使用修改版Dancing link X （舞蹈链）求解aquarium游戏
如果把舞蹈表的所有行的消除条件，改成覆盖总值达到n后消除，而不是覆盖总值达到1后消除，并且覆盖的行值也不是1，那会怎么样？
就变成了多值覆盖游戏！（其实这不就是舞蹈链的重复覆盖特殊情况了嘛）
正好这里有个游戏要解：https://www.puzzle-aquarium.com/里面的aquarium游戏，规则是：
1、游戏棋盘被分成几块，每块被称为一个“水箱”
2、游戏中你可以给每个水箱灌一些水，也可以让它空着
3、同一个水箱内的水位是等高的。也即同一水箱内的同一行的单元格，要么都有水，要么都空着。
4、棋盘外面的数字是该行或该列，灌水的单元格总数。
说白了，每行每列所占的方格数必须正好等于右侧上面标明的限制数；同水箱内所有最高方格互相之间必须等高；同水箱内最高方格与最低方格内的所有方格必须全部填满。
如下所示：
图1.ID为69,467的谜面答案示例
这里同水箱有固定不同的方格排布，可以事先把不同高度的排布在横竖方向上的占有数量记录下来；像上图左下角有个水箱，对应到排布就是：
0分布；第9行有5个，第0列有1个，第1列有1个，第2列有1个，第3列有1个，第4列有1个；第8行有1个，第9行有5个，第0列有1个，第1列有2个，第2列有1个，第3列有1个，第4列有1个。
舞蹈表的列可以这样设计：不同的水箱占据；不同行的横方向方格占据数量；不同列的竖方向方格占据数量。这样就可以达成上述的解题思路。为了舞蹈表出解时能判读对应排布，这里把行的id和水箱ID、水高度对应起来。
主要的解决代码：
""" dominosa solver using Dancing Links. """ from functools import reduce from dlmatrix import DancingLinksMatrix from alg_x import AlgorithmX import math import time __author__ = 'Funcfans' chess = [] limits = [] presum = [] occupy = [] heights = [] currheights = [] heightIdxs = [] rowsize = 0 maxnum = 0 def insertToStr(origin, i, j, value): if(origin[i][j] != '+'): origin[i] = origin[i][:j] + value + origin[i][j+1:] def get_names(maxnum): cnt = 0 for i in range(maxnum): yield f'B({i})' # block cnt += 1 # base = maxnum presum.append([]) for i,limit in enumerate(limits[0]): if limit == 0:continue presum[0].append(base) yield (f'H({i},{limit})', limit) base += 1 # presum.append([]) for i,limit in enumerate(limits[1]): if limit == 0:continue presum[1].append(base) yield (f'V({i},{limit})', limit) base += 1 # def compute_row(value, maxnum): row = [] row.append(value) for d in range(2): for i,li in enumerate(occupy[d][value]): if li == 0:continue if li > limits[d][i]:return None row.append((presum[d][i],li)) return row # class PrintFirstSol: def __init__(self, r, c): self.r = r self.c = c def __call__(self, sol): subSubImg = ' ' * rowsize solImg = reduce(lambda a,b:a+'\n'+b,[subSubImg] * rowsize) blockheight = [0] * maxnum startHeight = {} for v in sol.keys(): #v.sort() blockheight[heights[v][0]] = heights[v][1] # # for i in range(rowsize)[::-1]: for j in range(rowsize): if chess[i][j] not in startHeight: startHeight[chess[i][j]] = i if startHeight[chess[i][j]] - blockheight[chess[i][j]] + 1 > i: chess[i][j] = -1 subSubImg = '-'.join(['+'] * (rowsize+1)) for i in range(rowsize): print(subSubImg) print('|',end='') for j in range(rowsize): if chess[i][j] == -1: print(' |',end='') else: print('*|',end='') print('') print(subSubImg) return True def main(fileName='aquarium69,467_3.txt'): currheights.clear() heights.clear() heightIdxs.clear() start = time.time() with open(fileName,'r') as f: chessStr = f.read() rowStrs = chessStr.split('\n') global rowsize, maxnum rowsize = len(rowStrs) - 2 colSize = len(rowStrs[0].split(' ')) maxnum = 0 for rowStr in rowStrs[:-2]: row = [] for colStr in rowStr.split(' '): maxnum = maxnum if int(colStr) < maxnum else int(colStr) row.append(int(colStr)) chess.append(row) #print(chess) list(map(lambda a:int(a),rowStrs[-1].split(' '))) limits.append(list(map(lambda a:int(a),rowStrs[-2].split(' ')))) limits.append(list(map(lambda a:int(a),rowStrs[-1].split(' ')))) maxnum += 1 d = DancingLinksMatrix(get_names(maxnum)) occupy.append([[0 for ___ in range(rowsize)] for __ in range(maxnum)]) occupy.append([[0 for ___ in range(rowsize)] for __ in range(maxnum)]) rowidx = 0 for i in range(maxnum): row = compute_row(i, maxnum) #print(f'row ({i},0) :', row) d.add_sparse_row(row, already_sorted=True) currheights.append(1) heightIdxs.append([i]) heights.append((i, 0)) rowidx += 1 for i,row in list(enumerate(chess))[::-1]: used = set() for j,col in enumerate(row): used.add(col) occupy[0][col][i] += 1 occupy[1][col][j] += 1 for col in used: row = compute_row(col, maxnum) if row == None or len(row) <= 0:continue #print(f'row ({col},{currheights[col]}) :', row) d.add_sparse_row(row, already_sorted=True) heights.append((col, currheights[col])) heightIdxs[col].append(rowidx) currheights[col] += 1 rowidx += 1 #print('heights :', heights) #print('heightIdxs :', heightIdxs) ansIdx=[0 ,62 ,63 ,25 ,37 ,52 ,32 ,54 ,40 ,17 ,34 ,19 ,24 ,13] #[0, 13, 17, 19, 24, 25, 32, 34, 37, 40, 52, 54, 62, 63] ansIdx.sort() #print(ansIdx) 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('aquarium2,680,806_8.txt')
这里的解决算法有些变化，需要修改之前的模板。
解出问题的算法：
""" Implementation of Donald Knuth's Algorithm X (http://arxiv.org/abs/cs/0011047). """ from dlmatrix import DancingLinksMatrix, iterate_cell, MatrixDisplayer import string __author__ = 'FunCfans' testRow = [0 ,62 ,63 ,25 ,37 ,52 ,32 ,54 ,40 ,17 ,34 ,19 ,24 ,13] class AlgorithmX: """Callable object implementing the Algorithm X.""" def __init__(self, matrix, callback, choose_min=True): """ Creates an Algorithm_X object that solves the problem encoded in matrix. :param matrix: The DL_Matrix instance. :param callback: The callback called on every solution. callback has to be a function receiving a dict argument {row_index: linked list of the row}, and can return a bool value. The solver keeps going on until the callback returns a True value. :param choose_min: If True, the column with the minimum number of 1s is chosen at each iteration, if False a random column is chosen. """ self.sol_dict = {} self.stop = False self.matrix = matrix self.callback = callback self.choose_min = choose_min self.deduce_cnt = 0 self.depth = 0 self.last_matrix = None self.delta_file = None def __call__(self): """Starts the search.""" #self.delta_file = open('delta_matrix.txt','w',encoding='utf-8') #self._print(self.matrix.header, 'start') self._search(0) #self.delta_file.close() def _print(self, currrow, op): self.deduce_cnt += 1 f = open('step/' + 'step%04d_' % (self.deduce_cnt) + op + '_' + str(currrow.indexes) + '_depth%d.txt'%(self.depth),'w',encoding='utf-8') printrow = {} rowcontent = '' content = 'curr ' + op + ' row : ' + str(currrow.indexes) + '\n' content = '' for col in iterate_cell(self.matrix.header, 'R'): content += 'col name : '+col.name+' col limit : ' + str(col.limit_num) for row in iterate_cell(col, 'D'): content += ' ' + str((row.indexes[0],row.limit_num)) + ',' printrow[row.indexes[0]] = row content += '\n' content += '\n' for k,v in printrow.items(): content += 'row %d : '%(k) + str((v.limit_num, v.indexes[1])) + '-->' qv = v.R while(qv != v): content += str((qv.limit_num, qv.indexes[1])) + '-->' qv = qv.R content += str((qv.limit_num, qv.indexes[1])) + '\n' f.write(content) f.close() if self.last_matrix == None: self.last_matrix = {} for col in iterate_cell(self.matrix.header, 'R'): self.last_matrix[(col.name, col.indexes[0])] = collist = set() for row in iterate_cell(col, 'D'): collist.add(row.indexes[0]) else: curr_matrix = {} for col in iterate_cell(self.matrix.header, 'R'): curr_matrix[(col.name, col.indexes[0])] = collist = set() for row in iterate_cell(col, 'D'): collist.add(row.indexes[0]) add_col = set() addv = set(curr_matrix.keys()).difference(set(self.last_matrix.keys())) if len(addv) > 0: self.delta_file.write('add_col : '+str(addv) + ' ') delv = set(self.last_matrix.keys()).difference(set(curr_matrix.keys())) if len(delv) > 0: self.delta_file.write('delete_col : '+str(delv) + ' ') for k,v in curr_matrix.items(): if k not in self.last_matrix:continue addv = v.difference(self.last_matrix[k]) if len(addv) > 0: self.delta_file.write('add_col_'+str(k)+' : '+str(addv)) self.delta_file.write(' ') for k,v in self.last_matrix.items(): if k not in curr_matrix:continue delv = self.last_matrix[k].difference(v) if len(delv) > 0: self.delta_file.write('delete_col_'+str(k)+' : '+str(delv)) self.delta_file.write(' ') self.delta_file.write('\n') self.last_matrix = curr_matrix def _search(self, k): # print(f"Size: {k}") # k is depth # print(f"Solution: {self.sol_dict}") # print("Matrix:") # print(self.matrix) if self.matrix.header.R == self.matrix.header: # matrix is empty, solution found if self.callback(self._create_sol(k)): self.stop = True return if self.choose_min: col = self.matrix.min_column() else: col = self.matrix.random_column() # cover column col # row = col.D rows = [] for row in iterate_cell(col, 'D'): rows.append(row) rows.sort(key=lambda x:x.limit_num,reverse=True) self.depth += 1 for row in rows: if col.limit_num < row.limit_num:continue isValid = True for j in iterate_cell(row, 'R'): if j.C.limit_num < j.limit_num: isValid = False break if not isValid: continue self.sol_dict[k] = row col.limit_num -= row.limit_num row.D.U = row.U row.U.D = row.D col.size -= 1 if col.limit_num == 0: self.matrix.cover(col) for j in iterate_cell(row, 'R'): j.C.limit_num -= j.limit_num j.D.U = j.U j.U.D = j.D j.C.size -= 1 if j.C.limit_num == 0: self.matrix.cover(j.C) execYou = True for j in iterate_cell(self.matrix.header, 'R'): if j.limit_num > j.sum_limit_num: execYou = False break if execYou: self._search(k + 1) if self.stop: return # uncover columns for j in iterate_cell(row, 'L'): if j.C.limit_num == 0: self.matrix.uncover(j.C) j.C.size += 1 j.D.U = j.U.D = j j.C.limit_num += j.limit_num if col.limit_num == 0: self.matrix.uncover(col) col.size += 1 row.D.U = row.U.D = row col.limit_num += row.limit_num self.depth -= 1 # def _create_sol(self, k): # creates a solution from the inner dict sol = {} for key, row in self.sol_dict.items(): if key >= k: continue tmp_list = [row.C.name] tmp_list.extend(r.C.name for r in iterate_cell(row, 'R')) sol[row.indexes[0]] = tmp_list return sol # def main(): from_dense = (lambda row:[i for i, el in enumerate(row) if el]) rows = [from_dense([0, 0, 1, 0, 1, 1, 0]), from_dense([1, 0, 0, 1, 0, 0, 1]), from_dense([0, 1, 1, 0, 0, 1, 0]), from_dense([1, 0, 0, 1, 0, 0, 0]), from_dense([0, 1, 0, 0, 0, 0, 1]), from_dense([0, 0, 0, 1, 1, 0, 1])] size = max(max(rows, key=max)) + 1 d = DancingLinksMatrix(string.ascii_uppercase[:size]) for row in rows: d.add_sparse_row(row, already_sorted=True) AlgorithmX(d, print)() # if __name__ == "__main__": main()
舞蹈表数据结构：
""" Implementation of Donald Knuth's Dancing Links Sparse Matrix as a circular doubly linked list. (http://arxiv.org/abs/cs/0011047) """ import random import numpy as np __author__ = "FunCfans" class CannotAddRowsError(Exception): pass # class EmptyDLMatrix(Exception): pass # class Cell: """ Inner cell, storing 4 pointers to neighbors, a pointer to the column header and the indexes associated. """ __slots__ = list("UDLRC") + ["indexes", "limit_num"] def __init__(self, limitNum=1): self.U = self.D = self.L = self.R = self self.C = None self.indexes = None self.limit_num = limitNum def __str__(self): return f"Node: {self.indexes}" def __repr__(self): return f"Cell[{self.indexes}]" class HeaderCell(Cell): """ Column Header cell, a special cell that stores also a name and a size member. """ __slots__ = ["size", "name", "is_first", "sum_limit_num"] def __init__(self, name, limitNum=1): super(HeaderCell, self).__init__(limitNum) self.size = 0 self.name = name self.is_first = False self.sum_limit_num = 0 class DancingLinksMatrix: """ Dancing Links sparse matrix implementation. It stores a circular doubly linked list of 1s, and another list of column headers. Every cell points to its upper, lower, left and right neighbors in a circular fashion. """ def __init__(self, columns): """ Creates a DL_Matrix. :param columns: it can be an integer or an iterable. If columns is an integer, columns columns are added to the matrix, named C0,...,CN where N = columns -1. If columns is an iterable, the number of columns and the names are deduced from the iterable, else TypeError is raised. The iterable may yield the names, or a tuple (name,primary). primary is a bool value that is True if the column is a primary one. If not specified, is assumed that the column is a primary one. :raises TypeError, if columns is not a number neither an iterable. """ self.header = HeaderCell("<H>") self.header.is_first = True self.rows = self.cols = 0 self.col_list = [] self._create_column_headers(columns) def _create_column_headers(self, columns): if isinstance(columns, int): columns = int(columns) column_names = ((f"C{i}", 1) for i in range(columns)) else: try: column_names = iter(columns) except TypeError: raise TypeError("Argument is not valid") prev = self.header # links every column in a for loop for name in column_names: primary = True if isinstance(name, tuple) or isinstance(name, list): name, limitNum = name else: limitNum = 1 cell = HeaderCell(name, limitNum) cell.indexes = (-1, self.cols) cell.is_first = False self.col_list.append(cell) if primary: prev.R = cell cell.L = prev prev = cell self.cols += 1 prev.R = self.header self.header.L = prev def add_sparse_row(self, row, already_sorted=False): """ Adds a sparse row to the matrix. The row is in format [ind_0, ..., ind_n] where 0 <= ind_i < dl_matrix.ncols. If called after end_add is executed, CannotAddRowsError is raised. :param row: a sequence of integers indicating the 1s in the row. :param already_sorted: True if the row is already sorted, default is False. Use it for performance optimization. :raises CannotAddRowsError if end_add was already called. """ if self.col_list is None: raise CannotAddRowsError() prev = None start = None if not already_sorted: row = sorted(row) cell = None for ind in row: if isinstance(ind, int): ind = (ind, 1) cell = Cell(ind[1]) cell.indexes = (self.rows, ind[0]) if prev: prev.R = cell cell.L = prev else: start = cell col = self.col_list[ind[0]] # link the cell with the previous one and with the right column # cells. last = col.U last.D = cell cell.U = last col.U = cell cell.D = col cell.C = col col.size += 1 prev = cell col.sum_limit_num += ind[1] start.L = cell cell.R = start self.rows += 1 def end_add(self): """ Called when there are no more rows to be inserted. Not strictly necessary, but it can save some memory. """ self.col_list = None def min_column(self): """ Returns the column header of the column with the minimum number of 1s. :return: A column header. :raises: EmptyDLMatrix if the matrix is empty. """ # noinspection PyUnresolvedReferences if self.header.R.is_first: raise EmptyDLMatrix() col_min = self.header.R for col in iterate_cell(self.header, 'R'): if not col.is_first and col.size < col_min.size: col_min = col return col_min def random_column(self): """ Returns a random column header. (The matrix header is never returned) :return: A column header. :raises: EmptyDLMatrix if the matrix is empty. """ col = self.header.R if col is self.header: raise EmptyDLMatrix() n = random.randint(0, self.cols - 1) for _ in range(n): col = col.R if col.is_first: col = col.R return col def __str__(self): names = [] m = np.zeros((self.rows, self.cols), dtype=np.uint8) rows, cols = set(), [] for col in iterate_cell(self.header, 'R'): cols.append(col.indexes[1]) # noinspection PyUnresolvedReferences names.append(col.name) for cell in iterate_cell(col, 'D'): ind = cell.indexes rows.add(ind[0]) m[ind] = 1 m = m[list(rows)][:, cols] return "\n".join([", ".join(names), str(m)]) @staticmethod def coverRow(r, isadd=False): for j in iterate_cell(r, 'R'): if j.C.limit_num < j.limit_num: return False for j in iterate_cell(r, 'R'): j.D.U = j.U j.U.D = j.D j.C.size -= 1 j.C.sum_limit_num -= j.limit_num if isadd: j.C.limit_num -= j.limit_num return True @staticmethod def checkCover(c): subLimitNum = {} for i in iterate_cell(c, 'D'): for j in iterate_cell(i, 'R'): idx = j.indexes[1] if idx not in subLimitNum: subLimitNum[idx] = 0 subLimitNum[idx] += j.limit_num if j.C.sum_limit_num - subLimitNum[idx] < j.C.limit_num: return False return True @staticmethod def cover(c, isadd=False): """ Covers the column c by removing the 1s in the column and also all the rows connected to them. :param c: The column header of the column that has to be covered. """ # print("Cover column", c.name) c.R.L = c.L c.L.R = c.R for i in iterate_cell(c, 'D'): DancingLinksMatrix.coverRow(i, isadd) return True @staticmethod def uncoverRow(r, isadd=False): for j in iterate_cell(r, 'L'): j.C.sum_limit_num += j.limit_num j.C.size += 1 j.D.U = j.U.D = j if isadd: j.C.limit_num += j.limit_num return True @staticmethod def uncover(c, isadd=False): """ Uncovers the column c by readding the 1s in the column and also all the rows connected to them. :param c: The column header of the column that has to be uncovered. """ # print("Uncover column", c.name) for i in iterate_cell(c, 'U'): DancingLinksMatrix.uncoverRow(i, isadd) c.R.L = c.L.R = c def iterate_cell(cell, direction): cur = getattr(cell, direction) while cur is not cell: yield cur cur = getattr(cur, direction) # TODO to be completed class MatrixDisplayer: def __init__(self, matrix): dic = {} for col in iterate_cell(matrix.header, 'R'): dic[col.indexes] = col for col in iterate_cell(matrix.header, 'R'): first = col.D dic[first.indexes] = first for cell in iterate_cell(first, 'D'): if cell is not col: dic[cell.indexes] = cell self.dic = dic self.rows = matrix.rows self.cols = matrix.cols def print_matrix(self): m = {} for i in range(-1, self.rows): for j in range(0, self.cols): cell = self.dic.get((i, j)) if cell: if i == -1: m[0, 2 * j] = cell.name+','+str(cell.limit_num) else: m[2 * (i + 1), 2 * j] = "X"+','+str(cell.limit_num) for i in range(-1, self.rows * 2): for j in range(0, self.cols * 2): print(m.get((i, j), " "), end="") print() if __name__ == "__main__": def from_dense(row): return [i for i, el in enumerate(row) if el] r = [from_dense([1, 0, 0, 1, 0, 0, 1]), from_dense([1, 0, 0, 1, 0, 0, 0]), from_dense([0, 0, 0, 1, 1, 0, 1]), from_dense([0, 0, 1, 0, 1, 1, 0]), from_dense([0, 1, 1, 0, 0, 1, 1]), from_dense([0, 1, 0, 0, 0, 0, 1])] d = DancingLinksMatrix("1234567") for row in r: d.add_sparse_row(row, already_sorted=True) d.end_add() p = MatrixDisplayer(d) p.print_matrix() # print(d.rows) # print(d.cols) # print(d) mc = d.min_column() # print(mc) d.cover(mc) # print(d) p.print_matrix()
为了能获取谜面，我们需要把谜面转换为文件，这里的谜面排布与star battle类似，稍微改一下代码就可以用了：
from lxml import etree from selenium import webdriver from selenium.webdriver.chrome.options import Options import copy deepcopy = copy.deepcopy monthlyUrl = 'https://www.puzzle-aquarium.com/?size=11' weeklyUrl = 'https://www.puzzle-aquarium.com/?size=10' dailyUrl = 'https://www.puzzle-aquarium.com/?size=9' url = 'https://www.puzzle-aquarium.com/?size=8' # def getChessByChrome(): path = r'D:\chromedriver.exe' chrome_options = Options() #后面的两个是固定写法 必须这么写 chrome_options.add_argument('--headless') chrome_options.add_argument('--disable-gpu') driver = webdriver.Chrome(executable_path=path,chrome_options=chrome_options) try: driver.set_page_load_timeout(30) driver.get(url) except Exception as e: print(e) source = driver.page_source driver.quit() return source # def getChessByFile(): with open('aquarium.html','r',encoding='utf-8') as f:source=f.read() return source # def solve(): if url.find('size=') == -1: size = 0 else: size = url.split('size=')[1] size = int(size) source = getChessByChrome() htree = etree.HTML(source) chessSize = len(htree.xpath('//div[@id="game"]/div/div')) puzzleId = htree.xpath('//div[@class="puzzleInfo"]/p/span/text()') if len(puzzleId) != 0: puzzleId = puzzleId[0] else: puzzleId = htree.xpath('//div[@class="puzzleInfo"]/p/text()')[0] chessSize = round(chessSize**0.5) chess = [[-1 for _ in range(chessSize)] for __ in range(chessSize)] borderss = [['' for _ in range(chessSize)] for __ in range(chessSize)] chessStr = '' maxBlockNumber = 0 # br: on the right; bl: on the left; bb: on the down; bt: on the up for i,className in enumerate(htree.xpath('//div[@id="game"]/div/div[contains(@class,"cell")]')): x = i // chessSize y = i % chessSize value = className.xpath('./@class')[0] if value[:4] != 'cell': continue value = value.replace('cell selectable','') value = value.replace('cell-off','') borderss[x][y] = value for i in range(chessSize): for j in range(chessSize): if chess[i][j] != -1: continue queue = [(i, j)] chess[i][j] = str(maxBlockNumber) while len(queue) > 0: oldQueue = deepcopy(queue) queue = [] for pos in oldQueue: x, y = pos[0], pos[1] # if x > 0 and borderss[x][y].find('bt') == -1 and chess[x-1][y] == -1: queue.append((x-1, y)) chess[x-1][y] = chess[i][j] # if x < chessSize - 1 and borderss[x][y].find('bb') == -1 and chess[x+1][y] == -1: queue.append((x+1, y)) chess[x+1][y] = chess[i][j] # if y > 0 and borderss[x][y].find('bl') == -1 and chess[x][y-1] == -1: queue.append((x, y-1)) chess[x][y-1] = chess[i][j] # if y < chessSize - 1 and borderss[x][y].find('br') == -1 and chess[x][y+1] == -1: queue.append((x, y+1)) chess[x][y+1] = chess[i][j] # maxBlockNumber += 1 chessStr = '\n'.join(' '.join(chessRow) for chessRow in chess) chessStr += '\n' + ' '.join(htree.xpath('//div[@class="cell task h"]/text()')) chessStr += '\n' + ' '.join(htree.xpath('//div[@class="cell task v"]/text()')) with open('aquarium' + puzzleId + '_' + str(size) + '.txt','w') as f:f.write(chessStr) # if __name__ == '__main__': solve()
抓取10*10 easy，ID为69,467的谜面，储存文件并执行上述代码，结果如下：
+-+-+-+-+-+-+-+-+-+-+ | | | |*|*|*|*|*|*|*| +-+-+-+-+-+-+-+-+-+-+ | | | | |*| | | | |*| +-+-+-+-+-+-+-+-+-+-+ | | | |*|*| | |*|*|*| +-+-+-+-+-+-+-+-+-+-+ | | |*|*|*| | |*|*|*| +-+-+-+-+-+-+-+-+-+-+ | | |*|*|*|*| | | |*| +-+-+-+-+-+-+-+-+-+-+ | |*|*|*|*|*|*| | | | +-+-+-+-+-+-+-+-+-+-+ |*|*|*|*|*|*|*| | | | +-+-+-+-+-+-+-+-+-+-+ |*|*|*| |*| |*|*|*| | +-+-+-+-+-+-+-+-+-+-+ |*| |*|*|*|*|*|*|*|*| +-+-+-+-+-+-+-+-+-+-+ | | | | | |*|*|*|*|*| +-+-+-+-+-+-+-+-+-+-+ the DLX runtime is : 24.061376333236694s
与示例谜底相同，大功告成！
（一定要有版本管理的习惯，不然会吧自己坑进去。这里为了debug有瑕疵的算法，没有保存就把原先可能是正确的代码给修掉了T_T）
