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+

// 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.
+
+//! Low-level API for sampling indices
+
+#[cfg(feature = "alloc")] use core::slice;
+
+#[cfg(feature = "alloc")] use alloc::vec::{self, Vec};
+// BTreeMap is not as fast in tests, but better than nothing.
+#[cfg(all(feature = "alloc", not(feature = "std")))]
+use alloc::collections::BTreeSet;
+#[cfg(feature = "std")] use std::collections::HashSet;
+
+#[cfg(feature = "std")]
+use crate::distributions::WeightedError;
+
+#[cfg(feature = "alloc")]
+use crate::{Rng, distributions::{uniform::SampleUniform, Distribution, Uniform}};
+
+#[cfg(feature = "serde1")]
+use serde::{Serialize, Deserialize};
+
+/// A vector of indices.
+///
+/// Multiple internal representations are possible.
+#[derive(Clone, Debug)]
+#[cfg_attr(feature = "serde1", derive(Serialize, Deserialize))]
+pub enum IndexVec {
+    #[doc(hidden)]
+    U32(Vec<u32>),
+    #[doc(hidden)]
+    USize(Vec<usize>),
+}
+
+impl IndexVec {
+    /// Returns the number of indices
+    #[inline]
+    pub fn len(&self) -> usize {
+        match *self {
+            IndexVec::U32(ref v) => v.len(),
+            IndexVec::USize(ref v) => v.len(),
+        }
+    }
+
+    /// Returns `true` if the length is 0.
+    #[inline]
+    pub fn is_empty(&self) -> bool {
+        match *self {
+            IndexVec::U32(ref v) => v.is_empty(),
+            IndexVec::USize(ref v) => v.is_empty(),
+        }
+    }
+
+    /// Return the value at the given `index`.
+    ///
+    /// (Note: we cannot implement [`std::ops::Index`] because of lifetime
+    /// restrictions.)
+    #[inline]
+    pub fn index(&self, index: usize) -> usize {
+        match *self {
+            IndexVec::U32(ref v) => v[index] as usize,
+            IndexVec::USize(ref v) => v[index],
+        }
+    }
+
+    /// Return result as a `Vec<usize>`. Conversion may or may not be trivial.
+    #[inline]
+    pub fn into_vec(self) -> Vec<usize> {
+        match self {
+            IndexVec::U32(v) => v.into_iter().map(|i| i as usize).collect(),
+            IndexVec::USize(v) => v,
+        }
+    }
+
+    /// Iterate over the indices as a sequence of `usize` values
+    #[inline]
+    pub fn iter(&self) -> IndexVecIter<'_> {
+        match *self {
+            IndexVec::U32(ref v) => IndexVecIter::U32(v.iter()),
+            IndexVec::USize(ref v) => IndexVecIter::USize(v.iter()),
+        }
+    }
+}
+
+impl IntoIterator for IndexVec {
+    type Item = usize;
+    type IntoIter = IndexVecIntoIter;
+
+    /// Convert into an iterator over the indices as a sequence of `usize` values
+    #[inline]
+    fn into_iter(self) -> IndexVecIntoIter {
+        match self {
+            IndexVec::U32(v) => IndexVecIntoIter::U32(v.into_iter()),
+            IndexVec::USize(v) => IndexVecIntoIter::USize(v.into_iter()),
+        }
+    }
+}
+
+impl PartialEq for IndexVec {
+    fn eq(&self, other: &IndexVec) -> bool {
+        use self::IndexVec::*;
+        match (self, other) {
+            (&U32(ref v1), &U32(ref v2)) => v1 == v2,
+            (&USize(ref v1), &USize(ref v2)) => v1 == v2,
+            (&U32(ref v1), &USize(ref v2)) => {
+                (v1.len() == v2.len()) && (v1.iter().zip(v2.iter()).all(|(x, y)| *x as usize == *y))
+            }
+            (&USize(ref v1), &U32(ref v2)) => {
+                (v1.len() == v2.len()) && (v1.iter().zip(v2.iter()).all(|(x, y)| *x == *y as usize))
+            }
+        }
+    }
+}
+
+impl From<Vec<u32>> for IndexVec {
+    #[inline]
+    fn from(v: Vec<u32>) -> Self {
+        IndexVec::U32(v)
+    }
+}
+
+impl From<Vec<usize>> for IndexVec {
+    #[inline]
+    fn from(v: Vec<usize>) -> Self {
+        IndexVec::USize(v)
+    }
+}
+
+/// Return type of `IndexVec::iter`.
+#[derive(Debug)]
+pub enum IndexVecIter<'a> {
+    #[doc(hidden)]
+    U32(slice::Iter<'a, u32>),
+    #[doc(hidden)]
+    USize(slice::Iter<'a, usize>),
+}
+
+impl<'a> Iterator for IndexVecIter<'a> {
+    type Item = usize;
+
+    #[inline]
+    fn next(&mut self) -> Option<usize> {
+        use self::IndexVecIter::*;
+        match *self {
+            U32(ref mut iter) => iter.next().map(|i| *i as usize),
+            USize(ref mut iter) => iter.next().cloned(),
+        }
+    }
+
+    #[inline]
+    fn size_hint(&self) -> (usize, Option<usize>) {
+        match *self {
+            IndexVecIter::U32(ref v) => v.size_hint(),
+            IndexVecIter::USize(ref v) => v.size_hint(),
+        }
+    }
+}
+
+impl<'a> ExactSizeIterator for IndexVecIter<'a> {}
+
+/// Return type of `IndexVec::into_iter`.
+#[derive(Clone, Debug)]
+pub enum IndexVecIntoIter {
+    #[doc(hidden)]
+    U32(vec::IntoIter<u32>),
+    #[doc(hidden)]
+    USize(vec::IntoIter<usize>),
+}
+
+impl Iterator for IndexVecIntoIter {
+    type Item = usize;
+
+    #[inline]
+    fn next(&mut self) -> Option<Self::Item> {
+        use self::IndexVecIntoIter::*;
+        match *self {
+            U32(ref mut v) => v.next().map(|i| i as usize),
+            USize(ref mut v) => v.next(),
+        }
+    }
+
+    #[inline]
+    fn size_hint(&self) -> (usize, Option<usize>) {
+        use self::IndexVecIntoIter::*;
+        match *self {
+            U32(ref v) => v.size_hint(),
+            USize(ref v) => v.size_hint(),
+        }
+    }
+}
+
+impl ExactSizeIterator for IndexVecIntoIter {}
+
+
+/// Randomly sample exactly `amount` distinct indices from `0..length`, and
+/// return them in random order (fully shuffled).
+///
+/// This method is used internally by the slice sampling methods, but it can
+/// sometimes be useful to have the indices themselves so this is provided as
+/// an alternative.
+///
+/// The implementation used is not specified; we automatically select the
+/// fastest available algorithm for the `length` and `amount` parameters
+/// (based on detailed profiling on an Intel Haswell CPU). Roughly speaking,
+/// complexity is `O(amount)`, except that when `amount` is small, performance
+/// is closer to `O(amount^2)`, and when `length` is close to `amount` then
+/// `O(length)`.
+///
+/// Note that performance is significantly better over `u32` indices than over
+/// `u64` indices. Because of this we hide the underlying type behind an
+/// abstraction, `IndexVec`.
+///
+/// If an allocation-free `no_std` function is required, it is suggested
+/// to adapt the internal `sample_floyd` implementation.
+///
+/// Panics if `amount > length`.
+pub fn sample<R>(rng: &mut R, length: usize, amount: usize) -> IndexVec
+where R: Rng + ?Sized {
+    if amount > length {
+        panic!("`amount` of samples must be less than or equal to `length`");
+    }
+    if length > (::core::u32::MAX as usize) {
+        // We never want to use inplace here, but could use floyd's alg
+        // Lazy version: always use the cache alg.
+        return sample_rejection(rng, length, amount);
+    }
+    let amount = amount as u32;
+    let length = length as u32;
+
+    // Choice of algorithm here depends on both length and amount. See:
+    // https://github.com/rust-random/rand/pull/479
+    // We do some calculations with f32. Accuracy is not very important.
+
+    if amount < 163 {
+        const C: [[f32; 2]; 2] = [[1.6, 8.0 / 45.0], [10.0, 70.0 / 9.0]];
+        let j = if length < 500_000 { 0 } else { 1 };
+        let amount_fp = amount as f32;
+        let m4 = C[0][j] * amount_fp;
+        // Short-cut: when amount < 12, floyd's is always faster
+        if amount > 11 && (length as f32) < (C[1][j] + m4) * amount_fp {
+            sample_inplace(rng, length, amount)
+        } else {
+            sample_floyd(rng, length, amount)
+        }
+    } else {
+        const C: [f32; 2] = [270.0, 330.0 / 9.0];
+        let j = if length < 500_000 { 0 } else { 1 };
+        if (length as f32) < C[j] * (amount as f32) {
+            sample_inplace(rng, length, amount)
+        } else {
+            sample_rejection(rng, length, amount)
+        }
+    }
+}
+
+/// Randomly sample exactly `amount` distinct indices from `0..length`, and
+/// return them in an arbitrary order (there is no guarantee of shuffling or
+/// ordering). The weights are to be provided by the input function `weights`,
+/// which will be called once for each index.
+///
+/// This method is used internally by the slice sampling methods, but it can
+/// sometimes be useful to have the indices themselves so this is provided as
+/// an alternative.
+///
+/// This implementation uses `O(length + amount)` space and `O(length)` time
+/// if the "nightly" feature is enabled, or `O(length)` space and
+/// `O(length + amount * log length)` time otherwise.
+///
+/// Panics if `amount > length`.
+#[cfg(feature = "std")]
+#[cfg_attr(doc_cfg, doc(cfg(feature = "std")))]
+pub fn sample_weighted<R, F, X>(
+    rng: &mut R, length: usize, weight: F, amount: usize,
+) -> Result<IndexVec, WeightedError>
+where
+    R: Rng + ?Sized,
+    F: Fn(usize) -> X,
+    X: Into<f64>,
+{
+    if length > (core::u32::MAX as usize) {
+        sample_efraimidis_spirakis(rng, length, weight, amount)
+    } else {
+        assert!(amount <= core::u32::MAX as usize);
+        let amount = amount as u32;
+        let length = length as u32;
+        sample_efraimidis_spirakis(rng, length, weight, amount)
+    }
+}
+
+
+/// Randomly sample exactly `amount` distinct indices from `0..length`, and
+/// return them in an arbitrary order (there is no guarantee of shuffling or
+/// ordering). The weights are to be provided by the input function `weights`,
+/// which will be called once for each index.
+///
+/// This implementation uses the algorithm described by Efraimidis and Spirakis
+/// in this paper: https://doi.org/10.1016/j.ipl.2005.11.003
+/// It uses `O(length + amount)` space and `O(length)` time if the
+/// "nightly" feature is enabled, or `O(length)` space and `O(length
+/// + amount * log length)` time otherwise.
+///
+/// Panics if `amount > length`.
+#[cfg(feature = "std")]
+fn sample_efraimidis_spirakis<R, F, X, N>(
+    rng: &mut R, length: N, weight: F, amount: N,
+) -> Result<IndexVec, WeightedError>
+where
+    R: Rng + ?Sized,
+    F: Fn(usize) -> X,
+    X: Into<f64>,
+    N: UInt,
+    IndexVec: From<Vec<N>>,
+{
+    if amount == N::zero() {
+        return Ok(IndexVec::U32(Vec::new()));
+    }
+
+    if amount > length {
+        panic!("`amount` of samples must be less than or equal to `length`");
+    }
+
+    struct Element<N> {
+        index: N,
+        key: f64,
+    }
+    impl<N> PartialOrd for Element<N> {
+        fn partial_cmp(&self, other: &Self) -> Option<core::cmp::Ordering> {
+            self.key.partial_cmp(&other.key)
+        }
+    }
+    impl<N> Ord for Element<N> {
+        fn cmp(&self, other: &Self) -> core::cmp::Ordering {
+             // partial_cmp will always produce a value,
+             // because we check that the weights are not nan
+            self.partial_cmp(other).unwrap()
+        }
+    }
+    impl<N> PartialEq for Element<N> {
+        fn eq(&self, other: &Self) -> bool {
+            self.key == other.key
+        }
+    }
+    impl<N> Eq for Element<N> {}
+
+    #[cfg(feature = "nightly")]
+    {
+        let mut candidates = Vec::with_capacity(length.as_usize());
+        let mut index = N::zero();
+        while index < length {
+            let weight = weight(index.as_usize()).into();
+            if !(weight >= 0.) {
+                return Err(WeightedError::InvalidWeight);
+            }
+
+            let key = rng.gen::<f64>().powf(1.0 / weight);
+            candidates.push(Element { index, key });
+
+            index += N::one();
+        }
+
+        // Partially sort the array to find the `amount` elements with the greatest
+        // keys. Do this by using `select_nth_unstable` to put the elements with
+        // the *smallest* keys at the beginning of the list in `O(n)` time, which
+        // provides equivalent information about the elements with the *greatest* keys.
+        let (_, mid, greater)
+            = candidates.select_nth_unstable(length.as_usize() - amount.as_usize());
+
+        let mut result: Vec<N> = Vec::with_capacity(amount.as_usize());
+        result.push(mid.index);
+        for element in greater {
+            result.push(element.index);
+        }
+        Ok(IndexVec::from(result))
+    }
+
+    #[cfg(not(feature = "nightly"))]
+    {
+        use alloc::collections::BinaryHeap;
+
+        // Partially sort the array such that the `amount` elements with the largest
+        // keys are first using a binary max heap.
+        let mut candidates = BinaryHeap::with_capacity(length.as_usize());
+        let mut index = N::zero();
+        while index < length {
+            let weight = weight(index.as_usize()).into();
+            if !(weight >= 0.) {
+                return Err(WeightedError::InvalidWeight);
+            }
+
+            let key = rng.gen::<f64>().powf(1.0 / weight);
+            candidates.push(Element { index, key });
+
+            index += N::one();
+        }
+
+        let mut result: Vec<N> = Vec::with_capacity(amount.as_usize());
+        while result.len() < amount.as_usize() {
+            result.push(candidates.pop().unwrap().index);
+        }
+        Ok(IndexVec::from(result))
+    }
+}
+
+/// Randomly sample exactly `amount` indices from `0..length`, using Floyd's
+/// combination algorithm.
+///
+/// The output values are fully shuffled. (Overhead is under 50%.)
+///
+/// This implementation uses `O(amount)` memory and `O(amount^2)` time.
+fn sample_floyd<R>(rng: &mut R, length: u32, amount: u32) -> IndexVec
+where R: Rng + ?Sized {
+    // For small amount we use Floyd's fully-shuffled variant. For larger
+    // amounts this is slow due to Vec::insert performance, so we shuffle
+    // afterwards. Benchmarks show little overhead from extra logic.
+    let floyd_shuffle = amount < 50;
+
+    debug_assert!(amount <= length);
+    let mut indices = Vec::with_capacity(amount as usize);
+    for j in length - amount..length {
+        let t = rng.gen_range(0..=j);
+        if floyd_shuffle {
+            if let Some(pos) = indices.iter().position(|&x| x == t) {
+                indices.insert(pos, j);
+                continue;
+            }
+        } else if indices.contains(&t) {
+            indices.push(j);
+            continue;
+        }
+        indices.push(t);
+    }
+    if !floyd_shuffle {
+        // Reimplement SliceRandom::shuffle with smaller indices
+        for i in (1..amount).rev() {
+            // invariant: elements with index > i have been locked in place.
+            indices.swap(i as usize, rng.gen_range(0..=i) as usize);
+        }
+    }
+    IndexVec::from(indices)
+}
+
+/// Randomly sample exactly `amount` indices from `0..length`, using an inplace
+/// partial Fisher-Yates method.
+/// Sample an amount of indices using an inplace partial fisher yates method.
+///
+/// This allocates the entire `length` of indices and randomizes only the first `amount`.
+/// It then truncates to `amount` and returns.
+///
+/// This method is not appropriate for large `length` and potentially uses a lot
+/// of memory; because of this we only implement for `u32` index (which improves
+/// performance in all cases).
+///
+/// Set-up is `O(length)` time and memory and shuffling is `O(amount)` time.
+fn sample_inplace<R>(rng: &mut R, length: u32, amount: u32) -> IndexVec
+where R: Rng + ?Sized {
+    debug_assert!(amount <= length);
+    let mut indices: Vec<u32> = Vec::with_capacity(length as usize);
+    indices.extend(0..length);
+    for i in 0..amount {
+        let j: u32 = rng.gen_range(i..length);
+        indices.swap(i as usize, j as usize);
+    }
+    indices.truncate(amount as usize);
+    debug_assert_eq!(indices.len(), amount as usize);
+    IndexVec::from(indices)
+}
+
+trait UInt: Copy + PartialOrd + Ord + PartialEq + Eq + SampleUniform
+    + core::hash::Hash + core::ops::AddAssign {
+    fn zero() -> Self;
+    fn one() -> Self;
+    fn as_usize(self) -> usize;
+}
+impl UInt for u32 {
+    #[inline]
+    fn zero() -> Self {
+        0
+    }
+
+    #[inline]
+    fn one() -> Self {
+        1
+    }
+
+    #[inline]
+    fn as_usize(self) -> usize {
+        self as usize
+    }
+}
+impl UInt for usize {
+    #[inline]
+    fn zero() -> Self {
+        0
+    }
+
+    #[inline]
+    fn one() -> Self {
+        1
+    }
+
+    #[inline]
+    fn as_usize(self) -> usize {
+        self
+    }
+}
+
+/// Randomly sample exactly `amount` indices from `0..length`, using rejection
+/// sampling.
+///
+/// Since `amount <<< length` there is a low chance of a random sample in
+/// `0..length` being a duplicate. We test for duplicates and resample where
+/// necessary. The algorithm is `O(amount)` time and memory.
+///
+/// This function  is generic over X primarily so that results are value-stable
+/// over 32-bit and 64-bit platforms.
+fn sample_rejection<X: UInt, R>(rng: &mut R, length: X, amount: X) -> IndexVec
+where
+    R: Rng + ?Sized,
+    IndexVec: From<Vec<X>>,
+{
+    debug_assert!(amount < length);
+    #[cfg(feature = "std")]
+    let mut cache = HashSet::with_capacity(amount.as_usize());
+    #[cfg(not(feature = "std"))]
+    let mut cache = BTreeSet::new();
+    let distr = Uniform::new(X::zero(), length);
+    let mut indices = Vec::with_capacity(amount.as_usize());
+    for _ in 0..amount.as_usize() {
+        let mut pos = distr.sample(rng);
+        while !cache.insert(pos) {
+            pos = distr.sample(rng);
+        }
+        indices.push(pos);
+    }
+
+    debug_assert_eq!(indices.len(), amount.as_usize());
+    IndexVec::from(indices)
+}
+
+#[cfg(test)]
+mod test {
+    use super::*;
+
+    #[test]
+    #[cfg(feature = "serde1")]
+    fn test_serialization_index_vec() {
+        let some_index_vec = IndexVec::from(vec![254_usize, 234, 2, 1]);
+        let de_some_index_vec: IndexVec = bincode::deserialize(&bincode::serialize(&some_index_vec).unwrap()).unwrap();
+        match (some_index_vec, de_some_index_vec) {
+            (IndexVec::U32(a), IndexVec::U32(b)) => {
+                assert_eq!(a, b);
+            },
+            (IndexVec::USize(a), IndexVec::USize(b)) => {
+                assert_eq!(a, b);
+            },
+            _ => {panic!("failed to seralize/deserialize `IndexVec`")}
+        }
+    }
+
+    #[cfg(feature = "alloc")] use alloc::vec;
+
+    #[test]
+    fn test_sample_boundaries() {
+        let mut r = crate::test::rng(404);
+
+        assert_eq!(sample_inplace(&mut r, 0, 0).len(), 0);
+        assert_eq!(sample_inplace(&mut r, 1, 0).len(), 0);
+        assert_eq!(sample_inplace(&mut r, 1, 1).into_vec(), vec![0]);
+
+        assert_eq!(sample_rejection(&mut r, 1u32, 0).len(), 0);
+
+        assert_eq!(sample_floyd(&mut r, 0, 0).len(), 0);
+        assert_eq!(sample_floyd(&mut r, 1, 0).len(), 0);
+        assert_eq!(sample_floyd(&mut r, 1, 1).into_vec(), vec![0]);
+
+        // These algorithms should be fast with big numbers. Test average.
+        let sum: usize = sample_rejection(&mut r, 1 << 25, 10u32).into_iter().sum();
+        assert!(1 << 25 < sum && sum < (1 << 25) * 25);
+
+        let sum: usize = sample_floyd(&mut r, 1 << 25, 10).into_iter().sum();
+        assert!(1 << 25 < sum && sum < (1 << 25) * 25);
+    }
+
+    #[test]
+    #[cfg_attr(miri, ignore)] // Miri is too slow
+    fn test_sample_alg() {
+        let seed_rng = crate::test::rng;
+
+        // We can't test which algorithm is used directly, but Floyd's alg
+        // should produce different results from the others. (Also, `inplace`
+        // and `cached` currently use different sizes thus produce different results.)
+
+        // A small length and relatively large amount should use inplace
+        let (length, amount): (usize, usize) = (100, 50);
+        let v1 = sample(&mut seed_rng(420), length, amount);
+        let v2 = sample_inplace(&mut seed_rng(420), length as u32, amount as u32);
+        assert!(v1.iter().all(|e| e < length));
+        assert_eq!(v1, v2);
+
+        // Test Floyd's alg does produce different results
+        let v3 = sample_floyd(&mut seed_rng(420), length as u32, amount as u32);
+        assert!(v1 != v3);
+
+        // A large length and small amount should use Floyd
+        let (length, amount): (usize, usize) = (1 << 20, 50);
+        let v1 = sample(&mut seed_rng(421), length, amount);
+        let v2 = sample_floyd(&mut seed_rng(421), length as u32, amount as u32);
+        assert!(v1.iter().all(|e| e < length));
+        assert_eq!(v1, v2);
+
+        // A large length and larger amount should use cache
+        let (length, amount): (usize, usize) = (1 << 20, 600);
+        let v1 = sample(&mut seed_rng(422), length, amount);
+        let v2 = sample_rejection(&mut seed_rng(422), length as u32, amount as u32);
+        assert!(v1.iter().all(|e| e < length));
+        assert_eq!(v1, v2);
+    }
+
+    #[cfg(feature = "std")]
+    #[test]
+    fn test_sample_weighted() {
+        let seed_rng = crate::test::rng;
+        for &(amount, len) in &[(0, 10), (5, 10), (10, 10)] {
+            let v = sample_weighted(&mut seed_rng(423), len, |i| i as f64, amount).unwrap();
+            match v {
+                IndexVec::U32(mut indices) => {
+                    assert_eq!(indices.len(), amount);
+                    indices.sort_unstable();
+                    indices.dedup();
+                    assert_eq!(indices.len(), amount);
+                    for &i in &indices {
+                        assert!((i as usize) < len);
+                    }
+                },
+                IndexVec::USize(_) => panic!("expected `IndexVec::U32`"),
+            }
+        }
+    }
+
+    #[test]
+    fn value_stability_sample() {
+        let do_test = |length, amount, values: &[u32]| {
+            let mut buf = [0u32; 8];
+            let mut rng = crate::test::rng(410);
+
+            let res = sample(&mut rng, length, amount);
+            let len = res.len().min(buf.len());
+            for (x, y) in res.into_iter().zip(buf.iter_mut()) {
+                *y = x as u32;
+            }
+            assert_eq!(
+                &buf[0..len],
+                values,
+                "failed sampling {}, {}",
+                length,
+                amount
+            );
+        };
+
+        do_test(10, 6, &[8, 0, 3, 5, 9, 6]); // floyd
+        do_test(25, 10, &[18, 15, 14, 9, 0, 13, 5, 24]); // floyd
+        do_test(300, 8, &[30, 283, 150, 1, 73, 13, 285, 35]); // floyd
+        do_test(300, 80, &[31, 289, 248, 154, 5, 78, 19, 286]); // inplace
+        do_test(300, 180, &[31, 289, 248, 154, 5, 78, 19, 286]); // inplace
+
+        do_test(1_000_000, 8, &[
+            103717, 963485, 826422, 509101, 736394, 807035, 5327, 632573,
+        ]); // floyd
+        do_test(1_000_000, 180, &[
+            103718, 963490, 826426, 509103, 736396, 807036, 5327, 632573,
+        ]); // rejection
+    }
+}
+

+