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The classic textbook example of the use of backtracking is the eight queens puzzle, that asks for all arrangements of eight chess queens on a standard chessboard so that no queen attacks any other. In the common backtracking approach, the partial candidates are arrangements of k queens in the first k rows of the board, all in different rows and ...
In this basic backtracking algorithm, consistency is defined as the satisfaction of all constraints whose variables are all assigned. Several variants of backtracking exist. Backmarking improves the efficiency of checking consistency. Backjumping allows saving part of the search by backtracking "more than one variable" in some cases.
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.
In 1874, S. Günther proposed a method using determinants to find solutions. [1] J.W.L. Glaisher refined Gunther's approach. In 1972, Edsger Dijkstra used this problem to illustrate the power of what he called structured programming. He published a highly detailed description of a depth-first backtracking algorithm. [2]
Instead of backtracking, the algorithm makes some further elaboration, proving that the evaluations =, =, and = are not part of any solution. As a result, the current evaluation of x 1 x 2 {\displaystyle x_{1}x_{2}} is not part of any solution, and the algorithm can directly backjump to x 2 {\displaystyle x_{2}} , trying a new value for it.
The Dancing Links algorithm solving a polycube puzzle. In computer science, dancing links (DLX) is a technique for adding and deleting a node from a circular doubly linked list. It is particularly useful for efficiently implementing backtracking algorithms, such as Knuth's Algorithm X for the exact cover problem. [1]
In backtracking algorithms, look ahead is the generic term for a subprocedure that attempts to foresee the effects of choosing a branching variable to evaluate one of its values. The two main aims of look-ahead are to choose a variable to evaluate next and to choose the order of values to assign to it.
Roughly speaking, we run two-way backtracking a few times, then use the learning rate we get from then unchanged, except if the function value increases. Here is precisely how it is done. One choose in advance a number N {\displaystyle N} , and a number m ≤ N {\displaystyle m\leq N} .