骑士之旅高效解决方案

Question

我已经在prolog中构建了一个代码来找到一系列合法的动作,其中骑士只落在棋盘的每个方格(8x8)上一次.

我使用了如下逻辑:有8种类型的骑士动作:

• 正确1下来2
• 左1下2
• 右2下1
• 左2下来1
• 正确1上2
• 离开1上2
• 正确2上升1
• 离开2上升1

正确1倒2动作:

 move(X,Y) :- 
    C_X is X mod 8,      
        R_X is X // 8,       
        C_Y is C_X + 1,      % 1 right
        C_Y < 8,           
        R_Y is R_X + 2,      % 2 down
    R_Y < 8,
        Y is R_Y * 8 + C_Y,
    Y >= 0,
    X >= 0,
    X < 64,
    Y < 64.

对于所有8种类型的动作都重复这一点

问题是我的代码效率不高,找到正确的路径需要太多的步骤.有谁知道解决这个问题的有效方法？

Answer 1

为了能够在可行的时间内解决8x8 Knight的巡回难题,Warnsdorff的规则可能是必须的.

我在B-Prolog中创建了一个程序,可以很快地解决这个问题.如果你需要将程序放在其他Prolog中 - 翻译它或者只是使用它的一些想法并不难.

knight_moves(X, Y, NewX, NewY) :-
    ( NewX is X - 1, NewY is Y - 2 
    ; NewX is X - 1, NewY is Y + 2 
    ; NewX is X + 1, NewY is Y - 2
    ; NewX is X + 1, NewY is Y + 2
    ; NewX is X - 2, NewY is Y - 1 
    ; NewX is X - 2, NewY is Y + 1 
    ; NewX is X + 2, NewY is Y - 1
    ; NewX is X + 2, NewY is Y + 1 ).

possible_knight_moves(R, C, X, Y, Visits, NewX, NewY) :-
    knight_moves(X, Y, NewX, NewY),
    NewX > 0, NewX =< R,
    NewY > 0, NewY =< C,
    \+ (NewX, NewY) in Visits.

possible_moves_count(R, C, X, Y, Visits, Count) :-
    findall(_, possible_knight_moves(R, C, X, Y, Visits, _NewX, _NewY), Moves),
    length(Moves, Count).

:- table warnsdorff(+,+,+,+,+,-,-,min).
warnsdorff(R, C, X, Y, Visits, NewX, NewY, Score) :-
    possible_knight_moves(R, C, X, Y, Visits, NewX, NewY),
    possible_moves_count(R, C, NewX, NewY, [(NewX, NewY) | Visits], Score).

knight(R, C, X, Y, Visits, Path) :-
    length(Visits, L),
    L =:= R * C - 1,
    NewVisits = [(X, Y) | Visits],
    reverse(NewVisits, Path).

knight(R, C, X, Y, Visits, Path) :-
    length(Visits, L),
    L < R * C - 1,
    warnsdorff(R, C, X, Y, Visits, NewX, NewY, _Score),
    NewVisits = [(X, Y) | Visits],
    knight(R, C, NewX, NewY, NewVisits, Path).


| ?- time(knight(8, 8, 1, 1, [], Path)).

CPU time 0.0 seconds.

Path = [(1,1),(2,3),(1,5),(2,7),(4,8),(6,7),(8,8),(7,6),(6,8),(8,7),(7,5),(8,3),(7,1),(5,2),(3,1),(1,2),(2,4),(1,6),(2,8),(3,6),(1,7),(3,8),(5,7),(7,8),(8,6),(7,4),(8,2),(6,1),(7,3),(8,1),(6,2),(4,1),(2,2),(1,4),(2,6),(1,8),(3,7),(5,8),(7,7),(8,5),(6,6),(4,7),(3,5),(5,6),(6,4),(4,3),(5,5),(6,3),(5,1),(7,2),(8,4),(6,5),(4,4),(3,2),(5,3),(4,5),(3,3),(2,1),(1,3),(2,5),(4,6),(3,4),(4,2),(5,4)]
yes