metric: Add some general interfaces for metric spaces

2 files changed, 533 insertions, 0 deletions

diff --git a/src/main.rs b/src/main.rs
index f328e4d..0d7989b 100644
--- a/src/main.rs
+++ b/src/main.rs
@@ -1 +1,3 @@
+pub mod metric;
+
 fn main() {}
diff --git a/src/metric.rs b/src/metric.rs
new file mode 100644
index 0000000..e067771
--- /dev/null
+++ b/src/metric.rs
@@ -0,0 +1,531 @@
+//! [Metric spaces](https://en.wikipedia.org/wiki/Metric_space).
+
+use ordered_float::OrderedFloat;
+
+use std::cmp::Ordering;
+use std::collections::BinaryHeap;
+use std::iter::FromIterator;
+
+/// An [order embedding](https://en.wikipedia.org/wiki/Order_embedding) for distances.
+///
+/// Implementations of this trait must satisfy, for all non-negative distances `x` and `y`:
+///
+/// * `x == Self::from(x).into()`
+/// * `x <= y` iff `Self::from(x) <= Self::from(y)`
+///
+/// This trait exists to optimize the common case where distances can be compared more efficiently
+/// than their exact values can be computed.  For example, taking the square root can be avoided
+/// when comparing Euclidean distances (see [SquaredDistance]).
+pub trait Distance: Copy + From<f64> + Into<f64> + Ord {}
+
+/// A raw numerical distance.
+#[derive(Debug, Clone, Copy, Eq, Ord, PartialEq, PartialOrd)]
+pub struct RawDistance(OrderedFloat<f64>);
+
+impl From<f64> for RawDistance {
+    fn from(value: f64) -> Self {
+        Self(value.into())
+    }
+}
+
+impl From<RawDistance> for f64 {
+    fn from(value: RawDistance) -> Self {
+        value.0.into_inner()
+    }
+}
+
+impl Distance for RawDistance {}
+
+/// A squared distance, to avoid computing square roots unless absolutely necessary.
+#[derive(Debug, Clone, Copy, Eq, Ord, PartialEq, PartialOrd)]
+pub struct SquaredDistance(OrderedFloat<f64>);
+
+impl SquaredDistance {
+    /// Create a SquaredDistance from an already squared value.
+    pub fn from_squared(value: f64) -> Self {
+        Self(value.into())
+    }
+}
+
+impl From<f64> for SquaredDistance {
+    fn from(value: f64) -> Self {
+        Self::from_squared(value * value)
+    }
+}
+
+impl From<SquaredDistance> for f64 {
+    fn from(value: SquaredDistance) -> Self {
+        value.0.into_inner().sqrt()
+    }
+}
+
+impl Distance for SquaredDistance {}
+
+/// A [metric space](https://en.wikipedia.org/wiki/Metric_space).
+pub trait Metric<T: ?Sized = Self> {
+    /// The type used to represent distances.  Use [RawDistance] to compare the actual values
+    /// directly, or another type if comparisons can be implemented more efficiently.
+    type Distance: Distance;
+
+    /// Computes the distance between this point and another point.  This function must satisfy
+    /// three conditions:
+    ///
+    /// * `x.distance(y) == 0` iff `x == y` (identity of indiscernibles)
+    /// * `x.distance(y) == y.distance(x)` (symmetry)
+    /// * `x.distance(z) <= x.distance(y) + y.distance(z)` (triangle inequality)
+    fn distance(&self, other: &T) -> Self::Distance;
+}
+
+/// Blanket [Metric] implementation for references.
+impl<'a, 'b, T, U: Metric<T>> Metric<&'a T> for &'b U {
+    type Distance = U::Distance;
+
+    fn distance(&self, other: &&'a T) -> Self::Distance {
+        (*self).distance(other)
+    }
+}
+
+/// The standard [Euclidean distance](https://en.wikipedia.org/wiki/Euclidean_distance) metric.
+impl Metric for [f64] {
+    type Distance = SquaredDistance;
+
+    fn distance(&self, other: &Self) -> Self::Distance {
+        debug_assert!(self.len() == other.len());
+
+        let mut sum = 0.0;
+        for i in 0..self.len() {
+            let diff = self[i] - other[i];
+            sum += diff * diff;
+        }
+
+        Self::Distance::from_squared(sum)
+    }
+}
+
+/// A nearest neighbor to a target.
+#[derive(Clone, Copy, Debug, PartialEq)]
+pub struct Neighbor<T> {
+    /// The found item.
+    pub item: T,
+    /// The distance from the target.
+    pub distance: f64,
+}
+
+impl<T> Neighbor<T> {
+    /// Create a new Neighbor.
+    pub fn new(item: T, distance: f64) -> Self {
+        Self { item, distance }
+    }
+}
+
+/// A candidate nearest neighbor found during a search.
+#[derive(Debug)]
+struct Candidate<T, D> {
+    item: T,
+    distance: D,
+}
+
+impl<T, D: Distance> Candidate<T, D> {
+    fn new<U>(target: U, item: T) -> Self
+    where
+        U: Metric<T, Distance = D>,
+    {
+        let distance = target.distance(&item);
+        Self { item, distance }
+    }
+
+    fn into_neighbor(self) -> Neighbor<T> {
+        Neighbor::new(self.item, self.distance.into())
+    }
+}
+
+impl<T, D: Distance> PartialOrd for Candidate<T, D> {
+    fn partial_cmp(&self, other: &Self) -> Option<Ordering> {
+        self.distance.partial_cmp(&other.distance)
+    }
+}
+
+impl<T, D: Distance> Ord for Candidate<T, D> {
+    fn cmp(&self, other: &Self) -> Ordering {
+        self.distance.cmp(&other.distance)
+    }
+}
+
+impl<T, D: Distance> PartialEq for Candidate<T, D> {
+    fn eq(&self, other: &Self) -> bool {
+        self.distance.eq(&other.distance)
+    }
+}
+
+impl<T, D: Distance> Eq for Candidate<T, D> {}
+
+/// Accumulates nearest neighbor search results.
+pub trait Neighborhood<T, U: Metric<T>> {
+    /// Returns the target of the nearest neighbor search.
+    fn target(&self) -> U;
+
+    /// Check whether a distance is within this neighborhood.
+    fn contains(&self, distance: f64) -> bool {
+        distance < 0.0 || self.contains_distance(distance.into())
+    }
+
+    /// Check whether a distance is within this neighborhood.
+    fn contains_distance(&self, distance: U::Distance) -> bool;
+
+    /// Consider a new candidate neighbor.
+    fn consider(&mut self, item: T) -> U::Distance;
+}
+
+/// A [Neighborhood] with at most one result.
+#[derive(Debug)]
+struct SingletonNeighborhood<T, U: Metric<T>> {
+    /// The target of the nearest neighbor search.
+    target: U,
+    /// The current threshold distance to the farthest result.
+    threshold: Option<U::Distance>,
+    /// The current nearest neighbor, if any.
+    candidate: Option<Candidate<T, U::Distance>>,
+}
+
+impl<T, U> SingletonNeighborhood<T, U>
+where
+    U: Copy + Metric<T>,
+{
+    /// Create a new single metric result tracker.
+    ///
+    /// * `target`: The target fo the nearest neighbor search.
+    /// * `threshold`: The maximum allowable distance.
+    fn new(target: U, threshold: Option<f64>) -> Self {
+        Self {
+            target,
+            threshold: threshold.map(U::Distance::from),
+            candidate: None,
+        }
+    }
+
+    /// Consider a candidate.
+    fn push(&mut self, candidate: Candidate<T, U::Distance>) -> U::Distance {
+        let distance = candidate.distance;
+
+        if self.contains_distance(distance) {
+            self.threshold = Some(distance);
+            self.candidate = Some(candidate);
+        }
+
+        distance
+    }
+
+    /// Convert this result into an optional neighbor.
+    fn into_option(self) -> Option<Neighbor<T>> {
+        self.candidate.map(Candidate::into_neighbor)
+    }
+}
+
+impl<T, U> Neighborhood<T, U> for SingletonNeighborhood<T, U>
+where
+    U: Copy + Metric<T>,
+{
+    fn target(&self) -> U {
+        self.target
+    }
+
+    fn contains_distance(&self, distance: U::Distance) -> bool {
+        self.threshold.map(|t| distance <= t).unwrap_or(true)
+    }
+
+    fn consider(&mut self, item: T) -> U::Distance {
+        self.push(Candidate::new(self.target, item))
+    }
+}
+
+/// A [Neighborhood] of up to `k` results, using a binary heap.
+#[derive(Debug)]
+struct HeapNeighborhood<T, U: Metric<T>> {
+    /// The target of the nearest neighbor search.
+    target: U,
+    /// The number of nearest neighbors to find.
+    k: usize,
+    /// The current threshold distance to the farthest result.
+    threshold: Option<U::Distance>,
+    /// A max-heap of the best candidates found so far.
+    heap: BinaryHeap<Candidate<T, U::Distance>>,
+}
+
+impl<T, U> HeapNeighborhood<T, U>
+where
+    U: Copy + Metric<T>,
+{
+    /// Create a new metric result tracker.
+    ///
+    /// * `target`: The target fo the nearest neighbor search.
+    /// * `k`: The number of nearest neighbors to find.
+    /// * `threshold`: The maximum allowable distance.
+    fn new(target: U, k: usize, threshold: Option<f64>) -> Self {
+        Self {
+            target,
+            k,
+            threshold: threshold.map(U::Distance::from),
+            heap: BinaryHeap::with_capacity(k),
+        }
+    }
+
+    /// Consider a candidate.
+    fn push(&mut self, candidate: Candidate<T, U::Distance>) -> U::Distance {
+        let distance = candidate.distance;
+
+        if self.contains_distance(distance) {
+            let heap = &mut self.heap;
+
+            if heap.len() == self.k {
+                heap.pop();
+            }
+
+            heap.push(candidate);
+
+            if heap.len() == self.k {
+                self.threshold = self.heap.peek().map(|c| c.distance)
+            }
+        }
+
+        distance
+    }
+
+    /// Convert these results into a vector of neighbors.
+    fn into_vec(self) -> Vec<Neighbor<T>> {
+        self.heap
+            .into_sorted_vec()
+            .into_iter()
+            .map(Candidate::into_neighbor)
+            .collect()
+    }
+}
+
+impl<T, U> Neighborhood<T, U> for HeapNeighborhood<T, U>
+where
+    U: Copy + Metric<T>,
+{
+    fn target(&self) -> U {
+        self.target
+    }
+
+    fn contains_distance(&self, distance: U::Distance) -> bool {
+        self.k > 0 && self.threshold.map(|t| distance <= t).unwrap_or(true)
+    }
+
+    fn consider(&mut self, item: T) -> U::Distance {
+        self.push(Candidate::new(self.target, item))
+    }
+}
+
+/// A [nearest neighbor search](https://en.wikipedia.org/wiki/Nearest_neighbor_search) index.
+///
+/// Type parameters:
+/// * `T`: The search result type.
+/// * `U`: The query type.
+pub trait NearestNeighbors<T, U: Metric<T> = T> {
+    /// Returns the nearest neighbor to `target` (or `None` if this index is empty).
+    fn nearest(&self, target: &U) -> Option<Neighbor<&T>> {
+        self.search(SingletonNeighborhood::new(target, None))
+            .into_option()
+    }
+
+    /// Returns the nearest neighbor to `target` within the distance `threshold`, if one exists.
+    fn nearest_within(&self, target: &U, threshold: f64) -> Option<Neighbor<&T>> {
+        self.search(SingletonNeighborhood::new(target, Some(threshold)))
+            .into_option()
+    }
+
+    /// Returns the up to `k` nearest neighbors to `target`.
+    fn k_nearest(&self, target: &U, k: usize) -> Vec<Neighbor<&T>> {
+        self.search(HeapNeighborhood::new(target, k, None))
+            .into_vec()
+    }
+
+    /// Returns the up to `k` nearest neighbors to `target` within the distance `threshold`.
+    fn k_nearest_within(&self, target: &U, k: usize, threshold: f64) -> Vec<Neighbor<&T>> {
+        self.search(HeapNeighborhood::new(target, k, Some(threshold)))
+            .into_vec()
+    }
+
+    /// Search for nearest neighbors and add them to a neighborhood.
+    fn search<'a, 'b, N>(&'a self, neighborhood: N) -> N
+    where
+        T: 'a,
+        U: 'b,
+        N: Neighborhood<&'a T, &'b U>;
+}
+
+/// A [NearestNeighbors] implementation that does exhaustive search.
+#[derive(Debug)]
+pub struct ExhaustiveSearch<T>(Vec<T>);
+
+impl<T> ExhaustiveSearch<T> {
+    /// Create an empty ExhaustiveSearch index.
+    pub fn new() -> Self {
+        Self(Vec::new())
+    }
+
+    /// Add a new item to the index.
+    pub fn push(&mut self, item: T) {
+        self.0.push(item);
+    }
+}
+
+impl<T> FromIterator<T> for ExhaustiveSearch<T> {
+    fn from_iter<I: IntoIterator<Item = T>>(items: I) -> Self {
+        Self(items.into_iter().collect())
+    }
+}
+
+impl<T> IntoIterator for ExhaustiveSearch<T> {
+    type Item = T;
+    type IntoIter = std::vec::IntoIter<T>;
+
+    fn into_iter(self) -> Self::IntoIter {
+        self.0.into_iter()
+    }
+}
+
+impl<T> Extend<T> for ExhaustiveSearch<T> {
+    fn extend<I: IntoIterator<Item = T>>(&mut self, iter: I) {
+        for value in iter {
+            self.push(value);
+        }
+    }
+}
+
+impl<T, U: Metric<T>> NearestNeighbors<T, U> for ExhaustiveSearch<T> {
+    fn search<'a, 'b, N>(&'a self, mut neighborhood: N) -> N
+    where
+        T: 'a,
+        U: 'b,
+        N: Neighborhood<&'a T, &'b U>,
+    {
+        for e in &self.0 {
+            neighborhood.consider(e);
+        }
+        neighborhood
+    }
+}
+
+#[cfg(test)]
+pub mod tests {
+    use super::*;
+
+    use rand::prelude::*;
+
+    #[derive(Clone, Copy, Debug, PartialEq)]
+    pub struct Point(pub [f64; 3]);
+
+    impl Metric for Point {
+        type Distance = SquaredDistance;
+
+        fn distance(&self, other: &Self) -> Self::Distance {
+            self.0.distance(&other.0)
+        }
+    }
+
+    /// Test a [NearestNeighbors] impl.
+    pub fn test_nearest_neighbors<T, F>(from_iter: F)
+    where
+        T: NearestNeighbors<Point>,
+        F: Fn(Vec<Point>) -> T,
+    {
+        test_empty(&from_iter);
+        test_pythagorean(&from_iter);
+        test_random_points(&from_iter);
+    }
+
+    fn test_empty<T, F>(from_iter: &F)
+    where
+        T: NearestNeighbors<Point>,
+        F: Fn(Vec<Point>) -> T,
+    {
+        let points = Vec::new();
+        let index = from_iter(points);
+        let target = Point([0.0, 0.0, 0.0]);
+        assert_eq!(index.nearest(&target), None);
+        assert_eq!(index.nearest_within(&target, 1.0), None);
+        assert!(index.k_nearest(&target, 0).is_empty());
+        assert!(index.k_nearest(&target, 3).is_empty());
+        assert!(index.k_nearest_within(&target, 0, 1.0).is_empty());
+        assert!(index.k_nearest_within(&target, 3, 1.0).is_empty());
+    }
+
+    fn test_pythagorean<T, F>(from_iter: &F)
+    where
+        T: NearestNeighbors<Point>,
+        F: Fn(Vec<Point>) -> T,
+    {
+        let points = vec![
+            Point([3.0, 4.0, 0.0]),
+            Point([5.0, 0.0, 12.0]),
+            Point([0.0, 8.0, 15.0]),
+            Point([1.0, 2.0, 2.0]),
+            Point([2.0, 3.0, 6.0]),
+            Point([4.0, 4.0, 7.0]),
+        ];
+        let index = from_iter(points);
+        let target = Point([0.0, 0.0, 0.0]);
+
+        assert_eq!(
+            index.nearest(&target),
+            Some(Neighbor::new(&Point([1.0, 2.0, 2.0]), 3.0))
+        );
+
+        assert_eq!(index.nearest_within(&target, 2.0), None);
+        assert_eq!(
+            index.nearest_within(&target, 4.0),
+            Some(Neighbor::new(&Point([1.0, 2.0, 2.0]), 3.0))
+        );
+
+        assert!(index.k_nearest(&target, 0).is_empty());
+        assert_eq!(
+            index.k_nearest(&target, 3),
+            vec![
+                Neighbor::new(&Point([1.0, 2.0, 2.0]), 3.0),
+                Neighbor::new(&Point([3.0, 4.0, 0.0]), 5.0),
+                Neighbor::new(&Point([2.0, 3.0, 6.0]), 7.0),
+            ]
+        );
+
+        assert!(index.k_nearest(&target, 0).is_empty());
+        assert_eq!(
+            index.k_nearest_within(&target, 3, 6.0),
+            vec![
+                Neighbor::new(&Point([1.0, 2.0, 2.0]), 3.0),
+                Neighbor::new(&Point([3.0, 4.0, 0.0]), 5.0),
+            ]
+        );
+        assert_eq!(
+            index.k_nearest_within(&target, 3, 8.0),
+            vec![
+                Neighbor::new(&Point([1.0, 2.0, 2.0]), 3.0),
+                Neighbor::new(&Point([3.0, 4.0, 0.0]), 5.0),
+                Neighbor::new(&Point([2.0, 3.0, 6.0]), 7.0),
+            ]
+        );
+    }
+
+    fn test_random_points<T, F>(from_iter: &F)
+    where
+        T: NearestNeighbors<Point>,
+        F: Fn(Vec<Point>) -> T,
+    {
+        let mut points = Vec::new();
+        for _ in 0..255 {
+            points.push(Point([random(), random(), random()]));
+        }
+        let target = Point([random(), random(), random()]);
+
+        let eindex = ExhaustiveSearch::from_iter(points.clone());
+        let index = from_iter(points);
+
+        assert_eq!(index.k_nearest(&target, 3), eindex.k_nearest(&target, 3));
+    }
+
+    #[test]
+    fn test_exhaustive_index() {
+        test_nearest_neighbors(ExhaustiveSearch::from_iter);
+    }
+}