使用改良版多值覆盖Dancing link X （舞蹈链）求解aquarium游戏

Posted 2020-11-25 dgutfly

　　在上一篇文章中，我们通过改造了dancing link代码解出了aquarium游戏，并输出了正确答案。

　　但是之前的代码感觉有些慢，10*10的谜面都要跑24秒，而且感觉之前的dancing link代码有些不完善（存在重复查询问题）。这一篇文章介绍如何改良多值覆盖dancing link模板代码，还有如何在整体上优化这个游戏的解题流程。

　　之前的代码是从所有列中选择可能性最少的列进行突破，以减少查询宽度；但是在查询过程中发现了问题：之前查询过的较高占据值的行可能会再次被查询到，从而浪费不少时间。这里就需要在每个列的查询过程做额外处理：先从值最大的行开始消除，并且退出对这个列的遍历之前都不还原这些行。

　　之前遍历序列可能是这样：[1,2,3] [1,2,4] [1,3,2] [1,3,4] [1,4,2] [1,4,3]，现在的序列则是：[1,2,3] [1,2,4] [1,3,4] [2,3,4]，查询次数少了一些（别小看这少掉的2次·，反应到查询树里面影响可能显著，特别是最初的几层）

　　还有一些重要的优化：如果消除的行过多，导致最终能选的行总值无法满足要求，也要及时返回。为了随时随地查验总值，防止过度消除，在headerCell里面添加一个属性：xum_limit_num，并根据这个思路进行优化：

"""
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 "
".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()

dlmatrix.py

　　还有这个：

"""
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) + ‘
‘
        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 += ‘
‘
        content += ‘
‘
        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])) + ‘
‘
        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(‘
‘)
            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
            if col.sum_limit_num < col.limit_num:break
            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.sum_limit_num -= row.limit_num
            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.sum_limit_num -= j.limit_num
                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
            for j in iterate_cell(row, ‘L‘):
                if j.C.limit_num == 0:
                    self.matrix.uncover(j.C)
                j.C.limit_num += j.limit_num
            if col.limit_num == 0:
                self.matrix.uncover(col)
            col.limit_num += row.limit_num
            del self.sol_dict[k]
            # uncover columns

        for row in rows[::-1]:
            for j in iterate_cell(row, ‘L‘):
                j.C.size += 1
                j.D.U = j.U.D = j
                j.C.sum_limit_num += j.limit_num
            col.size += 1
            row.D.U = row.U.D = row
            col.sum_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()

alg_x.py

　　利用这个解上一篇文章里面提到的10*10 easy，ID为69,467的谜面，运行时间为16秒。这说明了这个优化有效果。

　　但是，从原理上来讲，这个解法只是锦上添花而已。重要的是事前无效解的裁剪。

　　这道题目有这俩要求：

　　1.水箱内同高度的方格状态必须一致；

　　2.水箱内的方格必须自底向上填满。

　　就是说在指定数量条件下，还要满足这些条件。可以利用这些条件反应到单行中的所有可能性，初步排除错误解：

　　1.水箱内同高度的方格状态必须一致。这意味着某些占据长度过长的水箱段必须涂上去：

　　比如如下阵型：（下面水箱编号是0,0,0,0,1,1,1,2,2,3）

 

图1

 

　　可以涂上这种阵型：

 

图2

 

　　也可以这样涂：

 

 

图3

　　但是无论怎么涂，最左边的那4格都必须涂上（如果所有的阵型固定点都是白色，则亦可判定不可填涂），可以将最终阵型的对应位置填上；

　　这里最左边必须涂的原理还不止是这个：这里有10格，限制数量是7，如果去掉了严格大于(10-7)=3的水箱段，其他的水箱段加起来也达不到条件。

　　也有那种水箱占宽超过限制数量的：（下面水箱编号是0,0,0,0,0,1,1,1,2,2,2,2,2,3,3）

图4

 

 

　　像这种，除了最右边的1个水箱段，其他3个水箱段都要排除。

　　2.水箱内的方格必须自底向上填满，这意味着某些方格可以预先确定。与横段不同，竖段无需通过暴力搜索找到必经之路

　　看下栗子：（自底向上是8个0，5个1，2个2）

图5

 

 

　　如果把最下段去掉，其他段加起来也不够限制，因此下段至少需要加一些水到最底下。那要加多少格水呢，答案当然是9-(15-8)=2

　　再看以下栗子：（自底向上是5个0，3个1，5个2，2个3）

图6

 

 

　　可以看出，所有段严格大于2的段，上方都要用不到的区域，比如最下段，5个方格，最上面5-2=3个是用不了的。

　　初期阵型优化完毕之后再执行操作，可以减少许多不必要的搜索。

　　代码如下：

"""
dominosa solver using Dancing Links.
"""
from functools import reduce
from dlmatrix import DancingLinksMatrix
from alg_x import AlgorithmX
import math
import time
import copy
deepcopy = copy.deepcopy
__author__ = ‘Funcfans‘
chess = []
valid = []
limits = []
presum = []
occupy = []
heights = []
currheights = []
heightIdxs = []
heightRange = []
haveBlock = {}
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 printAnswer():
    subSubImg = ‘ ‘ * rowsize
    solImg = reduce(lambda a,b:a+‘
‘+b,[subSubImg] * rowsize)
    blockheight = [0] * maxnum
    startHeight = {}
    for i in range(rowsize)[::-1]:
        for j in range(rowsize):
            if valid[i][j] == -1:
                chess[i][j] = -1
            if valid[i][j] == 1:
                continue
            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)
#
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]):
        presum[0].append(base)
        if limit != 0:
            yield (f‘H({i},{limit})‘, limit)
            base += 1
    #
    presum.append([])
    for i,limit in enumerate(limits[1]):
        presum[1].append(base)
        if limit != 0:
            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+‘
‘+b,[subSubImg] * rowsize)
        blockheight = [0] * maxnum
        startHeight = [-1] * maxnum
        reverseBlock = {}
        for k,v in haveBlock.items():
            reverseBlock[v] = k
        for k,v in sol.items():
            v.sort()
            v[0] = ‘B(‘ + str(reverseBlock[int(v[0].replace(‘B(‘,‘‘).replace(‘)‘,‘‘))]) + ‘)‘
            blockheight[heights[k][0]] = heights[k][1]
            #
        #
        for i in range(rowsize)[::-1]:
            for j in range(rowsize):
                if valid[i][j] == 1:
                    continue
                if valid[i][j] == -1:
                    chess[i][j] = -1
                    continue
                if startHeight[chess[i][j]] == -1:
                    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 valid[i][j] == 1:
                    print(‘*|‘,end=‘‘)
                elif chess[i][j] == -1:
                    print(‘ |‘,end=‘‘)
                else:
                    print(‘*|‘,end=‘‘)
            print(‘‘)
        print(subSubImg)
        return True
#
allow = 0
def tryHorizontal(rowNum, startPos, subValid):
    have = {}
    totalvalid = rowsize
    totallimit = limits[0][rowNum]
    for i in range(rowsize)[::-1]:
        if valid[rowNum][i] != 0:
            totalvalid -= 1
            if valid[rowNum][i] == 1:
                totallimit -= 1
            continue
        value = chess[rowNum][i]
        if value not in have:
            have[value] = [i]
        else:
            have[value].append(i)
    items = []
    for k,v in have.items():
        if len(v) > totallimit:
            for j in v:
                subValid[j] = -1
            totalvalid -= len(v)
        else:
            items.append((k, v))
    items.sort(key=lambda a:len(a[1]),reverse=True)
    for i in items:
        v = i[1]
        if len(v) > totallimit:
            for j in v:
                subValid[j] = -1
            totalvalid -= len(v)
            del items[items.index(i)]
        elif len(v) > totalvalid - totallimit:
            for j in v:
                subValid[j] = 1
            totalvalid -= len(v)
            totallimit -= len(v)
            del items[items.index(i)]
    for i in range(2**len(items)):
        tryValid = [0] * rowsize
        mylimit = 0
        for j in range(len(items)):
            if (i // (2**j)) % 2 == 1:
                for k in items[j][1]:
                    tryValid[k] = 1
                mylimit += len(items[j][1])
            else:
                for k in items[j][1]:
                    tryValid[k] = -1
        if mylimit == totallimit:
            for j in range(rowsize):
                if tryValid[j] == 0:continue
                elif subValid[j] == -2:continue
                elif subValid[j] == 0:subValid[j] = tryValid[j]
                elif subValid[j] != tryValid[j]:subValid[j] = -2
    return subValid
#
def fill():
    isChange = False
    for i in range(rowsize):
        for checkValue in (-1,1):
            checkList = valid[i]
            if checkValue == -1:
                youlimit = limits[0][i]
            else:
                youlimit = rowsize - limits[0][i]
            if checkList.count(-checkValue) == youlimit:
                for j in range(rowsize):
                    if valid[i][j] != -checkValue:
                        valid[i][j] = checkValue
                        if valid[i][j] != checkValue:
                            isChange = True
            #
            checkList = list(map(lambda q:q[i], valid))
            if checkValue == -1:
                youlimit = limits[1][i]
            else:
                youlimit = rowsize - limits[1][i]
            if checkList.count(-checkValue) == youlimit:
                for j in range(rowsize):
                    if valid[j][i] != -checkValue:
                        valid[j][i] = checkValue
                        if valid[j][i] != checkValue:
                            isChange = True
            #
        #
    return isChange
#
def tryVertical(colNum, startPos, subValid):
    have = {}
    totalvalid = rowsize
    totallimit = limits[1][colNum]
    for i in range(rowsize)[::-1]:
        if valid[i][colNum] != 0:
            totalvalid -= 1
            if valid[i][colNum] == 1:
                totallimit -= 1
            continue
        value = chess[i][colNum]
        if value not in have:
            have[value] = [i]
        else:
            have[value].append(i)
    for k,v in have.items():
        for i in v[totallimit:]:
            subValid[i] = -1
        if totallimit-totalvalid+len(v) > 0:
            for i in v[:totallimit-totalvalid+len(v)]:
                subValid[i] = 1
    return subValid
#
def fill_up_down(blockNum, rowNum, value):
    #print(‘blockNum :‘,blockNum,‘rowNum :‘,rowNum,‘value :‘,value)
    queryCnts = range(rowNum+1)[::-1] if value == -1 else range(rowNum, rowsize)
    for k in queryCnts:
        isContinue = False
        for l in range(rowsize):
            if blockNum != chess[k][l]:continue
            if valid[k][l] == value:break
            else:
                valid[k][l] = value
                retry = True
                if value == -1:
                    if heightRange[blockNum][0] < k:
                        heightRange[blockNum][0] = k
                else:
                    if heightRange[blockNum][1] > k:
                        heightRange[blockNum][1] = k
#
def trySolve():
    retry = True
    while retry:
        retry = False
        for i in range(rowsize):
            resultSet = tryHorizontal(i, 0, [0] * rowsize)
            ‘‘‘
            print(‘limit num :‘,limits[0][i])
            print(‘resultSet :‘,resultSet)
            print(‘hori valid :‘,valid[i])
            print(‘hori chess :‘,chess[i])
            ‘‘‘
            for j in range(rowsize):
                blockNum = chess[i][j]
                if resultSet[j] in (1,-1):
                    if valid[i][j] == resultSet[j]:continue
                    fill_up_down(blockNum, i, resultSet[j])
                    retry = True
        for i in range(rowsize):
            resultSet = tryVertical(i, rowsize-1, [0] * rowsize)
            for j in range(rowsize):
                blockNum = chess[j][i]
                if resultSet[j] in (1,-1):
                    if valid[j][i] == resultSet[j]:continue
                    fill_up_down(blockNum, j, resultSet[j])
                    retry = True
        if not retry:retry = fill()
        #print(‘------‘)
    #
    ‘‘‘
    for i in range(rowsize):
        for j in range(rowsize):
            if valid[i][j] == -1:
                print(‘X‘,end=‘‘)
            elif valid[i][j] == 0:
                print(‘ ‘,end=‘‘)
            else:
                print(‘*‘,end=‘‘)
        print()
    ‘‘‘
#
def main(fileName=‘aquarium69,467_3.txt‘):
    currheights.clear()
    heights.clear()
    heightIdxs.clear()
    heightRange.clear()
    haveBlock.clear()
    start = time.time()
    with open(fileName,‘r‘) as f:
        chessStr = f.read()
    rowStrs = chessStr.split(‘
‘)
    global rowsize, maxnum
    rowsize = len(rowStrs) - 2
    colSize = len(rowStrs[0].split(‘ ‘))
    maxnum = 0
    for rowStr in rowStrs[:-2]:
        row = []
        validRow = []
        for colStr in rowStr.split(‘ ‘):
            maxnum = maxnum if int(colStr) < maxnum else int(colStr)
            row.append(int(colStr))
            validRow.append(0)
        chess.append(row)
        valid.append(validRow)
    #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
    for _ in range(maxnum):heightRange.append([-1,rowsize])
    occupy.append([[0 for ___ in range(rowsize)] for __ in range(maxnum)])
    occupy.append([[0 for ___ in range(rowsize)] for __ in range(maxnum)])
    rowidx = 0
    trySolve()
    isSolve = True
    for i in range(rowsize):
        for j in range(rowsize):
            if valid[i][j] == 0:
                isSolve = False
            if valid[i][j] == 1:
                limits[0][i] -= 1
                limits[1][j] -= 1
    if isSolve:
        printAnswer()
        end = time.time()
        print(‘the DLX runtime is : ‘ + str(end-start) + ‘s‘)
        return
    #print(presum)
    
    startIdx = 0
    for i in range(rowsize):
        for j in range(rowsize):
            if valid[i][j] == 0:
                if chess[i][j] not in haveBlock:
                    haveBlock[chess[i][j]] = startIdx
                    chess[i][j] = startIdx
                    startIdx += 1
                else:
                    chess[i][j] = haveBlock[chess[i][j]]
    #print(haveBlock)
    d = DancingLinksMatrix(get_names(startIdx))
    for i in range(maxnum):
        currheights.append(0)
        heightIdxs.append([])
    for i in haveBlock.values():
        row = compute_row(i, startIdx)
        #print(‘row :‘, row)
        d.add_sparse_row(row, already_sorted=True)
        heightIdxs[i].append(rowidx)
        heights.append((i, 0))
        rowidx += 1
        currheights[i] += 1
    for i,row in list(enumerate(chess))[::-1]:
        used = set()
        for j,col in enumerate(row):
            if valid[i][j] != 0:continue
            used.add(col)
            occupy[0][col][i] += 1
            occupy[1][col][j] += 1
        for col in used:
            row = compute_row(col, maxnum)# or i <= heightRange[col][0] or i >= heightRange[col][1]
            #print(‘row :‘, row)
            heightIdxs[col].append(rowidx)
            heights.append((col, currheights[col]))
            d.add_sparse_row(row, already_sorted=True)
            rowidx += 1
            currheights[col] += 1
    #print(‘heightRange :‘, heightRange)
    #print(‘heightIdxs :‘, heightIdxs)
    print(‘rowidx :‘, rowidx)
    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‘)
    #main(‘aquarium5,434,697_7.txt‘)
    #main(‘aquariumSpecial Daily  17-12-2019_9.txt‘)
    main()

aquarium.py

　　运行10*10 easy，ID为69,467的谜面，时间为0.005000591278076172秒，效果显著！

　　运行15*15 hard，ID为5,434,697的谜面，运行结果：

+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| | |*|*|*|*|*|*|*|*| | | | | |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| | | | | |*|*|*|*| | | | | |*|
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| | | | | | |*|*|*|*| | | | | |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
|*|*|*| | | |*|*|*|*|*|*|*|*|*|
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| |*| | | |*|*| |*|*|*|*| | | |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| | |*|*|*|*|*|*|*|*|*|*|*| | |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
|*| |*|*| |*|*|*|*|*|*|*|*|*| |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
|*|*|*|*|*| | |*|*|*|*|*|*|*| |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
|*|*| |*|*|*| |*|*|*|*|*|*|*|*|
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
|*|*|*|*|*|*|*|*|*| | |*|*|*|*|
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
|*|*|*|*|*|*|*|*|*|*|*|*| |*|*|
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| | | |*| |*|*|*|*|*|*|*|*|*|*|
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| | | | | | | | |*|*| | |*|*|*|
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
|*| | | | |*|*| |*| |*|*|*|*|*|
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
|*| | | | | | |*| | | | | | | |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
the DLX runtime is : 0.0890049934387207s

　　把答案填入对应谜面：

图7.只要掌握了有效的优化技巧，就算是高难谜面也解给你看

　　大功告成！