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use crate::indexing::SpIndex;
use crate::sparse::permutation::PermOwnedI;
use crate::sparse::symmetric::is_symmetric;
use crate::sparse::CsMatViewI;
use std::collections::vec_deque::VecDeque;
pub struct Ordering<I> {
/// The computed permutation
pub perm: PermOwnedI<I>,
/// Indices inside the permutation delimiting connected components
pub connected_parts: Vec<usize>,
}
pub mod start {
use crate::indexing::SpIndex;
use crate::sparse::CsMatViewI;
/// This trait abstracts over possible strategies to choose a starting
/// vertex for the Cutihll-McKee algorithm. Common strategies are provided.
///
/// You can implement this trait yourself to enable custom strategies,
/// e.g. for predetermined starting vertices.
/// If you do that, please let us now by filing an issue in the repo,
/// since we would like to know which strategies are common in the wild,
/// so we can consider implementing them in the library.
pub trait Strategy<N, I, Iptr>
where
N: PartialEq,
I: SpIndex,
Iptr: SpIndex,
{
/// **Contract:** This function must always be called with at least one
/// unvisited vertex left.
fn find_start_vertex(
&mut self,
visited: &[bool],
degrees: &[usize],
mat: &CsMatViewI<N, I, Iptr>,
) -> usize;
}
/// This strategy chooses some next available vertex as starting vertex.
pub struct Next();
impl<N, I, Iptr> Strategy<N, I, Iptr> for Next
where
N: PartialEq,
I: SpIndex,
Iptr: SpIndex,
{
fn find_start_vertex(
&mut self,
visited: &[bool],
_degrees: &[usize],
_mat: &CsMatViewI<N, I, Iptr>,
) -> usize {
visited
.iter()
.enumerate()
.find_map(|(i, &a)| if a { None } else { Some(i) })
.expect(
"There should always be a unvisited vertex left to choose",
)
}
}
/// This strategy chooses a vertex of minimum degree as starting vertex.
pub struct MinimumDegree();
impl<N, I, Iptr> Strategy<N, I, Iptr> for MinimumDegree
where
N: PartialEq,
I: SpIndex,
Iptr: SpIndex,
{
fn find_start_vertex(
&mut self,
visited: &[bool],
degrees: &[usize],
_mat: &CsMatViewI<N, I, Iptr>,
) -> usize {
visited
.iter()
.enumerate()
.filter(|(_i, &a)| !a)
.min_by_key(|(i, _a)| degrees[*i])
.map(|(i, _a)| i)
.expect(
"There should always be a unvisited vertex left to choose",
)
}
}
/// This strategy employs an pseudoperipheral vertex finder as described by
/// George and Liu. It is the most expensive strategy to compute, but
/// typically results in the narrowest bandwidth.
#[derive(Default)]
pub struct PseudoPeripheral();
impl PseudoPeripheral {
#[inline]
pub fn new() -> Self {
Self::default()
}
/// Computes the rooted level structure rooted at `root`, returning the
/// index of vertex of the last level with minimum degree, called
/// "contender", and the height of the rls.
fn rls_contender_and_height<N, I, Iptr>(
&mut self,
root: usize,
degrees: &[usize],
mat: &CsMatViewI<N, I, Iptr>,
) -> (usize, usize)
where
N: PartialEq,
I: SpIndex,
Iptr: SpIndex,
{
// One might wonder: "Why are we not reusing the rooted level
// structure (rls) we build here, isn't it basically the same thing
// we build in the rcm again, afterwards?""
// The answer: Yes, but, No.
//
// The rooted level structure here differs from the one built
// afterwards by its order. The required order is very nasty
// indeed: the position of a vertex in its level depends primarily
// on the position of its neighboring vertex in the previous level,
// and secondarily on its degree.
//
// Still, one may think: "Well, then just keep the rls around, and
// sort it if its root is choosen as starting vertex". This is
// easier said than done, let's consider some strategies to do that:
// 1. Sort it without any additionally stored information. That
// would require going through the entire vec again, one by one.
// This would erase the overhead of allocating and then
// deallocating the rls, but making this pass performant requires
// non-trivial, error-prone code. Overall, this strategies
// perfomance gains can at most be minimal.
// 2. Store additional information, like the delimeters of levels,
// neighbouring vertex delimeters, etc. That would require doing
// additional work (computing and storing the information) while
// building the rls, and does not speed up sorting afterwards
// significantly, as the levels still need to be sorted serially.
// Overall, this strategy comes at a significant cost in memory,
// and it's performance improvements are debatable at best.
// 3. Maybe just build any rls in a way that makes it a valid rcm
// odering? That would be optimal if we always find a
// pseudoperipheral vertex on first try. Unfortunately, we rarely
// do, typical are a few swaps, meaning this strategy, overall,
// comes with a loss of performance.
//
// So, thats why we discard the rls. One may feel free to try on his
// own.
let nb_vertices = degrees.len();
// This is ok, if we are given a valid root we can never reach an
// invalid vertex.
let mut visited = vec![false; nb_vertices];
let mut rls = Vec::with_capacity(nb_vertices);
// Start out by pushing the root.
visited[root] = true;
rls.push(root);
let mut rls_index = 0;
// For calculating the height.
let mut height = 0;
let mut current_level_countdown = 1;
let mut next_level_countup = 0;
// The last levels len is used to compute the contender in the end.
let mut last_level_len = 1;
while rls_index < rls.len() {
let parent = rls[rls_index];
current_level_countdown -= 1;
let outer = mat.outer_view(parent.index()).unwrap();
for &neighbor in outer.indices() {
if !visited[neighbor.index()] {
visited[neighbor.index()] = true;
next_level_countup += 1;
rls.push(neighbor.index());
}
}
if current_level_countdown == 0 {
if next_level_countup > 0 {
last_level_len = next_level_countup;
}
current_level_countdown = next_level_countup;
next_level_countup = 0;
height += 1;
}
rls_index += 1;
}
// Choose the contender.
let rls_len = rls.len();
let last_level_start_index = rls_len - last_level_len;
let contender = rls[last_level_start_index..rls_len]
.iter()
.min_by_key(|i| degrees[i.index()])
.copied()
.unwrap();
// Return the node of the last level with minimal degree along with
// the rls's height.
(contender, height)
}
}
impl<N, I, Iptr> Strategy<N, I, Iptr> for PseudoPeripheral
where
N: PartialEq,
I: SpIndex,
Iptr: SpIndex,
{
fn find_start_vertex(
&mut self,
visited: &[bool],
degrees: &[usize],
mat: &CsMatViewI<N, I, Iptr>,
) -> usize {
// Choose the next available vertex as currrent starting vertex.
let mut current = visited
.iter()
.enumerate()
.find_map(|(i, &a)| if a { None } else { Some(i) })
.expect(
"There should always be a unvisited vertex left to choose",
);
// Isolated vertices are by definition pseudoperipheral.
if degrees[current] == 0 {
return current;
}
let (mut contender, mut current_height) =
self.rls_contender_and_height(current, degrees, mat);
// This loop always terminates, typically within very few
// iterations.
// This essentially comes from the fact that no same vertex can be
// choosen as `current` twice, as the height of the rls of `current`
// must always strictly increase for the loop to continue.
loop {
let (contender_contender, contender_height) =
self.rls_contender_and_height(contender, degrees, mat);
if contender_height > current_height {
current_height = contender_height;
current = contender;
contender = contender_contender;
} else {
return current;
}
}
}
}
}
pub mod order {
use super::Ordering;
use crate::indexing::SpIndex;
use crate::sparse::permutation::PermOwnedI;
/// Internal trait for working with Cuthill-McKee
///
/// This trait is very deeply integrated with the inner workings of the
/// Cuthill-McKee algorithm implemented here. It is conceptually only an
/// enum, specifying if the Cuthill-McKee ordering should be built in
/// reverse order.
///
/// No method on this trait should ever be called by the consumer.
//
// This is a trait, not an enum, because monomorphization is absolutely
// critical for performance. Also having the directions manage their state
// themselves enables some optimizations.
pub trait DirectedOrdering<I: SpIndex> {
/// Prepares this directed ordering for working with the specified
/// number of vertices.
// Seperated from `fn new`, as it requires `nb_vertices` as parameter,
// which the consumer would have to supply otherwise, which he can't be
// trusted to do corretly.
fn prepare(&mut self, nb_vertices: usize);
/// Adds a new `vertex_index` as computed in the algorithms main loop.
fn add_transposition(&mut self, vertex_index: usize);
/// Adds an index indicating the start of a new connected component.
fn add_component_delimeter(&mut self, index: usize);
/// Transforms this directed ordering into an ordering to return from
/// the algorithm.
// Actually implementing `From` or `Into` results in coherence errors.
fn into_ordering(self) -> Ordering<I>;
}
/// Indicates the Cuthill-McKee ordering should be built in forward order.
#[derive(Default)]
pub struct Forward<I: SpIndex> {
/// The permutation computed by the algorithm.
perm: Vec<I>,
/// Delimeting connected components inside `perm`.
connected_parts: Vec<usize>,
}
impl<I: SpIndex> Forward<I> {
/// Creates a new instance of this conceptual enum variant.
#[inline]
pub fn new() -> Self {
Self::default()
}
}
impl<I: SpIndex> DirectedOrdering<I> for Forward<I> {
#[inline]
fn prepare(&mut self, nb_vertices: usize) {
self.perm.reserve(nb_vertices);
self.connected_parts.reserve(nb_vertices / 16 + 1);
}
#[inline]
fn add_transposition(&mut self, vertex_index: usize) {
self.perm.push(I::from_usize(vertex_index));
}
#[inline]
fn add_component_delimeter(&mut self, index: usize) {
self.connected_parts.push(index);
}
#[inline]
fn into_ordering(self) -> Ordering<I> {
debug_assert!(crate::perm_is_valid(&self.perm));
Ordering {
perm: PermOwnedI::new_trusted(self.perm),
connected_parts: self.connected_parts,
}
}
}
/// Indicates the Cuthill-McKee ordering should be built in reverse order.
#[derive(Default)]
pub struct Reversed<I: SpIndex> {
/// The permutation computed by the algorithm, written in reverse order.
perm: Vec<I>,
/// Will be transformed to contain indices delimeting componenets in
/// `perm`.
connected_parts: Vec<usize>,
/// The total number of vertices in the matrix.
nb_vertices: usize,
/// Counting with the algorithms main loop, should always be in sync
/// with `perm_index`.
// Is a seperate variable to reduce unnecessary argument passing.
count: usize,
}
impl<I: SpIndex> Reversed<I> {
/// Creates a new instance of this conceptual enum variant.
// This is not optimal, as it leads to close-to-invalid states if not
// used correctly. A solution using some kind of "uninitialized"
// wrapper type however seems to be overkill, especially since all the
// uglieness is under the hood and not triggerable unless explicitly
// asked for.
#[inline]
pub fn new() -> Self {
Self::default()
}
}
impl<I: SpIndex> DirectedOrdering<I> for Reversed<I> {
#[inline]
fn prepare(&mut self, nb_vertices: usize) {
// Missed optimization: Work with MaybeUninit here.
self.perm = vec![I::default(); nb_vertices];
self.connected_parts = Vec::with_capacity(nb_vertices / 16 + 1);
self.nb_vertices = nb_vertices;
}
#[inline]
fn add_transposition(&mut self, vertex_index: usize) {
self.perm[self.nb_vertices - self.count - 1] =
I::from_usize(vertex_index);
self.count += 1;
}
#[inline]
fn add_component_delimeter(&mut self, index: usize) {
self.connected_parts.push(index);
}
#[inline]
fn into_ordering(self) -> Ordering<I> {
let nb_vertices = self.nb_vertices;
let mut connected_parts = self.connected_parts;
// Reverse-Inverse the connected parts, to fit with the reversed
// order.
connected_parts
.iter_mut()
.for_each(|i| *i = nb_vertices - *i);
connected_parts.reverse();
debug_assert!(crate::perm_is_valid(&self.perm));
Ordering {
perm: PermOwnedI::new_trusted(self.perm),
connected_parts,
}
}
}
}
/// A customized Cuthill-McKee algorithm.
///
/// Runs a customized Cuthill-McKee algorithm on the given matrix, returning a
/// permutation reducing its bandwidth.
///
/// The strategy employed to find starting vertices is critical for the quallity
/// of the reordering computed. This library implements several common
/// strategies, like `PseudoPeripheral` and `MinimumDegree`, but also allows
/// users to implement custom strategies if needed.
///
/// # Arguments
///
/// - `mat` - The matrix to compute a permutation for.
///
/// - `starting_strategy` - The strategy to use for choosing a starting vertex.
///
/// - `directed_ordering` - The order of the computed ordering, should either be
/// `Forward` or `Reverse`.
pub fn cuthill_mckee_custom<N, I, Iptr, S, D>(
mat: CsMatViewI<N, I, Iptr>,
mut starting_strategy: S,
mut directed_ordering: D,
) -> Ordering<I>
where
N: PartialEq,
I: SpIndex,
Iptr: SpIndex,
S: start::Strategy<N, I, Iptr>,
D: order::DirectedOrdering<I>,
{
debug_assert!(is_symmetric(&mat));
assert_eq!(mat.cols(), mat.rows());
let nb_vertices = mat.cols();
let degrees = mat.degrees();
let max_neighbors = degrees.iter().max().copied().unwrap_or(0);
// This will be transformed into the actual `Ordering` in the end,
// contains the permuntation and component delimeters.
directed_ordering.prepare(nb_vertices);
// This is the 'working data', into which new neighboring, sorted vertices
// are inserted, the next vertex to process is popped from here.
let mut deque = VecDeque::with_capacity(nb_vertices);
// This are all new neighbors of the currently processed vertex, they are
// collected here to be sorted prior to being appended to 'deque'.
// The alternative of immediately pushing to deque and sorting there
// surprisingly performs worse.
let mut neighbors = Vec::with_capacity(max_neighbors);
// Storing which vertices have already been visited.
let mut visited = vec![false; nb_vertices];
for perm_index in 0..nb_vertices {
// Find the next index to process, choosing a new starting vertex if
// necessary.
let current_vertex = deque.pop_front().unwrap_or_else(|| {
// We found a new connected component, starting at this iteration.
directed_ordering.add_component_delimeter(perm_index);
// Find a new starting vertex, using the given strategy.
let new_start_vertex = starting_strategy.find_start_vertex(
&visited, °rees, &mat
);
assert!(
!visited[new_start_vertex],
"Vertex returned by starting strategy should always be unvisited"
);
new_start_vertex
});
// Add the next transposition to the ordering.
directed_ordering.add_transposition(current_vertex);
visited[current_vertex.index()] = true;
// Find, sort, and push all new neighbors of the current vertex.
let outer = mat.outer_view(current_vertex.index()).unwrap();
neighbors.clear();
for &neighbor in outer.indices() {
if !visited[neighbor.index()] {
neighbors.push((degrees[neighbor.index()], neighbor));
visited[neighbor.index()] = true;
}
}
// Missed optimization: match small sizes explicitly, sort using sorting
// networks. This especially makes sense if swaps are predictably
// compiled into cmov instructions, which they aren't currently, see
// https://github.com/rust-lang/rust/issues/53823. For more information
// on how to do sorting networks efficiently see
// https://arxiv.org/pdf/1505.01962.pdf.
neighbors.sort_unstable_by_key(|&(deg, _)| deg);
for (_deg, neighbor) in &neighbors {
deque.push_back(neighbor.index());
}
}
directed_ordering.add_component_delimeter(nb_vertices);
directed_ordering.into_ordering()
}
/// The reverse Cuthill-McKee algorithm.
///
/// Runs the reverse Cuthill-McKee algorithm on the given matrix, returning a
/// permutation reducing its bandwidth.
///
/// This version of the algorithm chooses pseudoperipheral vertices as starting
/// vertices, and builds a reversed ordering. This is the most common
/// configuration of the algorithm.
///
/// This library also exposes a costomizable version of the algorithm,
/// [`cuthill_mckee_custom`](cuthill_mckee_custom).
///
/// Implemented as:
/// ```text
/// cuthill_mckee_custom(
/// mat, start::PseudoPeripheral::new(), order::Reversed::new()
/// )
/// ```
pub fn reverse_cuthill_mckee<N, I, Iptr>(
mat: CsMatViewI<N, I, Iptr>,
) -> Ordering<I>
where
N: PartialEq,
I: SpIndex,
Iptr: SpIndex,
{
cuthill_mckee_custom(
mat,
start::PseudoPeripheral::default(),
order::Reversed::new(),
)
}
#[cfg(test)]
mod test {
use super::{cuthill_mckee_custom, order, reverse_cuthill_mckee, start};
use crate::sparse::permutation::Permutation;
use crate::sparse::CsMat;
fn unconnected_graph_lap() -> CsMat<f64> {
// Take the laplacian matrix of the following graph
// (no border conditions):
//
// 0 - 4 - 2 6
// | \ | / | |
// 8 - 5 - 3 9
// | / | \ | |
// 1 - A - B 7
//
// The laplacian matrix structure is (with x = -1)
// 0 1 2 3 4 5 6 7 8 9 A B
// | 3 x x x | 0
// | 3 x x x | 1
// | 3 x x x | 2
// L = | x 3 x x | 3
// | x x 3 x | 4
// | x x x x x 8 x x x | 5
// | 1 x | 6
// | 1 x | 7
// | x x x 3 | 8
// | x x 2 | 9
// | x x 3 x | A
// | x x x 3 | B
let x = -1.;
#[rustfmt::skip]
let lap_mat = CsMat::new(
(12, 12),
vec![0, 4, 8, 12, 16, 20, 29, 31, 33, 37, 40, 44, 48],
vec![0, 4, 5, 8,
1, 5, 8, 10,
2, 3, 4, 5,
2, 3, 5, 11,
0, 2, 4, 5,
0, 1, 2, 3, 4, 5, 8, 10, 11,
6, 9,
7, 9,
0, 1, 5, 8,
6, 7, 9,
1, 5, 10, 11,
3, 5, 10, 11],
vec![3., x, x, x,
3., x, x, x,
3., x, x, x,
x, 3., x, x,
x, x, 3., x,
x, x, x, x, x, 8., x, x, x,
1., x,
1., x,
x, x, x, 3.,
x, x, 2.,
x, x, 3., x,
x, x, x, 3.],
);
lap_mat
}
#[test]
fn reverse_cuthill_mckee_unconnected_graph_lap_components() {
let lap_mat = unconnected_graph_lap();
let ordering = reverse_cuthill_mckee(lap_mat.view());
assert_eq!(&ordering.connected_parts, &[0, 3, 12],);
}
#[test]
fn reverse_cuthill_mckee_unconnected_graph_lap_perm() {
let lap_mat = unconnected_graph_lap();
let ordering = reverse_cuthill_mckee(lap_mat.view());
// This is just one posible permutation. Might be silently broken, e. g.
// through changes in unstable sorting.
let correct_perm =
Permutation::new(vec![7, 9, 6, 11, 10, 3, 1, 2, 5, 8, 4, 0]);
assert_eq!(&ordering.perm.vec(), &correct_perm.vec());
}
#[test]
fn reverse_cuthill_mckee_eye() {
let mat = CsMat::<f64>::eye(3);
let ordering = reverse_cuthill_mckee(mat.view());
let correct_perm = Permutation::new(vec![2, 1, 0]);
assert_eq!(&ordering.perm.vec(), &correct_perm.vec());
}
#[test]
fn cuthill_mckee_eye() {
let mat = CsMat::<f64>::eye(3);
let ordering = cuthill_mckee_custom(
mat.view(),
start::PseudoPeripheral::new(),
order::Forward::new(),
);
let correct_perm = Permutation::new(vec![0, 1, 2]);
assert_eq!(&ordering.perm.vec(), &correct_perm.vec());
}
}