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Some of the better-known exact cover problems include tiling, the n queens problem, and Sudoku. The name dancing links , which was suggested by Donald Knuth , stems from the way the algorithm works, as iterations of the algorithm cause the links to "dance" with partner links so as to resemble an "exquisitely choreographed dance."
In a typical backtracking solution to this problem, one could define a partial candidate as a list of integers c = (c[1], c[2], …, c[k]), for any k between 0 and n, that are to be assigned to the first k variables x[1], x[2], …, x[k]. The root candidate would then be the empty list (). The first and next procedures would then be
While backtracking always goes up one level in the search tree when all values for a variable have been tested, backjumping may go up more levels. In this article, a fixed order of evaluation of variables x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}} is used, but the same considerations apply to a dynamic order of evaluation.
Nauck also extended the puzzle to the n queens problem, with n queens on a chessboard of n×n squares. Since then, many mathematicians, including Carl Friedrich Gauss, have worked on both the eight queens puzzle and its generalized n-queens version. In 1874, S. Günther proposed a method using determinants to find solutions. [1]
Some hobbyists have developed computer programs that will solve Sudoku puzzles using a backtracking algorithm, which is a type of brute force search. [3] Backtracking is a depth-first search (in contrast to a breadth-first search), because it will completely explore one branch to a possible solution before moving to another branch.
Backtracking is the process of traversing the tree in preorder, depth first. Any systematic rule for choosing column c in this procedure will find all solutions, but some rules work much better than others. To reduce the number of iterations, Knuth suggests that the column-choosing algorithm select a column with the smallest number of 1s in it.
Min-Conflicts solves the N-Queens Problem by selecting a column from the chess board for queen reassignment. The algorithm searches each potential move for the number of conflicts (number of attacking queens), shown in each square. The algorithm moves the queen to the square with the minimum number of conflicts, breaking ties randomly.
Cassowary constraint solver, an open source project for constraint satisfaction (accessible from C, Java, Python and other languages). Comet, a commercial programming language and toolkit Gecode , an open source portable toolkit written in C++ developed as a production-quality and highly efficient implementation of a complete theoretical ...