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Implement Graph Coloring using Backtracking (#737)
* feat: implement Graph Coloring * ref: refactor implementation - Use `Vec<Vec<bool>>` to represent the adjacency matrix to make the code more readable and slightly more efficient in terms of memory usage. - Use `std::mem::take` to avoid unnecessary cloning while maintaining the original behavior of the algorithm - Add some edge tests * ref: refactor implementation - Add custom error `GraphColoringError` to handle the exceptional cases where the graph is empty or not squared - Adjust the `is_valid_color` method to make the implementation handle the directed graphs - Add tests for directed graphs * chore: rename `adj_matrix` to `adjacency_matrix` * chore: rename custom error variants * chore: add some more tests * chore: use 0 as first color --------- Co-authored-by: Piotr Idzik <[email protected]>
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‎src/backtracking/graph_coloring.rs

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//! This module provides functionality for generating all possible colorings of a undirected (or directed) graph
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//! given a certain number of colors. It includes the GraphColoring struct and methods
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//! for validating color assignments and finding all valid colorings.
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/// Represents potential errors when coloring on an adjacency matrix.
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#[derive(Debug, PartialEq, Eq)]
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pub enum GraphColoringError {
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// Indicates that the adjacency matrix is empty
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EmptyAdjacencyMatrix,
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// Indicates that the adjacency matrix is not squared
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ImproperAdjacencyMatrix,
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}
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/// Generates all possible valid colorings of a graph.
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///
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/// # Arguments
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///
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/// * `adjacency_matrix` - A 2D vector representing the adjacency matrix of the graph.
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/// * `num_colors` - The number of colors available for coloring the graph.
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///
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/// # Returns
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///
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/// * A `Result` containing an `Option` with a vector of solutions or a `GraphColoringError` if
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/// there is an issue with the matrix.
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pub fn generate_colorings(
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adjacency_matrix: Vec<Vec<bool>>,
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num_colors: usize,
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) -> Result<Option<Vec<Vec<usize>>>, GraphColoringError> {
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GraphColoring::new(adjacency_matrix)?.find_solutions(num_colors)
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}
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/// A struct representing a graph coloring problem.
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struct GraphColoring {
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// The adjacency matrix of the graph
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adjacency_matrix: Vec<Vec<bool>>,
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// The current colors assigned to each vertex
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vertex_colors: Vec<usize>,
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// Vector of all valid color assignments for the vertices found during the search
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solutions: Vec<Vec<usize>>,
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}
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impl GraphColoring {
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/// Creates a new GraphColoring instance.
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///
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/// # Arguments
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///
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/// * `adjacency_matrix` - A 2D vector representing the adjacency matrix of the graph.
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///
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/// # Returns
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///
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/// * A new instance of GraphColoring or a `GraphColoringError` if the matrix is empty or non-square.
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fn new(adjacency_matrix: Vec<Vec<bool>>) -> Result<Self, GraphColoringError> {
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let num_vertices = adjacency_matrix.len();
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// Check if the adjacency matrix is empty
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if num_vertices == 0 {
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return Err(GraphColoringError::EmptyAdjacencyMatrix);
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}
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// Check if the adjacency matrix is square
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if adjacency_matrix.iter().any(|row| row.len() != num_vertices) {
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return Err(GraphColoringError::ImproperAdjacencyMatrix);
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}
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Ok(GraphColoring {
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adjacency_matrix,
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vertex_colors: vec![usize::MAX; num_vertices],
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solutions: Vec::new(),
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})
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}
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/// Returns the number of vertices in the graph.
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fn num_vertices(&self) -> usize {
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self.adjacency_matrix.len()
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}
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/// Checks if a given color can be assigned to a vertex without causing a conflict.
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///
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/// # Arguments
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///
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/// * `vertex` - The index of the vertex to be colored.
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/// * `color` - The color to be assigned to the vertex.
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///
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/// # Returns
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///
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/// * `true` if the color can be assigned to the vertex, `false` otherwise.
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fn is_color_valid(&self, vertex: usize, color: usize) -> bool {
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for neighbor in 0..self.num_vertices() {
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// Check outgoing edges from vertex and incoming edges to vertex
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if (self.adjacency_matrix[vertex][neighbor] || self.adjacency_matrix[neighbor][vertex])
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&& self.vertex_colors[neighbor] == color
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{
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return false;
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}
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}
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true
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}
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/// Recursively finds all valid colorings for the graph.
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///
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/// # Arguments
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///
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/// * `vertex` - The current vertex to be colored.
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/// * `num_colors` - The number of colors available for coloring the graph.
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fn find_colorings(&mut self, vertex: usize, num_colors: usize) {
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if vertex == self.num_vertices() {
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self.solutions.push(self.vertex_colors.clone());
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return;
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}
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for color in 0..num_colors {
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if self.is_color_valid(vertex, color) {
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self.vertex_colors[vertex] = color;
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self.find_colorings(vertex + 1, num_colors);
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self.vertex_colors[vertex] = usize::MAX;
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}
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}
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}
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/// Finds all solutions for the graph coloring problem.
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///
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/// # Arguments
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///
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/// * `num_colors` - The number of colors available for coloring the graph.
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///
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/// # Returns
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///
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/// * A `Result` containing an `Option` with a vector of solutions or a `GraphColoringError`.
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fn find_solutions(
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&mut self,
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num_colors: usize,
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) -> Result<Option<Vec<Vec<usize>>>, GraphColoringError> {
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self.find_colorings(0, num_colors);
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if self.solutions.is_empty() {
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Ok(None)
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} else {
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Ok(Some(std::mem::take(&mut self.solutions)))
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}
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}
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}
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#[cfg(test)]
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mod tests {
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use super::*;
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macro_rules! test_graph_coloring {
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($($name:ident: $test_case:expr,)*) => {
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$(
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#[test]
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fn $name() {
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let (adjacency_matrix, num_colors, expected) = $test_case;
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let actual = generate_colorings(adjacency_matrix, num_colors);
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assert_eq!(actual, expected);
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}
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)*
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};
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}
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test_graph_coloring! {
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test_complete_graph_with_3_colors: (
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vec![
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vec![false, true, true, true],
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vec![true, false, true, false],
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vec![true, true, false, true],
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vec![true, false, true, false],
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],
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3,
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Ok(Some(vec![
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vec![0, 1, 2, 1],
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vec![0, 2, 1, 2],
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vec![1, 0, 2, 0],
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vec![1, 2, 0, 2],
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vec![2, 0, 1, 0],
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vec![2, 1, 0, 1],
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]))
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),
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test_linear_graph_with_2_colors: (
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vec![
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vec![false, true, false, false],
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vec![true, false, true, false],
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vec![false, true, false, true],
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vec![false, false, true, false],
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],
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2,
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Ok(Some(vec![
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vec![0, 1, 0, 1],
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vec![1, 0, 1, 0],
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]))
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),
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test_incomplete_graph_with_insufficient_colors: (
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vec![
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vec![false, true, true],
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vec![true, false, true],
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vec![true, true, false],
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],
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1,
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Ok(None::<Vec<Vec<usize>>>)
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),
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test_empty_graph: (
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vec![],
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1,
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Err(GraphColoringError::EmptyAdjacencyMatrix)
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),
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test_non_square_matrix: (
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vec![
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vec![false, true, true],
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vec![true, false, true],
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],
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3,
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Err(GraphColoringError::ImproperAdjacencyMatrix)
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),
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test_single_vertex_graph: (
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vec![
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vec![false],
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],
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1,
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Ok(Some(vec![
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vec![0],
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]))
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),
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test_bipartite_graph_with_2_colors: (
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vec![
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vec![false, true, false, true],
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vec![true, false, true, false],
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vec![false, true, false, true],
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vec![true, false, true, false],
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],
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2,
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Ok(Some(vec![
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vec![0, 1, 0, 1],
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vec![1, 0, 1, 0],
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]))
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),
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test_large_graph_with_3_colors: (
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vec![
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vec![false, true, true, false, true, true, false, true, true, false],
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vec![true, false, true, true, false, true, true, false, true, true],
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vec![true, true, false, true, true, false, true, true, false, true],
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vec![false, true, true, false, true, true, false, true, true, false],
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vec![true, false, true, true, false, true, true, false, true, true],
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vec![true, true, false, true, true, false, true, true, false, true],
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vec![false, true, true, false, true, true, false, true, true, false],
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vec![true, false, true, true, false, true, true, false, true, true],
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vec![true, true, false, true, true, false, true, true, false, true],
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vec![false, true, true, false, true, true, false, true, true, false],
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],
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3,
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Ok(Some(vec![
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vec![0, 1, 2, 0, 1, 2, 0, 1, 2, 0],
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vec![0, 2, 1, 0, 2, 1, 0, 2, 1, 0],
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vec![1, 0, 2, 1, 0, 2, 1, 0, 2, 1],
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vec![1, 2, 0, 1, 2, 0, 1, 2, 0, 1],
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vec![2, 0, 1, 2, 0, 1, 2, 0, 1, 2],
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vec![2, 1, 0, 2, 1, 0, 2, 1, 0, 2],
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]))
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),
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test_disconnected_graph: (
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vec![
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vec![false, false, false],
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vec![false, false, false],
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vec![false, false, false],
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],
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2,
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Ok(Some(vec![
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vec![0, 0, 0],
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vec![0, 0, 1],
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vec![0, 1, 0],
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vec![0, 1, 1],
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vec![1, 0, 0],
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vec![1, 0, 1],
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vec![1, 1, 0],
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vec![1, 1, 1],
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]))
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),
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test_no_valid_coloring: (
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vec![
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vec![false, true, true],
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vec![true, false, true],
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vec![true, true, false],
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],
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2,
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Ok(None::<Vec<Vec<usize>>>)
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),
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test_more_colors_than_nodes: (
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vec![
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vec![true, true],
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vec![true, true],
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],
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3,
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Ok(Some(vec![
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vec![0, 1],
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vec![0, 2],
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vec![1, 0],
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vec![1, 2],
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vec![2, 0],
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vec![2, 1],
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]))
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),
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test_no_coloring_with_zero_colors: (
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vec![
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vec![true],
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],
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0,
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Ok(None::<Vec<Vec<usize>>>)
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),
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test_complete_graph_with_3_vertices_and_3_colors: (
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vec![
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vec![false, true, true],
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vec![true, false, true],
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vec![true, true, false],
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],
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3,
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Ok(Some(vec![
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vec![0, 1, 2],
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vec![0, 2, 1],
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vec![1, 0, 2],
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vec![1, 2, 0],
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vec![2, 0, 1],
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vec![2, 1, 0],
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]))
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),
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test_directed_graph_with_3_colors: (
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vec![
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vec![false, true, false, true],
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vec![false, false, true, false],
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vec![true, false, false, true],
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vec![true, false, false, false],
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],
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3,
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Ok(Some(vec![
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vec![0, 1, 2, 1],
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vec![0, 2, 1, 2],
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vec![1, 0, 2, 0],
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vec![1, 2, 0, 2],
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vec![2, 0, 1, 0],
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vec![2, 1, 0, 1],
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]))
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),
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test_directed_graph_no_valid_coloring: (
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vec![
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vec![false, true, false, true],
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vec![false, false, true, true],
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vec![true, false, false, true],
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vec![true, false, false, false],
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],
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3,
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Ok(None::<Vec<Vec<usize>>>)
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),
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test_large_directed_graph_with_3_colors: (
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vec![
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vec![false, true, false, false, true, false, false, true, false, false],
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vec![false, false, true, false, false, true, false, false, true, false],
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vec![false, false, false, true, false, false, true, false, false, true],
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vec![true, false, false, false, true, false, false, true, false, false],
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vec![false, true, false, false, false, true, false, false, true, false],
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vec![false, false, true, false, false, false, true, false, false, true],
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vec![true, false, false, false, true, false, false, true, false, false],
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vec![false, true, false, false, false, true, false, false, true, false],
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vec![false, false, true, false, false, false, true, false, false, true],
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vec![true, false, false, false, true, false, false, true, false, false],
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],
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3,
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Ok(Some(vec![
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vec![0, 1, 2, 1, 2, 0, 1, 2, 0, 1],
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vec![0, 2, 1, 2, 1, 0, 2, 1, 0, 2],
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vec![1, 0, 2, 0, 2, 1, 0, 2, 1, 0],
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vec![1, 2, 0, 2, 0, 1, 2, 0, 1, 2],
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vec![2, 0, 1, 0, 1, 2, 0, 1, 2, 0],
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vec![2, 1, 0, 1, 0, 2, 1, 0, 2, 1]
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]))
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),
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}
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}

‎src/backtracking/mod.rs

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mod all_combination_of_size_k;
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mod graph_coloring;
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mod hamiltonian_cycle;
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mod knight_tour;
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mod n_queens;
@@ -8,6 +9,7 @@ mod rat_in_maze;
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mod sudoku;
910

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pub use all_combination_of_size_k::generate_all_combinations;
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pub use graph_coloring::generate_colorings;
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pub use hamiltonian_cycle::find_hamiltonian_cycle;
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pub use knight_tour::find_knight_tour;
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pub use n_queens::n_queens_solver;

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