Dancing links

TL;DR

This paper introduces two optimization tricks for depth-first search algorithms in combinatorial puzzles, improving efficiency and flexibility, demonstrated on tiling and N-queens problems with excellent results.

Contribution

The paper presents novel reversible list operations and a ghost square technique that enhance search efficiency and flexibility in solving combinatorial puzzles.

Findings

Efficient algorithms for tiling hexagons with hexiamonds.

Successful application to N-queens problem up to N=18.

Significant reduction in total computation time.

Abstract

The author presents two tricks to accelerate depth-first search algorithms for a class of combinatorial puzzle problems, such as tiling a tray by a fixed set of polyominoes. The first trick is to implement each assumption of the search with reversible local operations on doubly linked lists. By this trick, every step of the search affects the data incrementally. The second trick is to add a ghost square that represents the identity of each polyomino. Thus puts the rule that each polyomino be used once on the same footing as the rule that each square be covered once. The coding simplifies to a more abstract form which is equivalent to 0-1 integer programming. More significantly for the total computation time, the search can naturally switch between placing a fixed polyomino or covering a fixed square at different stages, according to a combined heuristic. Finally the author reports excellent performance for his algorithm for some familiar puzzles. These include tiling a hexagon by 19 hexiamonds and the N queens problem for N up to 18.