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If the size of the state space is finite, calculating the size of the state space is a combinatorial problem. [4] For example, in the Eight queens puzzle, the state space can be calculated by counting all possible ways to place 8 pieces on an 8x8 chessboard. This is the same as choosing 8 positions without replacement from a set of 64, or
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]
Animation of min-conflicts resolution of 8-queens. First stage assigns columns greedily minimizing conflicts, then solves. 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 ...
Backtracking is a class of algorithms for finding solutions to some computational problems, notably constraint satisfaction problems, that incrementally builds candidates to the solutions, and abandons a candidate ("backtracks") as soon as it determines that the candidate cannot possibly be completed to a valid solution.
One way to speed up a brute-force algorithm is to reduce the search space, that is, the set of candidate solutions, by using heuristics specific to the problem class. For example, in the eight queens problem the challenge is to place eight queens on a standard chessboard so that no queen attacks any other.
3.6 Space-partitioning trees. ... The specific problem is: ... a hash-based chained data structure that can persist state history over time;
The most famous problem of this type is the eight queens puzzle. Problems are further extended by asking how many possible solutions exist. Further generalizations apply the problem to NxN boards. [3] [4] An 8×8 chessboard can have 16 independent kings, 8 independent queens, 8 independent rooks, 14 independent bishops, or 32 independent ...
NP-complete special cases include the edge dominating set problem, i.e., the dominating set problem in line graphs. NP-complete variants include the connected dominating set problem and the maximum leaf spanning tree problem. [3]: ND2 Feedback vertex set [2] [3]: GT7 Feedback arc set [2] [3]: GT8 Graph coloring [2] [3]: GT4