I am given a task to parallelize a sudoku solution counting program. The problem was a part of Marathon of Parallel Programming (MOPP) called sudokount.

We are given an input sudoku puzzle of dimension size^4 (e.g 3×3 by 3×3), 3≤size≤8. The job is to count the print the number of valid solutions to a given sudoku puzzle. Input is given from stdin where 0 represents an empty cell and a number indicates a fixed number which was assigned to that cell.

To find the solution, I am using two methods,

1. Constraint propagation:

• Fill the number of possible digits (1-size^2) in each cell which has no fixed digit.

• For all the digits which have a fixed possibility, delete that number from its peers (cell's same row, column and box)

• After the deletion process is completed, find the cell which has minimum number of remaining possibilities (MRV)

For all digits in the MRV cell,

• Fix any one digit in the cell and continue eliminating (this time recursively) to eliminate that digit from its peers, and if an elimination resulted in a cell having only one possibility, recursively call eliminate on that cell with that one value. We would reach an invalid configuration

• If an elimination has emptied a cell, then we would backtrack and change the fixed digit in the MRV cell found earlier.

After elimination has been performed, push a copy of the puzzle into a stack to perform DFS(to avoid implicit recursion, as I want to parallelize the code).
Finally continue eliminating and pushing the puzzle into the stack until the stack becomes empty

The MRV heuristic costs O(n^2) however I implemented something similar using binary heap (min heap) to track which cell has minimum number of elements.

2. Twin pair:

If a two cells in the same peer (i.e two cells in the same row, column or box) have the same two possibilities, I would go on and delete those two possibilities from any other cell (having number of possibilities more than 2) in the peer set of those two cells.

I have a working solution in rust which implements Constraint propagation and MRV heuristic (with binary heap to keep track of cells with least number of possibilities) however it's sequential and damm slow compared to even the sequential C code.

Here are some code snippets:

First the sudoku struct

use std::sync::atomic::{AtomicUsize, Ordering};
pub static SOLUTION_COUNTER: AtomicUsize = AtomicUsize::new(0);
use std::collections::BinaryHeap;
use std::cmp::Ordering as OrdOrdering;

#[derive(Eq, PartialEq, Clone)]
struct CellPossibilities{
num_possibilities: usize,
coordinates: (usize, usize),
}

// Implement Ord so that cells with fewer possibilities are given higher priority
impl Ord for CellPossibilities {
fn cmp(&self, other: &Self) -> OrdOrdering {
// Note that we flip the order here for min-heap behavior
other.num_possibilities.cmp(&self.num_possibilities)
}
}

impl PartialOrd for CellPossibilities {
fn partial_cmp(&self, other: &Self) -> Option<OrdOrdering> {
Some(self.cmp(other))
}
}

pub struct SudokuPeers{
size: usize,
len: usize,
pub peers: Vec<Vec<Vec<(usize, usize)>>>,
}

impl SudokuPeers {
// build peer is builds a set of all the cells that are in the same row, column, or box as the cell at (x, y)
// it populates the peers field of the Sudoku struct
pub fn new(size: usize) -> Self {
let len = size * size;
// let sqrt = (size as f64).sqrt() as usize;
let mut peers = vec![vec![Vec::new(); len]; len];

for i in 0..len {
for j in 0..len {
// Add cells in the same row
for x in 0..len {
if x != i {
peers[i][j].push((x, j));
}
}
// Add cells in the same column
for y in 0..len {
if y != j {
peers[i][j].push((i, y));
}
}
// Add cells in the same box
let box_start_row = i - i % size;
let box_start_col = j - j % size;
for x in box_start_row..box_start_row + size {
for y in box_start_col..box_start_col + size {
// if x != i || y != j {
// if (x,y) != (i,j) && (x!=i || y!=j) {
if x!=i && y!=j {
peers[i][j].push((x, y));
}
}
}
}
}

SudokuPeers { size, len, peers }
}
}

#[derive(Clone)]
pub struct Sudoku{
size: usize,
len: usize,
pub peers: &'static SudokuPeers,
mrv: (usize,usize),
board: Vec<Vec<u64>>,
no_solution: u64,
// cell_possibilities: BinaryHeap<CellPossibilities>,
}

impl Sudoku {

pub fn new(size: usize) -> Sudoku {
let len = size * size;
let board = vec![vec![0; len]; len];
let peers = Box::leak(Box::new(SudokuPeers::new(size)));
let mrv: (usize, usize) = (0,0);
let no_solution = 0;
Sudoku { size, len, peers, mrv,  board, no_solution }
}

pub fn set_sol_counter(&mut self) {
}

pub fn get_mrv(&self) -> (usize, usize) {
self.mrv
}

pub fn get_peers(&self, i: usize, j: usize) -> &Vec<(usize, usize)> {
&self.peers.peers[i][j]
}

pub fn get_no_of_solutions(&self) -> u64 {
self.no_solution
}

pub fn clear_cell(&mut self, x: usize, y: usize) {
self.board[x][y] = 0;
}

pub fn print_peers(&self) {
for (x, row) in self.peers.peers.iter().enumerate() {
for (y, peers) in row.iter().enumerate() {
print!("Peers for cell ({}, {}): ", x, y);
for (x, y) in peers {
print!("({}, {}), ", x, y);
}
println!();
}
}
}

pub fn set_field(&mut self, x: usize, y: usize, value: usize) {
if value == 0 {
println!("Value must be between 1 and {}", self.len);
return;
}
self.board[x][y] |= 1 << (value - 1);
}

pub fn unset_field(&mut self, x: usize, y: usize, value: usize) {
self.board[x][y] &= !(1 << (value - 1));
}

pub fn get_field(&self, x: usize, y: usize) -> u64 {
self.board[x][y]
}

pub fn print(&self) {
for x in 0..self.len {
for y in 0..self.len {
print!("|");
let possibilities = self.get_possibilities(x, y);
let pos_len = possibilities.len();
for possibility in possibilities {
print!("{},", possibility);
}
if pos_len > 2 {
print!("|\t");
}else{
print!("|\t\t");
}
}
println!();
}
}

pub fn print_raw(&self) {
for row in self.board.iter() {
for col in row.iter() {
print!("{:016x} ", col);
}
println!();
}
}

pub fn get_num_possibilities(&self, x: usize, y: usize) -> usize {
self.board[x][y].count_ones() as usize
}

pub fn get_single_remaining_value(&self, x: usize, y: usize) -> usize {
let value = self.get_field(x, y);
return value.trailing_zeros() as usize + 1;
}

pub fn is_digit_present(&self, i:usize, j:usize, d:usize) -> bool {
let mut value = self.get_field(i, j);
value & (1 << (d - 1)) != 0
}

pub fn get_possibilities(&self, x: usize, y: usize) -> Vec<usize> {
let mut possibilities = Vec::new();
let value = self.get_field(x, y) as u64; // Cast to u64 to ensure sufficient size for shifts
for i in 1..=self.len {
// Ensure i does not exceed the number of bits in u64 to avoid overflow
if i > 64 {
break; // Or handle this case as appropriate for your application
}
if (value & (1u64 << (i - 1))) != 0 {
possibilities.push(i);
}
}
possibilities
}

pub fn min_rem_val (&mut self){
let mut min = usize::MAX;
let mut mrv = (usize::MAX,usize::MAX);
for i in 0..self.len {
for j in 0..self.len {
let num_possibilities = self.get_num_possibilities(i, j);
if num_possibilities > 1 && num_possibilities < min {
min = num_possibilities;
mrv = (i, j);
}
}
}
self.mrv = mrv;
}
// eliminate_all is a function which iterates over all peers of a cell
// and removes the value of the cell from the peer's possible values
// iff the value of the cell is already determined
// returns true if elimination was successful otherwise false
pub fn eliminate_all(&mut self) -> Result<bool, &'static str> {
let mut changed = false;
let mut min_possibilities = usize::MAX;

for x in 0..self.len {
for y in 0..self.len {
let num_possibilities = self.get_num_possibilities(x, y);
if num_possibilities == 1 {
let value = self.get_single_remaining_value(x, y);
let peers_of_curr_cell = self.peers.peers[x][y].iter(); // should consider not cloning
for (i, j) in peers_of_curr_cell {
if self.is_digit_present(*i, *j, value) {
self.unset_field(*i, *j, value);
if self.get_field(*i, *j) == 0 {
return Err("Invalid puzzle: elimination resulted in no possibilities");
}
changed = true;
}
}
}
}
}
self.min_rem_val();
Ok(changed)
}

pub fn eliminate(&mut self, x: usize, y: usize, value: usize) -> Result<bool, &'static str> {
// println!("Called Eliminate {} at {},{}", value, x, y);
// Check if the cell's value is already determined and matches the given value
if self.get_num_possibilities(x, y) != 1 || self.get_single_remaining_value(x, y) != value {
return Ok(false);
}

let mut changed = false;
// Iterate over all peers of the cell
for &(i, j) in &self.peers.peers[x][y] {
// Attempt to remove the value from the peer's possibilities
if self.is_digit_present(i, j, value) {
self.unset_field(i, j, value);
changed = true; // Mark that a change was made

// Check if the peer now has a single remaining possibility
if self.get_num_possibilities(i, j) == 0 {
return Err("Invalid puzzle: elimination resulted in no possibilities");
} else if self.get_num_possibilities(i, j) == 1 {
let new_value = self.get_single_remaining_value(i, j);
// Recursively eliminate the new value from the peers of this cell
self.eliminate(i, j, new_value)?;
}
}
}

Ok(changed)
}

pub fn delete_twins(&mut self) -> Result<bool, &'static str> {
let mut changed = false;
for x in 0..self.len {
for y in 0..self.len {
let possibilities = self.get_possibilities(x, y);
if possibilities.len() == 2 {
let peers = self.peers.peers[x][y].iter();
for &(i, j) in peers {
if self.get_possibilities(i, j) == possibilities {
let peer_peers = self.peers.peers[i][j].iter();
for &(ii, jj) in peer_peers {
if (ii, jj) != (x, y) && (ii, jj) != (i, j) {
let mut peer_possibilities = self.get_possibilities(ii, jj);
let original = peer_possibilities.clone();
peer_possibilities.retain(|&x| !possibilities.contains(&x));
if peer_possibilities.is_empty() {
return Err("Invalid puzzle: elimination resulted in no possibilities");
}
if peer_possibilities != original {
for num in peer_possibilities{
self.set_field(ii, jj, num);
changed = true;
}
}
}
}
}
}
}
}
}
Ok(changed)
}
}

And now the search functionality

use std::sync::atomic::Ordering;
use crate::sudoku::{Sudoku, SOLUTION_COUNTER}; // Replace with the actual path to your Sudoku struct and SOLUTION_COUNTER

pub fn search(sudoku: &mut Sudoku) {
match sudoku.eliminate_all(){
Ok(changed) => {
// println!("Elimination successfull, changed: {}", changed);
},
Err(err) => {
println!("Error: {}", err);
}
};

// eliminate all also populates the mrv field of the game
// println!("MRV: {:?} and value is {}", sudoku.get_mrv(), sudoku.get_field(sudoku.get_mrv().0, sudoku.get_mrv().1));
// now we can copy the game in the search function and start the search
let mut stack: Vec<Sudoku> = vec![sudoku.clone()];

while stack.len() > 0 {
// println!("Stack size = {}", stack.len());
let mut game: Sudoku = stack.pop().unwrap();
match game.delete_twins() {
Ok(changed) => {
// println!("Naked twins successfull, changed: {}", changed);
},
Err(err) => {
// println!("Error: {}", err);
}
}
game.min_rem_val();
let (x, y) = game.get_mrv();
// println!("MRV = {},{}", x, y);
if (x,y) == (usize::MAX, usize::MAX) { // No more MRV
println!("Solution found, total solutions = {}", SOLUTION_COUNTER.load(Ordering::SeqCst));
// println!("Got to solution:");
// game.print();
continue;
}
let possibilities = game.get_possibilities(x, y);
// println!("Possibilities len = {}", possibilities.len());

if possibilities.len() == 1 {
println!("\$\$\$\$ No possibilities found at {},{}", x, y);
continue;
}
for possibility in possibilities{
let mut puzzle = game.clone();
puzzle.clear_cell(x, y);
puzzle.set_field(x, y, possibility);
match puzzle.eliminate(x, y, possibility){
Ok(changed) => {
// println!("Elimination successfull at {},{} value={}, changed: {}", x, y, possibility, changed);
if changed {
// puzzle.print();
stack.push(puzzle.clone());
}
// puzzle.print();
},
Err(err) => {
// println!("Error: {}", err);
continue;
}
}
}
}
}

Is there any suggestion so that it can be improved? Code linked below

Code which implements MRV using heaps (without twins pairs) also has the reference C code

Code which implements MRV using normal O(n^2) search with naive twin pair implementation