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Exact Cover

This crate provides an implementation of Knuth's Algorithm X for solving exact cover problems.

The algorithms and data structures in this module, as well as the terminology of "options" and "items", come from Donald Knuth's The Art of Computer Programming, section 7.2.2.1 "Dancing Links"1.

  • No dependencies
  • 100% safe Rust
  • Choose whether to find a single solution or all possible solutions
  • Designed for flexibility (see below)

From a high level, Algorithm X is simply a backtracking exhaustive search algorithm. At a lower level, Algorithm X cleverly manipulates linked lists using the "dancing links" technique to make the backtracking more efficient. This crate implements the backtracking search as a provided trait method ExactCoverProblem::search, while the dancing links data structure and operations are implemented by the Dlx struct. A custom implementation of the ExactCoverProblem trait can make use of the dancing links structure provided by Dlx, control the order items are selected for covering, and apply additional filters to options for problems beyond pure exact cover.

A ready-to-use implementation using the minimum remaining values heuristic to select items (i.e. select the item covered by the fewest number of available options) is provided with MrvExactCoverSearch. Its code can be helpful as a starting point for custom implementations.

Examples

The following pure exact cover problem is often used as an example in texts by Knuth, and consists of 7 items and 6 options. For a more practical application, see examples/sudoku.rs.

# use xcov::*;
let mut builder = DlxBuilder::new(7, 0);
builder.add_option(&[      3,    5      ]);
builder.add_option(&[1,       4,       7]);
builder.add_option(&[   2, 3,       6   ]);
builder.add_option(&[1,       4,    6   ]);
builder.add_option(&[   2,             7]);
builder.add_option(&[         4, 5,    7]);

let mut problem = MrvExactCoverSearch::new(builder.build());
problem.search();
let solution = problem
   .current_solution()
   .expect("this example problem has a solution")
   .into_iter()
   .collect::<Vec<_>>();
assert_eq!(
   solution,
   [
      &[1, 4, 6][..],
      &[2, 7],
      &[3, 5],
   ]
);

// can look for additional solutions but this problem has none
assert!(!problem.search());
assert!(problem.current_solution().is_none());

Footnotes

  1. A pre-print is accessible at https://www-cs-faculty.stanford.edu/~knuth/fasc5c.ps.gz. ↩