rand/distr/

weighted_index.rs

// Copyright 2018 Developers of the Rand project.
//
// Licensed under the Apache License, Version 2.0 <LICENSE-APACHE or
// https://www.apache.org/licenses/LICENSE-2.0> or the MIT license
// <LICENSE-MIT or https://opensource.org/licenses/MIT>, at your
// option. This file may not be copied, modified, or distributed
// except according to those terms.

//! Weighted index sampling

use crate::distr::uniform::{SampleBorrow, SampleUniform, UniformSampler};
use crate::distr::Distribution;
use crate::Rng;
use core::fmt;

// Note that this whole module is only imported if feature="alloc" is enabled.
use alloc::vec::Vec;
use core::fmt::Debug;

#[cfg(feature = "serde")]
use serde::{Deserialize, Serialize};

/// A distribution using weighted sampling of discrete items.
///
/// Sampling a `WeightedIndex` distribution returns the index of a randomly
/// selected element from the iterator used when the `WeightedIndex` was
/// created. The chance of a given element being picked is proportional to the
/// weight of the element. The weights can use any type `X` for which an
/// implementation of [`Uniform<X>`] exists. The implementation guarantees that
/// elements with zero weight are never picked, even when the weights are
/// floating point numbers.
///
/// # Performance
///
/// Time complexity of sampling from `WeightedIndex` is `O(log N)` where
/// `N` is the number of weights. There are two alternative implementations with
/// different runtimes characteristics:
/// * [`rand_distr::weighted_alias`] supports `O(1)` sampling, but with much higher
///   initialisation cost.
/// * [`rand_distr::weighted_tree`] keeps the weights in a tree structure where sampling
///   and updating is `O(log N)`.
///
/// A `WeightedIndex<X>` contains a `Vec<X>` and a [`Uniform<X>`] and so its
/// size is the sum of the size of those objects, possibly plus some alignment.
///
/// Creating a `WeightedIndex<X>` will allocate enough space to hold `N - 1`
/// weights of type `X`, where `N` is the number of weights. However, since
/// `Vec` doesn't guarantee a particular growth strategy, additional memory
/// might be allocated but not used. Since the `WeightedIndex` object also
/// contains an instance of `X::Sampler`, this might cause additional allocations,
/// though for primitive types, [`Uniform<X>`] doesn't allocate any memory.
///
/// Sampling from `WeightedIndex` will result in a single call to
/// `Uniform<X>::sample` (method of the [`Distribution`] trait), which typically
/// will request a single value from the underlying [`RngCore`], though the
/// exact number depends on the implementation of `Uniform<X>::sample`.
///
/// # Example
///
/// ```
/// use rand::prelude::*;
/// use rand::distr::WeightedIndex;
///
/// let choices = ['a', 'b', 'c'];
/// let weights = [2,   1,   1];
/// let dist = WeightedIndex::new(&weights).unwrap();
/// let mut rng = rand::rng();
/// for _ in 0..100 {
///     // 50% chance to print 'a', 25% chance to print 'b', 25% chance to print 'c'
///     println!("{}", choices[dist.sample(&mut rng)]);
/// }
///
/// let items = [('a', 0.0), ('b', 3.0), ('c', 7.0)];
/// let dist2 = WeightedIndex::new(items.iter().map(|item| item.1)).unwrap();
/// for _ in 0..100 {
///     // 0% chance to print 'a', 30% chance to print 'b', 70% chance to print 'c'
///     println!("{}", items[dist2.sample(&mut rng)].0);
/// }
/// ```
///
/// [`Uniform<X>`]: crate::distr::Uniform
/// [`RngCore`]: crate::RngCore
/// [`rand_distr::weighted_alias`]: https://docs.rs/rand_distr/*/rand_distr/weighted_alias/index.html
/// [`rand_distr::weighted_tree`]: https://docs.rs/rand_distr/*/rand_distr/weighted_tree/index.html
#[derive(Debug, Clone, PartialEq)]
#[cfg_attr(feature = "serde", derive(Serialize, Deserialize))]
pub struct WeightedIndex<X: SampleUniform + PartialOrd> {
    cumulative_weights: Vec<X>,
    total_weight: X,
    weight_distribution: X::Sampler,
}

impl<X: SampleUniform + PartialOrd> WeightedIndex<X> {
    /// Creates a new a `WeightedIndex` [`Distribution`] using the values
    /// in `weights`. The weights can use any type `X` for which an
    /// implementation of [`Uniform<X>`] exists.
    ///
    /// Error cases:
    /// -   [`WeightError::InvalidInput`] when the iterator `weights` is empty.
    /// -   [`WeightError::InvalidWeight`] when a weight is not-a-number or negative.
    /// -   [`WeightError::InsufficientNonZero`] when the sum of all weights is zero.
    /// -   [`WeightError::Overflow`] when the sum of all weights overflows.
    ///
    /// [`Uniform<X>`]: crate::distr::uniform::Uniform
    pub fn new<I>(weights: I) -> Result<WeightedIndex<X>, WeightError>
    where
        I: IntoIterator,
        I::Item: SampleBorrow<X>,
        X: Weight,
    {
        let mut iter = weights.into_iter();
        let mut total_weight: X = iter
            .next()
            .ok_or(WeightError::InvalidInput)?
            .borrow()
            .clone();

        let zero = X::ZERO;
        if !(total_weight >= zero) {
            return Err(WeightError::InvalidWeight);
        }

        let mut weights = Vec::<X>::with_capacity(iter.size_hint().0);
        for w in iter {
            // Note that `!(w >= x)` is not equivalent to `w < x` for partially
            // ordered types due to NaNs which are equal to nothing.
            if !(w.borrow() >= &zero) {
                return Err(WeightError::InvalidWeight);
            }
            weights.push(total_weight.clone());

            if let Err(()) = total_weight.checked_add_assign(w.borrow()) {
                return Err(WeightError::Overflow);
            }
        }

        if total_weight == zero {
            return Err(WeightError::InsufficientNonZero);
        }
        let distr = X::Sampler::new(zero, total_weight.clone()).unwrap();

        Ok(WeightedIndex {
            cumulative_weights: weights,
            total_weight,
            weight_distribution: distr,
        })
    }

    /// Update a subset of weights, without changing the number of weights.
    ///
    /// `new_weights` must be sorted by the index.
    ///
    /// Using this method instead of `new` might be more efficient if only a small number of
    /// weights is modified. No allocations are performed, unless the weight type `X` uses
    /// allocation internally.
    ///
    /// In case of error, `self` is not modified. Error cases:
    /// -   [`WeightError::InvalidInput`] when `new_weights` are not ordered by
    ///     index or an index is too large.
    /// -   [`WeightError::InvalidWeight`] when a weight is not-a-number or negative.
    /// -   [`WeightError::InsufficientNonZero`] when the sum of all weights is zero.
    ///     Note that due to floating-point loss of precision, this case is not
    ///     always correctly detected; usage of a fixed-point weight type may be
    ///     preferred.
    ///
    /// Updates take `O(N)` time. If you need to frequently update weights, consider
    /// [`rand_distr::weighted_tree`](https://docs.rs/rand_distr/*/rand_distr/weighted_tree/index.html)
    /// as an alternative where an update is `O(log N)`.
    pub fn update_weights(&mut self, new_weights: &[(usize, &X)]) -> Result<(), WeightError>
    where
        X: for<'a> core::ops::AddAssign<&'a X>
            + for<'a> core::ops::SubAssign<&'a X>
            + Clone
            + Default,
    {
        if new_weights.is_empty() {
            return Ok(());
        }

        let zero = <X as Default>::default();

        let mut total_weight = self.total_weight.clone();

        // Check for errors first, so we don't modify `self` in case something
        // goes wrong.
        let mut prev_i = None;
        for &(i, w) in new_weights {
            if let Some(old_i) = prev_i {
                if old_i >= i {
                    return Err(WeightError::InvalidInput);
                }
            }
            if !(*w >= zero) {
                return Err(WeightError::InvalidWeight);
            }
            if i > self.cumulative_weights.len() {
                return Err(WeightError::InvalidInput);
            }

            let mut old_w = if i < self.cumulative_weights.len() {
                self.cumulative_weights[i].clone()
            } else {
                self.total_weight.clone()
            };
            if i > 0 {
                old_w -= &self.cumulative_weights[i - 1];
            }

            total_weight -= &old_w;
            total_weight += w;
            prev_i = Some(i);
        }
        if total_weight <= zero {
            return Err(WeightError::InsufficientNonZero);
        }

        // Update the weights. Because we checked all the preconditions in the
        // previous loop, this should never panic.
        let mut iter = new_weights.iter();

        let mut prev_weight = zero.clone();
        let mut next_new_weight = iter.next();
        let &(first_new_index, _) = next_new_weight.unwrap();
        let mut cumulative_weight = if first_new_index > 0 {
            self.cumulative_weights[first_new_index - 1].clone()
        } else {
            zero.clone()
        };
        for i in first_new_index..self.cumulative_weights.len() {
            match next_new_weight {
                Some(&(j, w)) if i == j => {
                    cumulative_weight += w;
                    next_new_weight = iter.next();
                }
                _ => {
                    let mut tmp = self.cumulative_weights[i].clone();
                    tmp -= &prev_weight; // We know this is positive.
                    cumulative_weight += &tmp;
                }
            }
            prev_weight = cumulative_weight.clone();
            core::mem::swap(&mut prev_weight, &mut self.cumulative_weights[i]);
        }

        self.total_weight = total_weight;
        self.weight_distribution = X::Sampler::new(zero, self.total_weight.clone()).unwrap();

        Ok(())
    }
}

/// A lazy-loading iterator over the weights of a `WeightedIndex` distribution.
/// This is returned by [`WeightedIndex::weights`].
pub struct WeightedIndexIter<'a, X: SampleUniform + PartialOrd> {
    weighted_index: &'a WeightedIndex<X>,
    index: usize,
}

impl<X> Debug for WeightedIndexIter<'_, X>
where
    X: SampleUniform + PartialOrd + Debug,
    X::Sampler: Debug,
{
    fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
        f.debug_struct("WeightedIndexIter")
            .field("weighted_index", &self.weighted_index)
            .field("index", &self.index)
            .finish()
    }
}

impl<X> Clone for WeightedIndexIter<'_, X>
where
    X: SampleUniform + PartialOrd,
{
    fn clone(&self) -> Self {
        WeightedIndexIter {
            weighted_index: self.weighted_index,
            index: self.index,
        }
    }
}

impl<X> Iterator for WeightedIndexIter<'_, X>
where
    X: for<'b> core::ops::SubAssign<&'b X> + SampleUniform + PartialOrd + Clone,
{
    type Item = X;

    fn next(&mut self) -> Option<Self::Item> {
        match self.weighted_index.weight(self.index) {
            None => None,
            Some(weight) => {
                self.index += 1;
                Some(weight)
            }
        }
    }
}

impl<X: SampleUniform + PartialOrd + Clone> WeightedIndex<X> {
    /// Returns the weight at the given index, if it exists.
    ///
    /// If the index is out of bounds, this will return `None`.
    ///
    /// # Example
    ///
    /// ```
    /// use rand::distr::WeightedIndex;
    ///
    /// let weights = [0, 1, 2];
    /// let dist = WeightedIndex::new(&weights).unwrap();
    /// assert_eq!(dist.weight(0), Some(0));
    /// assert_eq!(dist.weight(1), Some(1));
    /// assert_eq!(dist.weight(2), Some(2));
    /// assert_eq!(dist.weight(3), None);
    /// ```
    pub fn weight(&self, index: usize) -> Option<X>
    where
        X: for<'a> core::ops::SubAssign<&'a X>,
    {
        use core::cmp::Ordering::*;

        let mut weight = match index.cmp(&self.cumulative_weights.len()) {
            Less => self.cumulative_weights[index].clone(),
            Equal => self.total_weight.clone(),
            Greater => return None,
        };

        if index > 0 {
            weight -= &self.cumulative_weights[index - 1];
        }
        Some(weight)
    }

    /// Returns a lazy-loading iterator containing the current weights of this distribution.
    ///
    /// If this distribution has not been updated since its creation, this will return the
    /// same weights as were passed to `new`.
    ///
    /// # Example
    ///
    /// ```
    /// use rand::distr::WeightedIndex;
    ///
    /// let weights = [1, 2, 3];
    /// let mut dist = WeightedIndex::new(&weights).unwrap();
    /// assert_eq!(dist.weights().collect::<Vec<_>>(), vec![1, 2, 3]);
    /// dist.update_weights(&[(0, &2)]).unwrap();
    /// assert_eq!(dist.weights().collect::<Vec<_>>(), vec![2, 2, 3]);
    /// ```
    pub fn weights(&self) -> WeightedIndexIter<'_, X>
    where
        X: for<'a> core::ops::SubAssign<&'a X>,
    {
        WeightedIndexIter {
            weighted_index: self,
            index: 0,
        }
    }

    /// Returns the sum of all weights in this distribution.
    pub fn total_weight(&self) -> X {
        self.total_weight.clone()
    }
}

impl<X> Distribution<usize> for WeightedIndex<X>
where
    X: SampleUniform + PartialOrd,
{
    fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> usize {
        let chosen_weight = self.weight_distribution.sample(rng);
        // Find the first item which has a weight *higher* than the chosen weight.
        self.cumulative_weights
            .partition_point(|w| w <= &chosen_weight)
    }
}

/// Bounds on a weight
///
/// See usage in [`WeightedIndex`].
pub trait Weight: Clone {
    /// Representation of 0
    const ZERO: Self;

    /// Checked addition
    ///
    /// -   `Result::Ok`: On success, `v` is added to `self`
    /// -   `Result::Err`: Returns an error when `Self` cannot represent the
    ///     result of `self + v` (i.e. overflow). The value of `self` should be
    ///     discarded.
    #[allow(clippy::result_unit_err)]
    fn checked_add_assign(&mut self, v: &Self) -> Result<(), ()>;
}

macro_rules! impl_weight_int {
    ($t:ty) => {
        impl Weight for $t {
            const ZERO: Self = 0;
            fn checked_add_assign(&mut self, v: &Self) -> Result<(), ()> {
                match self.checked_add(*v) {
                    Some(sum) => {
                        *self = sum;
                        Ok(())
                    }
                    None => Err(()),
                }
            }
        }
    };
    ($t:ty, $($tt:ty),*) => {
        impl_weight_int!($t);
        impl_weight_int!($($tt),*);
    }
}
impl_weight_int!(i8, i16, i32, i64, i128, isize);
impl_weight_int!(u8, u16, u32, u64, u128, usize);

macro_rules! impl_weight_float {
    ($t:ty) => {
        impl Weight for $t {
            const ZERO: Self = 0.0;

            fn checked_add_assign(&mut self, v: &Self) -> Result<(), ()> {
                // Floats have an explicit representation for overflow
                *self += *v;
                Ok(())
            }
        }
    };
}
impl_weight_float!(f32);
impl_weight_float!(f64);

#[cfg(test)]
mod test {
    use super::*;

    #[cfg(feature = "serde")]
    #[test]
    fn test_weightedindex_serde() {
        let weighted_index = WeightedIndex::new([1, 2, 3, 4, 5, 6, 7, 8, 9, 10]).unwrap();

        let ser_weighted_index = bincode::serialize(&weighted_index).unwrap();
        let de_weighted_index: WeightedIndex<i32> =
            bincode::deserialize(&ser_weighted_index).unwrap();

        assert_eq!(
            de_weighted_index.cumulative_weights,
            weighted_index.cumulative_weights
        );
        assert_eq!(de_weighted_index.total_weight, weighted_index.total_weight);
    }

    #[test]
    fn test_accepting_nan() {
        assert_eq!(
            WeightedIndex::new([f32::NAN, 0.5]).unwrap_err(),
            WeightError::InvalidWeight,
        );
        assert_eq!(
            WeightedIndex::new([f32::NAN]).unwrap_err(),
            WeightError::InvalidWeight,
        );
        assert_eq!(
            WeightedIndex::new([0.5, f32::NAN]).unwrap_err(),
            WeightError::InvalidWeight,
        );

        assert_eq!(
            WeightedIndex::new([0.5, 7.0])
                .unwrap()
                .update_weights(&[(0, &f32::NAN)])
                .unwrap_err(),
            WeightError::InvalidWeight,
        )
    }

    #[test]
    #[cfg_attr(miri, ignore)] // Miri is too slow
    fn test_weightedindex() {
        let mut r = crate::test::rng(700);
        const N_REPS: u32 = 5000;
        let weights = [1u32, 2, 3, 0, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7];
        let total_weight = weights.iter().sum::<u32>() as f32;

        let verify = |result: [i32; 14]| {
            for (i, count) in result.iter().enumerate() {
                let exp = (weights[i] * N_REPS) as f32 / total_weight;
                let mut err = (*count as f32 - exp).abs();
                if err != 0.0 {
                    err /= exp;
                }
                assert!(err <= 0.25);
            }
        };

        // WeightedIndex from vec
        let mut chosen = [0i32; 14];
        let distr = WeightedIndex::new(weights.to_vec()).unwrap();
        for _ in 0..N_REPS {
            chosen[distr.sample(&mut r)] += 1;
        }
        verify(chosen);

        // WeightedIndex from slice
        chosen = [0i32; 14];
        let distr = WeightedIndex::new(&weights[..]).unwrap();
        for _ in 0..N_REPS {
            chosen[distr.sample(&mut r)] += 1;
        }
        verify(chosen);

        // WeightedIndex from iterator
        chosen = [0i32; 14];
        let distr = WeightedIndex::new(weights.iter()).unwrap();
        for _ in 0..N_REPS {
            chosen[distr.sample(&mut r)] += 1;
        }
        verify(chosen);

        for _ in 0..5 {
            assert_eq!(WeightedIndex::new([0, 1]).unwrap().sample(&mut r), 1);
            assert_eq!(WeightedIndex::new([1, 0]).unwrap().sample(&mut r), 0);
            assert_eq!(
                WeightedIndex::new([0, 0, 0, 0, 10, 0])
                    .unwrap()
                    .sample(&mut r),
                4
            );
        }

        assert_eq!(
            WeightedIndex::new(&[10][0..0]).unwrap_err(),
            WeightError::InvalidInput
        );
        assert_eq!(
            WeightedIndex::new([0]).unwrap_err(),
            WeightError::InsufficientNonZero
        );
        assert_eq!(
            WeightedIndex::new([10, 20, -1, 30]).unwrap_err(),
            WeightError::InvalidWeight
        );
        assert_eq!(
            WeightedIndex::new([-10, 20, 1, 30]).unwrap_err(),
            WeightError::InvalidWeight
        );
        assert_eq!(
            WeightedIndex::new([-10]).unwrap_err(),
            WeightError::InvalidWeight
        );
    }

    #[test]
    fn test_update_weights() {
        let data = [
            (
                &[10u32, 2, 3, 4][..],
                &[(1, &100), (2, &4)][..], // positive change
                &[10, 100, 4, 4][..],
            ),
            (
                &[1u32, 2, 3, 0, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7][..],
                &[(2, &1), (5, &1), (13, &100)][..], // negative change and last element
                &[1u32, 2, 1, 0, 5, 1, 7, 1, 2, 3, 4, 5, 6, 100][..],
            ),
        ];

        for (weights, update, expected_weights) in data.iter() {
            let total_weight = weights.iter().sum::<u32>();
            let mut distr = WeightedIndex::new(weights.to_vec()).unwrap();
            assert_eq!(distr.total_weight, total_weight);

            distr.update_weights(update).unwrap();
            let expected_total_weight = expected_weights.iter().sum::<u32>();
            let expected_distr = WeightedIndex::new(expected_weights.to_vec()).unwrap();
            assert_eq!(distr.total_weight, expected_total_weight);
            assert_eq!(distr.total_weight, expected_distr.total_weight);
            assert_eq!(distr.cumulative_weights, expected_distr.cumulative_weights);
        }
    }

    #[test]
    fn test_update_weights_errors() {
        let data = [
            (
                &[1i32, 0, 0][..],
                &[(0, &0)][..],
                WeightError::InsufficientNonZero,
            ),
            (
                &[10, 10, 10, 10][..],
                &[(1, &-11)][..],
                WeightError::InvalidWeight, // A weight is negative
            ),
            (
                &[1, 2, 3, 4, 5][..],
                &[(1, &5), (0, &5)][..], // Wrong order
                WeightError::InvalidInput,
            ),
            (
                &[1][..],
                &[(1, &1)][..], // Index too large
                WeightError::InvalidInput,
            ),
        ];

        for (weights, update, err) in data.iter() {
            let total_weight = weights.iter().sum::<i32>();
            let mut distr = WeightedIndex::new(weights.to_vec()).unwrap();
            assert_eq!(distr.total_weight, total_weight);
            match distr.update_weights(update) {
                Ok(_) => panic!("Expected update_weights to fail, but it succeeded"),
                Err(e) => assert_eq!(e, *err),
            }
        }
    }

    #[test]
    fn test_weight_at() {
        let data = [
            &[1][..],
            &[10, 2, 3, 4][..],
            &[1, 2, 3, 0, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7][..],
            &[u32::MAX][..],
        ];

        for weights in data.iter() {
            let distr = WeightedIndex::new(weights.to_vec()).unwrap();
            for (i, weight) in weights.iter().enumerate() {
                assert_eq!(distr.weight(i), Some(*weight));
            }
            assert_eq!(distr.weight(weights.len()), None);
        }
    }

    #[test]
    fn test_weights() {
        let data = [
            &[1][..],
            &[10, 2, 3, 4][..],
            &[1, 2, 3, 0, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7][..],
            &[u32::MAX][..],
        ];

        for weights in data.iter() {
            let distr = WeightedIndex::new(weights.to_vec()).unwrap();
            assert_eq!(distr.weights().collect::<Vec<_>>(), weights.to_vec());
        }
    }

    #[test]
    fn value_stability() {
        fn test_samples<X: Weight + SampleUniform + PartialOrd, I>(
            weights: I,
            buf: &mut [usize],
            expected: &[usize],
        ) where
            I: IntoIterator,
            I::Item: SampleBorrow<X>,
        {
            assert_eq!(buf.len(), expected.len());
            let distr = WeightedIndex::new(weights).unwrap();
            let mut rng = crate::test::rng(701);
            for r in buf.iter_mut() {
                *r = rng.sample(&distr);
            }
            assert_eq!(buf, expected);
        }

        let mut buf = [0; 10];
        test_samples(
            [1i32, 1, 1, 1, 1, 1, 1, 1, 1],
            &mut buf,
            &[0, 6, 2, 6, 3, 4, 7, 8, 2, 5],
        );
        test_samples(
            [0.7f32, 0.1, 0.1, 0.1],
            &mut buf,
            &[0, 0, 0, 1, 0, 0, 2, 3, 0, 0],
        );
        test_samples(
            [1.0f64, 0.999, 0.998, 0.997],
            &mut buf,
            &[2, 2, 1, 3, 2, 1, 3, 3, 2, 1],
        );
    }

    #[test]
    fn weighted_index_distributions_can_be_compared() {
        assert_eq!(WeightedIndex::new([1, 2]), WeightedIndex::new([1, 2]));
    }

    #[test]
    fn overflow() {
        assert_eq!(
            WeightedIndex::new([2, usize::MAX]),
            Err(WeightError::Overflow)
        );
    }
}

/// Errors returned by [`WeightedIndex::new`], [`WeightedIndex::update_weights`] and other weighted distributions
#[derive(Debug, Clone, Copy, PartialEq, Eq)]
// Marked non_exhaustive to allow a new error code in the solution to #1476.
#[non_exhaustive]
pub enum WeightError {
    /// The input weight sequence is empty, too long, or wrongly ordered
    InvalidInput,

    /// A weight is negative, too large for the distribution, or not a valid number
    InvalidWeight,

    /// Not enough non-zero weights are available to sample values
    ///
    /// When attempting to sample a single value this implies that all weights
    /// are zero. When attempting to sample `amount` values this implies that
    /// less than `amount` weights are greater than zero.
    InsufficientNonZero,

    /// Overflow when calculating the sum of weights
    Overflow,
}

#[cfg(feature = "std")]
impl std::error::Error for WeightError {}

impl fmt::Display for WeightError {
    fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
        f.write_str(match *self {
            WeightError::InvalidInput => "Weights sequence is empty/too long/unordered",
            WeightError::InvalidWeight => "A weight is negative, too large or not a valid number",
            WeightError::InsufficientNonZero => "Not enough weights > zero",
            WeightError::Overflow => "Overflow when summing weights",
        })
    }
}