puzzle team club做了一个网站，里面有好多解谜游戏可以玩。身为程序员玩这种游戏，就应该想一些使用程序解决的方法。
但是里面好多游戏还蛮难解的。。。我是说写出简明扼要的程序，并且像在ACM赛场一样执行和开发时间占优。所有游戏都需要用搜索完成毫无疑问。
但是不少游戏都涉及到图论，单纯的搜索很难一下想到方法，像slither link（连接回路）、hashi（搭桥）、nirikabe（围城）和slithes（连接）这几个游戏就是跟图论相关。
其他看起来不像图论的游戏，像skyscrapers（摩天大楼）、star battle（星星战斗），属于有限次数完全覆盖，很难一下想到简明的方法。
这里只求解能使用舞蹈链解决的，相对简单的dominosa游戏（链接：https://www.puzzle-dominosa.com/），这个游戏跟舞蹈链天生一对（误），跟数独的规则差不多，场上所有数字都必须用到1次，用到的数字用于连接数对（任意顺序数对算同种），所有数对必须并且只出现1次，属于精确覆盖题目。
这里就可以想怎么转化为舞蹈链了，这里数对的所有可能连接方式就是舞蹈链的行，而数对的限制以及占用数字的限制则是舞蹈链的列，一旦有一个列被使用，舞蹈链会把其他用到这个列的所有行去除，以达到快速剪枝的目的。
在这个游戏里面，数对可以横向连接（(i,j)-(i,j+1)），也可竖向连接（(i,j)-(i+1,j)），这两种连接方式分别是(r-1)c种和r(c-1)种，就是说行数有大约2cr；数字一般有0-(c-1)这几个数字，可以组成c(c+1)/2种数对（这个站点的dominosa游戏列数一般比行数大1），并且所有r*c个方格必须占据，只能占据1次，这意味着列数大约有3cr/2，并且这个站点的这种谜题只有1个解答，搜索宽度不会太长，最多只有20*19的谜题，执行时间不会太长。（这个站点还可以找到一个画方格游戏shikaku，也可以用精确覆盖，但是那个游戏长宽大得不像话）
基于这个思路，我们开始写Python代码（DLX模板是用的https://github.com/DavideCanton/DancingLinksX这个链接里面的代码，我们只是构造表而已）：
首先，构造舞蹈表的列头：
def get_names(r, c, m): cnt = 0 for i in range(r): for j in range(c): yield f'L({i},{j})' # location pair cnt = cnt + 1 for i in range(m+2): for j in range(i): yield f'O({j},{i-1})' # number pair cnt = cnt + 1
第一个二重循环对应占用数字条件列头，第一个二重循环对应占用数对条件列头
构造每个可能的数对对应条件约束：
def compute_row(i, j, a, b, isHorizontal, r, c): # H is 0 .. r*(c-1)-1 # V is r*(c-1) .. r*(c-1)+c*(r-1)-1 # O is r*c*2-r-c .. r*c*2-r-c+(m-1)*m//2-1 if a < b: t = a a = b b = t row = [] if isHorizontal: row.append(c*i+j) row.append(c*i+j+1) else: row.append(c*i+j) row.append(c*(i+1)+j) row.append(r*c+a*(a+1)//2+b) return row
完成舞蹈链遍历，还要把结果按照正确的格式输出。
为了连接字符串方便，这里声明个给单个或多个字符串赋值的函数：
def insertToStr(origin, i, j, value): if(origin[i][j] != '+'): origin[i] = origin[i][:j] + value + origin[i][j+1:]
为了能按照指定格式输出答案，我们这里声明一个类：
class PrintFirstSol: def __init__(self, r, c): self.r = r self.c = c def __call__(self, sol): maxNum = 0 chessWidth = 0 for v in sol.values(): k = list(map(lambda a:int(a),v[2].split('(')[1].replace(')','').split(','))) maxNum = maxNum if k[0] < maxNum else k[0] maxNum = maxNum if k[1] < maxNum else k[1] while maxNum > 0: chessWidth += 1 maxNum //= 10 #print(chessWidth) solImg = [] for _ in range(self.r*2+2): Str1 = '' for _ in range(self.c*(chessWidth+1)+2): Str1 += ' ' solImg.append(Str1) for v in sol.values(): v.sort() v[0] = list(map(lambda a:int(a),v[0].split('(')[1].replace(')','').split(','))) v[1] = list(map(lambda a:int(a),v[1].split('(')[1].replace(')','').split(','))) v[2] = v[2].split('(')[1].replace(')','').split(',') #print(v) i = v[0][0] j = v[0][1] if(v[0][1] != v[1][1]): for q in range(chessWidth): insertToStr(solImg, 2*i, (1+chessWidth)*j+1+q, '-') insertToStr(solImg, 2*i, (1+chessWidth)*(j+1)+1+q, '-') insertToStr(solImg, 2*i+2, (1+chessWidth)*j+1+q, '-') insertToStr(solImg, 2*i+2, (1+chessWidth)*(j+1)+1+q, '-') insertToStr(solImg, 2*i, (1+chessWidth)*(j+1), '-') insertToStr(solImg, 2*i+2, (1+chessWidth)*(j+1), '-') insertToStr(solImg, 2*i, (1+chessWidth)*j, '+') insertToStr(solImg, 2*i, (1+chessWidth)*(j+2), '+') insertToStr(solImg, 2*i+2, (1+chessWidth)*j, '+') insertToStr(solImg, 2*i+2, (1+chessWidth)*(j+2), '+') solImg[2*i+1] = solImg[2*i+1][:(1+chessWidth)*j]\ + ('|%' + str(chessWidth) + 'd')%(chess[i][j])\ + ' ' + ('%' + str(chessWidth) + 'd|')%(chess[i][j+1])\ + solImg[2*i+1][(1+chessWidth)*(j+2)+1:] else: for q in range(chessWidth): insertToStr(solImg, 2*i, (1+chessWidth)*j+1+q, '-') insertToStr(solImg, 2*i+4, (1+chessWidth)*j+1+q, '-') insertToStr(solImg, 2*i+2, (1+chessWidth)*j, '|') insertToStr(solImg, 2*i+2, (1+chessWidth)*(j+1), '|') insertToStr(solImg, 2*i, (1+chessWidth)*j, '+') insertToStr(solImg, 2*i, (1+chessWidth)*(j+1), '+') insertToStr(solImg, 2*i+4, (1+chessWidth)*j, '+') insertToStr(solImg, 2*i+4, (1+chessWidth)*(j+1), '+') solImg[2*i+1] = solImg[2*i+1][:(1+chessWidth)*j]\ + ('|%' + str(chessWidth) + 'd|')%(chess[i][j])\ + solImg[2*i+1][(1+chessWidth)*(j+1)+1:] solImg[2*i+3] = solImg[2*i+3][:(1+chessWidth)*j]\ + ('|%' + str(chessWidth) + 'd|')%(chess[i+1][j])\ + solImg[2*i+3][(1+chessWidth)*(j+1)+1:] # # for i in solImg: print(i) return True
主要的处理步骤在main()这个函数里面：
def main(): start = time.time() with open('dominosaChess.txt','r') as f: chessStr = f.read() rowStrs = chessStr.split('\n') rowsize = len(rowStrs) colSize = len(rowStrs[0].split(' ')) maxnum = 0 for rowStr in rowStrs: row = [] for colStr in rowStr.split(' '): maxnum = maxnum if int(colStr) < maxnum else int(colStr) row.append(int(colStr)) chess.append(row) #print(chess) d = DancingLinksMatrix(get_names(rowsize, colSize, maxnum)) for i in range(rowsize): for j in range(colSize-1): row = compute_row(i, j, chess[i][j], chess[i][j+1], True, rowsize, colSize) #print(row) d.add_sparse_row(row, already_sorted=True) # for i in range(rowsize-1): for j in range(colSize): row = compute_row(i, j, chess[i][j], chess[i+1][j], False, rowsize, colSize) #print(row) d.add_sparse_row(row, already_sorted=True) d.end_add() p = PrintFirstSol(rowsize, colSize) AlgorithmX(d, p)() end = time.time() print('the DLX runtime is : ' + str(end-start) + 's')
总的代码就是这样：
""" N-queens solver using Dancing Links. """ from dlmatrix import DancingLinksMatrix from alg_x import AlgorithmX import math import time __author__ = 'Funcfans' chess = [] def insertToStr(origin, i, j, value): if(origin[i][j] != '+'): origin[i] = origin[i][:j] + value + origin[i][j+1:] def get_names(r, c, m): cnt = 0 for i in range(r): for j in range(c): yield f'L({i},{j})' # location pair cnt = cnt + 1 for i in range(m+2): for j in range(i): yield f'O({j},{i-1})' # number pair cnt = cnt + 1 # def compute_row(i, j, a, b, isHorizontal, r, c): # H is 0 .. r*(c-1)-1 # V is r*(c-1) .. r*(c-1)+c*(r-1)-1 # O is r*c*2-r-c .. r*c*2-r-c+(m-1)*m//2-1 if a < b: t = a a = b b = t row = [] if isHorizontal: row.append(c*i+j) row.append(c*i+j+1) else: row.append(c*i+j) row.append(c*(i+1)+j) row.append(r*c+a*(a+1)//2+b) return row class PrintFirstSol: def __init__(self, r, c): self.r = r self.c = c def __call__(self, sol): maxNum = 0 chessWidth = 0 for v in sol.values(): k = list(map(lambda a:int(a),v[2].split('(')[1].replace(')','').split(','))) maxNum = maxNum if k[0] < maxNum else k[0] maxNum = maxNum if k[1] < maxNum else k[1] while maxNum > 0: chessWidth += 1 maxNum //= 10 #print(chessWidth) solImg = [] for _ in range(self.r*2+2): Str1 = '' for _ in range(self.c*(chessWidth+1)+2): Str1 += ' ' solImg.append(Str1) for v in sol.values(): v.sort() v[0] = list(map(lambda a:int(a),v[0].split('(')[1].replace(')','').split(','))) v[1] = list(map(lambda a:int(a),v[1].split('(')[1].replace(')','').split(','))) v[2] = v[2].split('(')[1].replace(')','').split(',') #print(v) i = v[0][0] j = v[0][1] if(v[0][1] != v[1][1]): for q in range(chessWidth): insertToStr(solImg, 2*i, (1+chessWidth)*j+1+q, '-') insertToStr(solImg, 2*i, (1+chessWidth)*(j+1)+1+q, '-') insertToStr(solImg, 2*i+2, (1+chessWidth)*j+1+q, '-') insertToStr(solImg, 2*i+2, (1+chessWidth)*(j+1)+1+q, '-') insertToStr(solImg, 2*i, (1+chessWidth)*(j+1), '-') insertToStr(solImg, 2*i+2, (1+chessWidth)*(j+1), '-') insertToStr(solImg, 2*i, (1+chessWidth)*j, '+') insertToStr(solImg, 2*i, (1+chessWidth)*(j+2), '+') insertToStr(solImg, 2*i+2, (1+chessWidth)*j, '+') insertToStr(solImg, 2*i+2, (1+chessWidth)*(j+2), '+') solImg[2*i+1] = solImg[2*i+1][:(1+chessWidth)*j]\ + ('|%' + str(chessWidth) + 'd')%(chess[i][j])\ + ' ' + ('%' + str(chessWidth) + 'd|')%(chess[i][j+1])\ + solImg[2*i+1][(1+chessWidth)*(j+2)+1:] else: for q in range(chessWidth): insertToStr(solImg, 2*i, (1+chessWidth)*j+1+q, '-') insertToStr(solImg, 2*i+4, (1+chessWidth)*j+1+q, '-') insertToStr(solImg, 2*i+2, (1+chessWidth)*j, '|') insertToStr(solImg, 2*i+2, (1+chessWidth)*(j+1), '|') insertToStr(solImg, 2*i, (1+chessWidth)*j, '+') insertToStr(solImg, 2*i, (1+chessWidth)*(j+1), '+') insertToStr(solImg, 2*i+4, (1+chessWidth)*j, '+') insertToStr(solImg, 2*i+4, (1+chessWidth)*(j+1), '+') solImg[2*i+1] = solImg[2*i+1][:(1+chessWidth)*j]\ + ('|%' + str(chessWidth) + 'd|')%(chess[i][j])\ + solImg[2*i+1][(1+chessWidth)*(j+1)+1:] solImg[2*i+3] = solImg[2*i+3][:(1+chessWidth)*j]\ + ('|%' + str(chessWidth) + 'd|')%(chess[i+1][j])\ + solImg[2*i+3][(1+chessWidth)*(j+1)+1:] # # for i in solImg: print(i) return True def main(): start = time.time() with open('dominosaChess.txt','r') as f: chessStr = f.read() rowStrs = chessStr.split('\n') rowsize = len(rowStrs) colSize = len(rowStrs[0].split(' ')) maxnum = 0 for rowStr in rowStrs: row = [] for colStr in rowStr.split(' '): maxnum = maxnum if int(colStr) < maxnum else int(colStr) row.append(int(colStr)) chess.append(row) #print(chess) d = DancingLinksMatrix(get_names(rowsize, colSize, maxnum)) for i in range(rowsize): for j in range(colSize-1): row = compute_row(i, j, chess[i][j], chess[i][j+1], True, rowsize, colSize) #print(row) d.add_sparse_row(row, already_sorted=True) # for i in range(rowsize-1): for j in range(colSize): row = compute_row(i, j, chess[i][j], chess[i+1][j], False, rowsize, colSize) #print(row) d.add_sparse_row(row, already_sorted=True) d.end_add() p = PrintFirstSol(rowsize, colSize) AlgorithmX(d, p)() end = time.time() print('the DLX runtime is : ' + str(end-start) + 's') if __name__ == "__main__": main()
这里在代码同目录下创建一个dominosaChess.txt文件，并且输入以下布局：
7 5 1 2 6 7 6 7 3 2 1 1 1 6 7 0 2 3 5 7 3 6 2 0 4 2 0 1 4 4 5 6 1 6 1 5 7 2 0 5 4 3 3 0 4 2 7 0 0 7 6 5 1 3 4 5 5 4 1 0 6 2 0 6 4 5 3 4 7 3 2 3
可以得到以下输出：
+---+---+-+-+---+-+ |7 5|1 2|6|7|6 7|3| +-+-+-+-+ | +-+-+ | |2|1 1|1|6|7|0|2|3| | +---+ +-+-+ | +-+ |5|7 3|6|2 0|4|2|0| +-+---+-+-+-+-+-+ | |1|4 4|5 6|1|6|1|5| | +-+-+---+ | | +-+ |7|2|0|5 4|3|3|0|4| +-+ | +---+-+-+-+ | |2|7|0|0 7|6|5 1|3| | +-+-+---+ +---+-+ |4|5 5|4 1|0|6 2|0| +-+-+-+-+-+-+---+ | |6 4|5 3|4 7|3 2|3| +---+---+---+---+-+ the DLX runtime is : 0.005000591278076172s
可以看出来，这种方法解dominosa比其他方法解要快很多。
把答案填在dominosa里面并提交：
大功告成！