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299/** Algorithms for converting 2D coordinates to and from the Hilbert index.
Here the Hilbert curve has been scaled and discretized, so that the
range {0, 1, ..., n^2 - 1} is mapped to coordinates
{0, 1, ..., n-1} x {0, 1, ..., n-1}. In the classical Hilbert curve,
the continuous interval [0,1] is mapped to the unit square [0,1]^2.
*/
use rand::{distributions::Uniform, Rng};
use rand_chacha::ChaCha8Rng;
use std::collections::VecDeque;
pub type Coordinates = (usize, usize);
type Matrix = Vec<i32>;
pub type Vector = Vec<i32>;
#[inline]
pub fn log2(n: usize) -> usize {
(n as f64).log2().floor() as usize
}
/// Create a matrix.
/// note that the representation Vec of Vec is not optimal.
pub fn make_matrix<R: rand::Rng>(n: usize, low: i32, high: i32, rng: &mut R) -> Matrix {
let range = Uniform::new(low, high);
(0..(n * n)).map(|_| rng.sample(&range)).collect()
}
/// Naive product
#[allow(non_snake_case)]
pub fn naive_matrix_vector_product(A: &Matrix, v: &Vector, output: &mut Vector, n: usize) {
// // TODO: put asserts here to make sure no bounds checking happens.
// assert_eq!(output.len(), n);
// assert_eq!(A.len(), n * n);
// assert_eq!(v.len(), n);
for i in 0..n {
for j in 0..n {
output[i] += A[flat_index(i, j, n)] * v[j];
}
}
}
/// Converts [i][j] into [n*i+j]
#[inline]
fn flat_index(i: usize, j: usize, n: usize) -> usize {
n * i + j
}
/// Flatten matrix A according to the provided Hilbert coordinates.
#[allow(non_snake_case)]
pub fn flatten_matrix(depth: usize, A: Vec<i32>, n: usize) -> Vector {
let mut flattened_A = vec![0; n * n];
for (t, (i, j)) in HilbertIter::new(depth) {
flattened_A[t] = A[flat_index(i, j, n)];
}
flattened_A
}
#[allow(non_snake_case)]
pub fn hilbert_matrix_vector_product(
flattened_A: &Vector,
v: &Vector,
output: &mut Vector,
hilbert_iter: &Vec<(usize, Coordinates)>,
) {
for (t, (i, j)) in hilbert_iter {
output[*i] += flattened_A[*t] * v[*j];
}
}
/// `hilbert_matrix_vector_product` but Hilbert index is an iterator.
#[allow(non_snake_case)]
pub fn hilbert_matrix_vector_product_iter(
flattened_A: &Vector,
v: &Vector,
output: &mut Vector,
depth: usize,
) {
for (t, (i, j)) in HilbertIter::new(depth) {
output[i] += flattened_A[t] * v[j];
}
}
struct HilbertIter {
/// Number of steps remaining
n: usize,
index: usize,
i: usize,
j: usize,
queue: VecDeque<(char, usize)>,
buffer: Option<(usize, Coordinates)>,
}
impl HilbertIter {
pub fn new(depth: usize) -> Self {
let n = 2usize.pow(depth as u32);
let n = n * n + 1;
let queue = VecDeque::from([('H', depth)]);
Self {
n,
index: 1,
i: 0,
j: 0,
queue,
buffer: Some((0, (0, 0))),
}
}
fn step(&mut self) {
while self.buffer.is_none() && !self.queue.is_empty() {
let (symbol, depth) = self.queue.pop_front().unwrap();
if depth == 0 {
let non_terminal = match symbol {
'↑' => {
self.i += 1;
true
}
'↓' => {
self.i -= 1;
true
}
'→' => {
self.j += 1;
true
}
'←' => {
self.j -= 1;
true
}
_ => false,
};
if non_terminal {
self.buffer = Some((self.index, (self.i, self.j)));
self.index += 1;
}
}
if depth > 0 {
match symbol {
'H' => {
self.queue.push_back(('A', depth - 1));
self.queue.push_back(('↑', depth - 1));
self.queue.push_back(('H', depth - 1));
self.queue.push_back(('→', depth - 1));
self.queue.push_back(('H', depth - 1));
self.queue.push_back(('↓', depth - 1));
self.queue.push_back(('B', depth - 1));
}
'A' => {
self.queue.push_back(('H', depth - 1));
self.queue.push_back(('→', depth - 1));
self.queue.push_back(('A', depth - 1));
self.queue.push_back(('↑', depth - 1));
self.queue.push_back(('A', depth - 1));
self.queue.push_back(('←', depth - 1));
self.queue.push_back(('C', depth - 1));
}
'B' => {
self.queue.push_back(('C', depth - 1));
self.queue.push_back(('←', depth - 1));
self.queue.push_back(('B', depth - 1));
self.queue.push_back(('↓', depth - 1));
self.queue.push_back(('B', depth - 1));
self.queue.push_back(('→', depth - 1));
self.queue.push_back(('H', depth - 1));
}
'C' => {
self.queue.push_back(('B', depth - 1));
self.queue.push_back(('↓', depth - 1));
self.queue.push_back(('C', depth - 1));
self.queue.push_back(('←', depth - 1));
self.queue.push_back(('C', depth - 1));
self.queue.push_back(('↑', depth - 1));
self.queue.push_back(('A', depth - 1));
}
_ => {
// # terminal up/down/left/right symbols
// # must be preserved until the end
self.queue.push_back((symbol, depth - 1));
}
};
}
}
}
}
impl Iterator for HilbertIter {
type Item = (usize, Coordinates);
fn next(&mut self) -> Option<Self::Item> {
if self.n > 0 {
self.n -= 1;
} else {
panic!("Hilbert iteration is out of bounds");
}
self.step();
self.buffer.take()
}
fn size_hint(&self) -> (usize, Option<usize>) {
(self.n, Some(self.n))
}
}
/// Generate (A, v) as inputs for matrix multiplication
pub fn setup_inputs(n: usize, rng: &mut ChaCha8Rng) -> (Vec<i32>, Vec<i32>) {
let range = Uniform::new(1, 11);
#[allow(non_snake_case)]
let A = make_matrix(n, 1, 11, rng);
let v: Vec<_> = (0..n).map(|_| rng.sample(&range)).collect();
assert_eq!(v.len(), n);
(A, v)
}
/// Setup (coordinates, flattened_A) for Hilbert multiplication
#[allow(non_snake_case)]
pub fn setup_hilbert(n: usize, A: Vec<i32>) -> (Vec<(usize, Coordinates)>, Vec<i32>) {
assert_eq!(n * n, A.len());
let depth: usize = log2(n);
let hilbert_iter: Vec<_> = HilbertIter::new(depth).collect();
println!("Hilbert matrix size: {}", hilbert_iter.len());
#[allow(non_snake_case)]
let flattened_A = flatten_matrix(depth, A, n);
(hilbert_iter, flattened_A)
}
/// Setup (depth, flattened_A) for Hilbert multiplication
#[allow(non_snake_case)]
pub fn setup_hilbert_iter(n: usize, A: Vec<i32>) -> (usize, Vec<i32>) {
assert_eq!(n * n, A.len());
let depth: usize = log2(n);
#[allow(non_snake_case)]
let flattened_A = flatten_matrix(depth, A, n);
(depth, flattened_A)
}
#[cfg(test)]
mod test {
use std::time::{self};
use insta::assert_yaml_snapshot;
use rand::distributions::Uniform;
use rand::Rng;
use rand_chacha::{rand_core::SeedableRng, ChaCha8Rng};
use timeit::timeit_loops;
use crate::{
hilbert_matrix_vector_product, hilbert_matrix_vector_product_iter, make_matrix,
naive_matrix_vector_product,
};
#[test]
fn test_equality_of_implementation() {
let mut rng = ChaCha8Rng::seed_from_u64(10);
let range = Uniform::new(1, 11);
let n: usize = 2usize.pow(8);
let start = time::Instant::now();
// A = [[random.randint(1, 10) for _ in range(n)] for _ in range(n)]
#[allow(non_snake_case)]
let A = make_matrix(n, 1, 11, &mut rng);
// v = [random.randint(1, 10) for _ in range(n)]
let v: Vec<_> = (0..n).map(|_| rng.sample(&range)).collect();
assert_eq!(v.len(), n);
let mut output1 = vec![0; n];
let mut output2 = vec![0; n];
let mut output3 = vec![0; n];
let end = time::Instant::now();
println!("Initial data generation: {}s", (end - start).as_secs_f32());
// Naive
let timeit_count = 3;
let _ = timeit_loops! {timeit_count,
{ naive_matrix_vector_product(&A, &v, &mut output1, n); }
};
// reorder data
#[allow(non_snake_case)]
let (depth, flattened_A) = super::setup_hilbert_iter(n, A.clone());
// Hilbert Product
let _ = timeit_loops! {timeit_count,
{hilbert_matrix_vector_product_iter(&flattened_A,&v, &mut output2, depth);}
};
// non-iterative version
let (hilbert_iter, flattened_a) = super::setup_hilbert(n, A);
let _ = timeit_loops! {timeit_count,
{hilbert_matrix_vector_product(&flattened_a,&v, &mut output3, &hilbert_iter);}
};
assert_eq!(output1, output2);
assert_eq!(output1, output3);
assert_yaml_snapshot!(output2);
}
}