Strided slices.

This library provides two types Strided and MutStrided as generalised forms of &[T] and &mut [T] respectively, where the elements are regularly spaced in memory, but not necessarily immediately adjacently.

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lib.rs:

Strided slices.

This library provides two types Stride and MutStride as generalised forms of &[T] and &mut [T] respectively, where the elements are regularly spaced in memory, but not necessarily immediately adjacently.

For example, given an underlying array [1, 2, 3, 4, 5], the elements [1, 3, 5] are a strided slice with stride 2, and [1, 4] has stride 3. Any slice can be regarded as a strided slice with stride 1.

This provides functionality through which one can safely and efficiently manipulate every nth element of a slice (even a mutable one) as close as possible to it being a conventional slice. This releases one from worries about stride bookkeeping, aliasing of &mut or any unsafe code.

Quick start

The work-horse functions are .substrides(n) and .substrides_mut(n), which return an iterator across a series of n new strided slices (shared and mutable, respectively), each of which points to every nth element, and each of which starts at the next successive offset. For example, the following has n = 3.

use strided::MutStride;

let mut v = [1u8, 2, 3, 4, 5];
let mut all = MutStride::new(&mut v);

let mut substrides = all.substrides_mut(3);

let a = substrides.next().unwrap();
let b = substrides.next().unwrap();
let c = substrides.next().unwrap();
assert!(substrides.next().is_none()); // there was exactly 3.

assert_eq!(a, MutStride::new(&mut [1, 4]));
assert_eq!(b, MutStride::new(&mut [2, 5]));
assert_eq!(c, MutStride::new(&mut [3]));

The common case of n = 2 has an abbreviation substrides2 (resp. substrides2_mut), which takes the liberty of returns a tuple rather than an iterator to make direct destructuring work. Continuing with the values above, left and right point to alternate elements, starting at index 0 and 1 of their parent slice respectively.

let (left, right) = all.substrides2_mut();

assert_eq!(left, MutStride::new(&mut [1, 3, 5]));
assert_eq!(right, MutStride::new(&mut [2, 4]));

A lot of the conventional slice functionality is available, such as indexing, iterators and slicing.

let (mut left, right) = all.substrides2_mut();
assert_eq!(left[2], 5);
assert!(right.get(10).is_none()); // out of bounds

left[2] += 10;
match left.get_mut(0) {
    Some(val) => *val -= 1,
    None => {}
}

assert_eq!(right.iter().fold(0, |sum, a| sum + *a), 2 + 4);
for val in left.iter_mut() {
    *val /= 2
}

Ownership and reborrow

MutStride has a method reborrow which has signature

impl<'a, T> MutStride<'a, T> {
    pub fn reborrow<'b>(&'b mut self) -> MutStride<'b, T> { ... }
}

That is, it allows temporarily viewing a strided slices as one with a shorter lifetime. This method is key because many of the methods on MutStride take self by-value and so consume ownership... which is rather unfortunate if one wants to use a strided slice multiple times.

The temporary returned by reborrow can be used with the consuming methods, which allows the parent slice to continuing being used after that temporary has disappeared. For example, all of the splitting and slicing methods on MutStride consume ownership, and so reborrow is necessary there to continue using, in this case, left.

let (mut left, right) = all.substrides2_mut();
assert_eq!(left.reborrow().slice_mut(1, 3), MutStride::new(&mut [3, 5]));
assert_eq!(left.reborrow().slice_from_mut(2), MutStride::new(&mut [5]));
assert_eq!(left.reborrow().slice_to_mut(2), MutStride::new(&mut [1, 3]));

// no reborrow:
assert_eq!(right.split_at_mut(1),
           (MutStride::new(&mut [2]), MutStride::new(&mut [4])));
// println!("{}", right); // error: use of moved value `right`.

These contortions are necessary to ensure that &muts cannot alias, while still maintaining flexibility: leaving elements with the maximum possible lifetime (i.e. that of the non-strided slices which they lie in). Theoretically they are necessary with &mut [] too, but the compiler inserts implicit reborrows and so one rarely needs to do them manually.

In practice, one should only need to insert reborrows if the compiler complains about the use of a moved value.

The shared Stride is equivalent to &[] and only handles & references, making ownership transfer and reborrow unnecessary, so all its methods act identically to those on &[].

Example

The fast Fourier transform (FFT) is a signal processing algorithm that performs a discrete Fourier transform (DFT) of length n in O(n log n) time. A DFT breaks a waveform into the sum of sines and cosines, and is an important part of many other algorithms due to certain nice properties of the Fourier transform.

The first FFT algorithm was the Cooley-Tukey algorithm. The decimation-in-time variant works by computing the FFT of equal-length subarrays of equally spaced elements and then combining these together into the desired result. This sort of spacing is exactly the striding provided by this library, and hence this library can be used to create an FFT algorithm in a very natural way.

Below is an implementation of the radix-2 case, that is, when the length n is a power of two. In this case, only two strided subarrays are necessary: exactly the alternating ones provided by substrides2. Note the use of reborrow to allow start and end to be used for the recursive fft calls and then again later in the loop.

extern crate strided;
extern crate num; // https://github.com/rust-lang/num
use std::f64;
use num::complex::{Complex, Complex64};
use strided::{MutStride, Stride};

/// Writes the forward DFT of `input` to `output`.
fn fft(input: Stride<Complex64>, mut output: MutStride<Complex64>) {
    // check it's a power of two.
    assert!(input.len() == output.len() && input.len().count_ones() == 1);

    // base case: the DFT of a single element is itself.
    if input.len() == 1 {
        output[0] = input[0];
        return
    }

    // split the input into two arrays of alternating elements ("decimate in time")
    let (evens, odds) = input.substrides2();
    // break the output into two halves (front and back, not alternating)
    let (mut start, mut end) = output.split_at_mut(input.len() / 2);

    // recursively perform two FFTs on alternating elements of the input, writing the
    // results into the first and second half of the output array respectively.
    fft(evens, start.reborrow());
    fft(odds, end.reborrow());

    // exp(-2πi/N)
    let twiddle = Complex::from_polar(&1.0, &(-2.0 * f64::consts::PI / input.len() as f64));

    let mut factor = Complex::new(1., 0.);

    // combine the subFFTs with the relations:
    //   X_k       = E_k + exp(-2πki/N) * O_k
    //   X_{k+N/2} = E_k - exp(-2πki/N) * O_k
    for (even, odd) in start.iter_mut().zip(end.iter_mut()) {
        let twiddled = factor * *odd;
        let e = *even;

        *even = e + twiddled;
        *odd = e - twiddled;
        factor = factor * twiddle;
    }
}

fn main() {
    let a = [Complex::new(2., 0.), Complex::new(1., 0.),
             Complex::new(2., 0.), Complex::new(1., 0.)];
    let mut b = [Complex::new(0., 0.); 4];

    fft(Stride::new(&a), MutStride::new(&mut b));
    println!("forward: {:?} -> {:?}", &a, &b);
}

The above definitely has complexity O(n log n), but it has a much larger constant factor than an optimised library like FFTW. (Strictly speaking output does not need to be a strided slice, since it is never split into alternating elements.)