sized + PartialEq {
/// Distances should be positive, finite `f64`s. It is undefined behavior to
/// return a negative, infinite, or `NaN` result.
///
/// Distance should satisfy the triangle inequality. That is, `a.distance(c)`
/// must be less or equal to than `a.distance(b) + b.distance(c)`.
fn distance(&self, other: &Self) -> f64;
/// If `d` is `0`, a point equal to the `self` should be returned. If `d` is equal
/// to `self.distance(other)`, a point equal to `other` should be returned.
/// Intermediate distances should be linearly interpolated between the two points,
/// so if `d` is equal to `self.distance(other) / 2.0`, the midpoint should be
/// returned.
/// It is undefined behavior to use a distance that is negative, `NaN`, or greater
/// than `self.distance(other)`.
fn move_towards(&self, other: &Self, d: f64) -> Self;
}
fn midpoint<P: Point>(a: &P, b: &P) -> P {
let d = a.distance(b);
a.move_towards(b, d / 2.0)
}
/// Implement `Point` in the normal `D` dimensional Euclidean way for all arrays of floats. For example, a 2D point
/// would be a `[f64; 2]`.
impl<const D: usize> Point for [f64; D] {
fn distance(&self, other: &Self) -> f64 {
self.iter()
.zip(other)
.map(|(a, b)| (*a - *b).powi(2))
.sum::<f64>()
.sqrt()
}
fn move_towards(&self, other: &Self, d: f64) -> Self {
let mut result = self.clone();
let distance = self.distance(other);
// Don't want to get a NaN in the division below
if distance == 0.0 {
return result;
}
let scale = d / self.distance(other);
for i in 0..D {
result[i] += scale * (other[i] - self[i]);
}
result
}
}
// A little helper to allow us to use comparative functions on `f64`s by asserting that
// `NaN` isn't present.
#[derive(Debug, Clone, PartialEq, PartialOrd)]
struct OrdF64(f64);
impl OrdF64 {
fn new(x: f64) -> Self {
assert!(!x.is_nan());
OrdF64(x)
}
}
impl Eq for OrdF64 {}
impl Ord for OrdF64 {
fn cmp(&self, other: &Self) -> Ordering {
self.partial_cmp(other).unwrap()
}
}
#[derive(Debug, Copy, Clone, PartialEq)]
struct Sphere<C> {
center: C,
radius: f64,
}
impl<C: Point> Sphere<C> {
fn nearest_distance(&self, p: &C) -> f64 {
let d = self.center.distance(p) - self.radius;
d.max(0.0)
}
fn farthest_distance(&self, p: &C) -> f64 {
self.center.distance(p) + self.radius
}
}
// Implementation of the "bouncing bubble" algorithm which essentially works like this:
// * Pick a point `a` that is farthest from `points[0]`
// * Pick a point `b` that is farthest from `a`
// * Use these two points to create an initial sphere centered at their midpoint and with
// enough radius to encompass them
// * While there is still a point outside of this sphere, move the sphere towards that
// point just enough to encompass that point, and grow the sphere radius by 1%
//
// This process will produce a non-optimal, but relatively snug fitting bounding sphere.
fn bounding_sphere<P: Point>(points: &[P]) -> Sphere<P> {
assert!(points.len() >= 2);
let a = &points
.iter()
.max_by_key(|a| OrdF64::new(points[0].distance(a)))
.unwrap();
let b = &points
.iter()
.max_by_key(|b| OrdF64::new(a.distance(b)))
.unwrap();
let mut center: P = midpoint(a, b);
let mut radius = center.distance(b).max(std::f64::EPSILON);
loop {
match points.iter().filter(|p| center.distance(p) > radius).next() {
None => break Sphere { center, radius },
Some(p) => {
let c_to_p = center.distance(&p);
let d = c_to_p - radius;
center = center.move_towards(p, d);
radius = radius * 1.01;
}
}
}
}
// Produce a partition of the given points with the following process:
// * Pick a point `a` that is farthest from `points[0]`
// * Pick a point `b` that is farthest from `a`
// * Partition the points into two groups: those closest to `a` and those closest to `b`
//
// This doesn't necessarily form the best partition, since `a` and `b` are not guaranteed
// to be the most distant pair of points, but it's usually sufficient.
fn partition<P: Point, V>(
mut points: Vec<P>,
mut values: Vec<V>,
) -> ((Vec<P>, Vec<V>), (Vec<P>, Vec<V>)) {
assert!(points.len() >= 2);
assert_eq!(points.len(), values.len());
let a_i = points
.iter()
.enumerate()
.max_by_key(|(_, a)| OrdF64::new(points[0].distance(a)))
.unwrap()
.0;
let b_i = points
.iter()
.enumerate()
.max_by_key(|(_, b)| OrdF64::new(points[a_i].distance(b)))
.unwrap()
.0;
let (a_i, b_i) = (a_i.max(b_i), a_i.min(b_i));
let (mut aps, mut avs) = (vec![points.swap_remove(a_i)], vec![values.swap_remove(a_i)]);
let (mut bps, mut bvs) = (vec![points.swap_remove(b_i)], vec![values.swap_remove(b_i)]);
for (p, v) in points.into_iter().zip(values) {
if aps[0].distance(&p) < bps[0].distance(&p) {
aps.push(p);
avs.push(v);
} else {
bps.push(p);
bvs.push(v);
}
}
((aps, avs), (bps, bvs))
}
#[derive(Debug, Clone)]
enum BallTreeInner<P, V> {
Empty,
Leaf(P, Vec<V>),
// The sphere is a bounding sphere that encompasses this node (both children)
Branch {
sphere: Sphere<P>,
a: Box<BallTreeInner<P, V>>,
b: Box<BallTreeInner<P, V>>,
count: usize,
},
}
impl<P: Point, V> Default for BallTreeInner<P, V> {
fn default() -> Self {
BallTreeInner::Empty
}
}
impl<P: Point, V> BallTreeInner<P, V> {
fn new(mut points: Vec<P>, values: Vec<V>) -> Self {
assert_eq!(
points.len(),
values.len(),
"Given two vectors of differing lengths. points: {}, values: {}",
points.len(),
values.len()
);
if points.is_empty() {
BallTreeInner::Empty
} else if points.iter().all(|p| p == &points[0]) {
BallTreeInner::Leaf(points.pop().unwrap(), values)
} else {
let count = points.len();
let sphere = bounding_sphere(&points);
let ((aps, avs), (bps, bvs)) = partition(points, values);
let (a_tree, b_tree) = (BallTreeInner::new(aps, avs), BallTreeInner::new(bps, bvs));
BallTreeInner::Branch { sphere, a: Box::new(a_tree), b: Box::new(b_tree), count }
}
}
fn nearest_distance(&self, p: &P) -> f64 {
match self {
BallTreeInner::Empty => std::f64::INFINITY,
// The distance to a leaf is the distance to the single point inside of it
BallTreeInner::Leaf(p0, _) => p.distance(p0),
// The distance to a branch is the distance to the edge of the bounding sphere
BallTreeInner::Branch { sphere, .. } => sphere.nearest_distance(p),
}
}
}
// We need a little wrapper to hold our priority queue elements for two reasons:
// * Rust's BinaryHeap is a max-heap, and we need a min-heap, so we invert the
// ordering
// * We only want to order based on the first element, so we need a custom
// implementation rather than deriving the order (which would require the value
// to be orderable which is not necessary).
#[derive(Debug, Copy, Clone)]
struct Item<T>(f64, T);
impl<T> PartialEq for Item<T> {
fn eq(&self, other: &Self) -> bool {
self.0 == other.0
}
}
impl<T> Eq for Item<T> {}
impl<T> PartialOrd for Item<T> {
fn partial_cmp(&self, other: &Self) -> Option<Ordering> {
self.0
.partial_cmp(&other.0)
.map(|ordering| ordering.reverse())
}
}
impl<T> Ord for Item<T> {
fn cmp(&self, other: &Self) -> Ordering {
self.partial_cmp(other).unwrap()
}
}
/// Iterator over the nearest neighbors.
// Maintain a priority queue of the nodes that are closest to the provided `point`. If we
// pop a leaf from the queue, that leaf is necessarily the next closest point. If we
// pop a branch from the queue, add its children. The priority of a node is its
// `distance` as defined above.
#[derive(Debug)]
pub struct Iter<'tree, 'query, P, V> {
point: &'query P,
balls: &'query mut BinaryHeap<Item<&'tree BallTreeInner<P, V>>>,
i: usize,
max_radius: f64,
}
impl<'tree, 'query, P: Point, V> Iterator for Iter<'tree, 'query, P, V> {
type Item = (&'tree P, f64, &'tree V);
fn next(&mut self) -> Option<Self::Item> {
while self.balls.len() > 0 {
// Peek in the leaf case, because we might need to visit this leaf multiple
// times (if it has multiple values).
if let Item(d, BallTreeInner::Leaf(p, vs)) = self.balls.peek().unwrap() {
if self.i < vs.len() && *d <= self.max_radius {
self.i += 1;
return Some((p, *d, &vs[self.i - 1]));
}
}
// Reset index for the next leaf we encounter
self.i = 0;
// Expand branch nodes
if let Item(_, BallTreeInner::Branch { a, b, .. }) = self.balls.pop().unwrap() {
let d_a = a.nearest_distance(self.point);
let d_b = b.nearest_distance(self.point);
if d_a <= self.max_radius {
self.balls.push(Item(d_a, a));
}
if d_b <= self.max_radius {
self.balls.push(Item(d_b, b));
}
}
}
None
}
}
/// A `BallTree` is a space-partitioning data-structure that allows for finding
/// nearest neighbors in logarithmic time.
///
/// It does this by partitioning data into a series of nested bounding spheres
/// ("balls" in the literature). Spheres are used because it is trivial to
/// compute the distance between a point and a sphere (distance to the sphere's
/// center minus thte radius). The key observation is that a potential neighbor
/// is necessarily closer than all neighbors that are located inside of a
/// bounding sphere that is farther than the aforementioned neighbor.
///
/// Graphically:
/// ```text
///
/// A -
/// | ---- distance(A, B) = 4
/// | - B distance(A, S) = 6
/// |
/// |
/// | S
/// --------
/// / G \
/// / C \
/// | D |
/// | F |
/// \ E /
/// \_________/
///```
///
/// In the diagram, `A` is closer to `B` than to `S`, and because `S` bounds
/// `C`, `D`, `E`, `F`, and `G`, it can be determined that `A` it is necessarily
/// closer to `B` than the other points without even computing exact distances
/// to them.
///
/// Ball trees are most commonly used as a form of predictive model where the
/// points are features and each point is associated with a value or label. Thus,
/// This implementation allows the user to associate a value with each point. If
/// this functionality is unneeded, `()` can be used as a value.
///
/// This implementation returns the nearest neighbors, their distances, and their
/// associated values. Returning the distances allows the user to perform some
/// sort of weighted interpolation of the neighbors for predictive purposes.
#[derive(Debug, Clone)]
pub struct BallTree<P, V>(BallTreeInner<P, V>);
impl<P: Point, V> Default for BallTree<P, V> {
fn default() -> Self {
BallTree(BallTreeInner::default())
}
}
impl<P: Point, V> BallTree<P, V> {
/// Construct this `BallTree`. Construction is somewhat expensive, so `BallTree`s
/// are best constructed once and then used repeatedly.
///
/// `panic` if `points.len() != values.len()`
pub fn new(points: Vec<P>, values: Vec<V>) -> Self {
BallTree(BallTreeInner::new(points, values))
}
/// Query this `BallTree`. The `Query` object provides a nearest-neighbor API and internally re-uses memory to avoid
/// allocations on repeated queries.
pub fn query(&self) -> Query<P, V> {
Query {
ball_tree: self,
balls: Default::default(),
}
}
}
/// A context for repeated nearest-neighbor queries that internally re-uses memory across queries.
#[derive(Debug, Clone)]
pub struct Query<'tree, P, V> {
ball_tree: &'tree BallTree<P, V>,
balls: BinaryHeap<Item<&'tree BallTreeInner<P, V>>>,
}
impl<'tree, P: Point, V> Query<'tree, P, V> {
/// Given a `point`, return an `Iterator` that yields neighbors from closest to
/// farthest. To get the K nearest neighbors, simply `take` K from the iterator.
///
/// The neighbor, its distance, and associated value are returned.
pub fn nn<'query>(
&'query mut self,
point: &'query P,
) -> Iter<'tree, 'query, P, V> {
self.nn_within(point, f64::INFINITY)
}
/// The same as `nn` but only consider neighbors whose distance is `<= max_radius`.
pub fn nn_within<'query>(
&'query mut self,
point: &'query P,
max_radius: f64,
) -> Iter<'tree, 'query, P, V> {
let balls = &mut self.balls;
balls.clear();
balls.push(Item(self.ball_tree.0.nearest_distance(point), &self.ball_tree.0));
Iter {
point,
balls,
i: 0,
max_radius,
}
}
/// What is the minimum radius that encompasses `k` neighbors of `point`?
pub fn min_radius<'query>(&'query mut self, point: &'query P, k: usize) -> f64 {
let mut total_count = 0;
let balls = &mut self.balls;
balls.clear();
balls.push(Item(self.ball_tree.0.nearest_distance(point), &self.ball_tree.0));
while let Some(Item(distance, node)) = balls.pop() {
match node {
BallTreeInner::Empty => {}
BallTreeInner::Leaf(_, vs) => {
total_count += vs.len();
if total_count >= k {
return distance;
}
}
BallTreeInner::Branch { sphere, a, b, count } => {
let next_distance = balls.peek().map(|Item(d, _)| *d).unwrap_or(f64::INFINITY);
if total_count + count < k && sphere.farthest_distance(point) < next_distance {
total_count += count;
} else {
balls.push(Item(a.nearest_distance(point), &a));
balls.push(Item(b.nearest_distance(point), &b));
}
}
}
}
f64::INFINITY
}
/// How many neighbors are `<= max_radius` of `point`?
pub fn count<'query>(&'query mut self, point: &'query P, max_radius: f64) -> usize {
let mut total = 0;
let balls = &mut self.balls;
balls.clear();
balls.push(Item(self.ball_tree.0.nearest_distance(point), &self.ball_tree.0));
while let Some(Item(nearest_distance, node)) = balls.pop() {
if nearest_distance > max_radius {
break;
}
match node {
BallTreeInner::Empty => {}
BallTreeInner::Leaf(_, vs) => {
total += vs.len();
}
BallTreeInner::Branch { a, b, count, sphere} => {
let next_distance = balls.peek().map(|Item(d, _)| *d).unwrap_or(f64::INFINITY).min(max_radius);
if sphere.farthest_distance(point) < next_distance {
total += count;
} else {
balls.push(Item(a.nearest_distance(point), &a));
balls.push(Item(b.nearest_distance(point), &b));
}
}
}
}
total
}
/// Return the size in bytes of the memory this `Query` is keeping internally to avoid allocation.
pub fn allocated_size(&self) -> usize {
self.balls.capacity() * std::mem::size_of::<Item<&'tree BallTreeInner<P, V>>>()
}
/// The `Query` object re-uses memory internally to avoid allocation. This method deallocates that memory.
pub fn deallocate_memory(&mut self) {
self.balls.clear();
self.balls.shrink_to_fit();
}
}
#[cfg(test)]
mod tests {
use super::*;
use rand::{Rng, SeedableRng};
use rand_chacha::ChaChaRng;
use std::collections::HashSet;
#[test]
fn test_3d_points() {
let mut rng: ChaChaRng = SeedableRng::seed_from_u64(0xcb42c94d23346e96);
macro_rules! random_small_f64 {
() => {
rng.gen_range(-100.0 ..= 100.0)
};
}
macro_rules! random_3d_point {
() => {
[
random_small_f64!(),
random_small_f64!(),
random_small_f64!(),
]
};
}
for i in 0..1000 {
let point_count: usize = if i < 100 {
rng.gen_range(1..=3)
} else if i < 500 {
rng.gen_range(1..=10)
} else {
rng.gen_range(1..=100)
};
let mut points = vec![];
let mut values = vec![];
for _ in 0..point_count {
let point = random_3d_point!();
let value = rng.gen::<u64>();
points.push(point);
values.push(value);
}
let tree = BallTree::new(points.clone(), values.clone());
let mut query = tree.query();
for _ in 0..100 {
let point = random_3d_point!();
let max_radius = rng.gen_range(0.0 ..= 110.0);
let expected_values = points
.iter()
.zip(&values)
.filter(|(p, _)| p.distance(&point) <= max_radius)
.map(|(_, v)| v)
.cloned()
.collect::<HashSet<_>>();
let mut found_values = HashSet::new();
let mut previous_d = 0.0;
for (p, d, v) in query.nn_within(&point, max_radius) {
assert_eq!(point.distance(p), d);
assert!(d >= previous_d);
assert!(d <= max_radius);
previous_d = d;
found_values.insert(*v);
}
assert_eq!(expected_values, found_values);
assert_eq!(found_values.len(), query.count(&point, max_radius));
let radius = query.min_radius(&point, expected_values.len());
let should_be_fewer = query.count(&point, radius * 0.99);
assert!(expected_values.is_empty() || should_be_fewer < expected_values.len(), "{} < {}", should_be_fewer, expected_values.len());
}
assert!(query.allocated_size() > 0);
// 2 (branching factor) * 8 (pointer size) * point count rounded up (max of 4 due to minimum vec sizing)
assert!(query.allocated_size() <= 2 * 8 * point_count.next_power_of_two().max(4));
query.deallocate_memory();
assert_eq!(query.allocated_size(), 0);
}
}
#[test]
fn test_point_array_impls() {
assert_eq!([5.0].distance(&[7.0]), 2.0);
assert_eq!([5.0].move_towards(&[3.0], 1.0), [4.0]);
assert_eq!([5.0, 3.0].distance(&[7.0, 5.0]), 2.0 * 2f64.sqrt());
assert_eq!(
[5.0, 3.0].move_towards(&[3.0, 1.0], 2f64.sqrt()),
[4.0, 2.0]
);
assert_eq!([0.0, 0.0, 0.0, 0.0].distance(&[2.0, 2.0, 2.0, 2.0]), 4.0);
assert_eq!(
[0.0, 0.0, 0.0, 0.0].move_towards(&[2.0, 2.0, 2.0, 2.0], 8.0),
[4.0, 4.0, 4.0, 4.0]
);
}
}