华容道求解器 klotski solver
好久好久没写代码了，正好前段时间打印了个华容道 (klotski) 玩玩，写个 solver 复健一下。
小时候一直有听说这玩意，好像也有同学带到学校过，短暂的下课或许上手试过，印象中很难。
人脑求解
手动尝试最有名的初始摆法，横刀立马，花了一个小时左右。稍微上手玩后可以发现一些规律，比如朝长边方向移动 2 格长的块后，一定是 2 块 1x1 的块填补空缺，又比如 1x1 的小块总是两两联合移动，否则后续难以进行。不过仅凭这些还不足以从繁多的可能中找到正解。经过一些尝试后，最后是靠盯住最大的 2x2 块，在保证它下一步能移动到更接近出口的方向的前提下，反推/猜测其他块的排布，然后再设法构造出这种盘面。就是这样一步一步推出，不过必然是绕了很多弯子，离最优解肯定差很远。
编程求解
观察到盘面的可能态总量并不大；对每个盘面，可能的移动方式也很少，一般是4种左右；最优解的步数并不算长 (100 左右)；仅需要找到 1 个最优解。很自然想到用 BFS 暴力求解。
为了方便编写，使用 2 维字符串数组 board 表示盘面状态，压平拼接成单个字符串做 hashKey，维护一个 hashTable (python 的字典类)，键为当前盘面，值为移动任意块1格，到达当前盘面的序列，记录搜索过的内容。
写好一些 helper function，比如 board 和 hashKey 的相互转换，检查是否求解完成，清晰画出盘面。接下来就是如何遍历节点的边 (即每个盘面中某块移动一格后所有可能的盘面)。移动总是发生在空格子周边，分成了两种情况编写：其一是块朝边长 1 的方向向空格移动 1 格；其二是对两个相连的空格，块朝边长 2 的方向向相连的空格移动 1 格。对4个方向和块沿移动方向的长度，分各种情况写好逻辑。
Fixing bugs, and it works.
一些可能的优化
首先是 hashTable 的值并不用保存整个移动列表，只需保存上一步的键，生成解法时只需链式查找即可，如此免去列表复制可以快速许多；用更快的语言，使用一些编码技巧，使用比字符串更简洁的数据做 hashKey，应该也能提高一些性能；然后就是省去 hashKey 和 board 列表的转换，只使用编码后的 hashKey 完成程序的逻辑，可读性会差一些，但相应的也会快一些。
以上这些都并未实现，查找出解也仅需 1s 左右，完全可以接受，所以就不做了。
普遍的步数计算方法，并不是计移动的格数，而是记方块移动的次数。像是 2x1 方块朝短边方向移动2格，或是 1x1 方块拐弯连续移动两格，都是记 1 步的。格数最优往往不是这种计法的最优，但多跑几个 loop，稍微加点代码计算方块移动次数就可以解决，不过懒得写了 (
最后附上代码：
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import operator
# 横刀立马
init = [['21', '22', '22', '21'],
['21', '22', '22', '21'],
['21', '12', '12', '21'],
['21', '11', '11', '21'],
['11', '00', '00', '11'],]
# 横竖皆将
# init = [["21", "22", "22", "21"],
# ["21", "22", "22", "21"],
# ["21", "12", "12", "11"],
# ["21", "12", "12", "11"],
# ["11", "00", "00", "11"]]
# possible moves
height = len(init)
width = len(init[0])
def nextState(board):
ret = []
for row in range(height):
for col in range(width):
if board[row][col] == '00':
# width = 1
# Up
i = row - 1
j = col
if i >= 0:
if board[i][j] == '11':
b = [l[:] for l in board]
b[row][col] = '11'
b[i][j] = '00'
key = toKey(b)
ret.append(key)
elif board[i][j] == '21':
b = [l[:] for l in board]
b[row][col] = '21'
b[i-1][j] = '00'
key = toKey(b)
ret.append(key)
# Down
i = row + 1
j = col
if i < height:
if board[i][j] == '11':
b = [l[:] for l in board]
b[row][col] = '11'
b[i][j] = '00'
key = toKey(b)
ret.append(key)
elif board[i][j] == '21':
b = [l[:] for l in board]
b[row][col] = '21'
b[i+1][j] = '00'
key = toKey(b)
ret.append(key)
# Left
i = row
j = col - 1
if j >= 0:
if board[i][j] == '11':
b = [l[:] for l in board]
b[row][col] = '11'
b[i][j] = '00'
key = toKey(b)
ret.append(key)
elif board[i][j] == '12':
b = [l[:] for l in board]
b[row][col] = '12'
b[i][j-1] = '00'
key = toKey(b)
ret.append(key)
# Right
i = row
j = col + 1
if j < width:
if board[i][j] == '11':
b = [l[:] for l in board]
b[row][col] = '11'
b[i][j] = '00'
key = toKey(b)
ret.append(key)
elif board[i][j] == '12':
b = [l[:] for l in board]
b[row][col] = '12'
b[i][j+1] = '00'
key = toKey(b)
ret.append(key)
# width = 2
# 2x1 blank
i = row + 1
j = col
if i < height and board[i][j] == '00':
if j-1 >= 0:
if board[i][j-1] == board[row][col-1] == '21':
legal = True
if row-1 >= 0 and i+1 < height:
if board[row-1][col-1] == board[i+1][j-1] == '21':
legal = False
if legal:
b = [l[:] for l in board]
b[i][j-1] = '00'
b[row][col-1] = '00'
b[row][col] = '21'
b[i][j] = '21'
key = toKey(b)
ret.append(key)
elif board[i][j-1] == board[row][col-1] == '22':
b = [l[:] for l in board]
b[i][j-2] = '00'
b[row][col-2] = '00'
b[i][j] = '22'
b[row][col] = '22'
key = toKey(b)
ret.append(key)
if j+1 < width:
if board[i][j+1] == board[row][col+1] == '21':
legal = True
if row-1 >= 0 and i+1 < height:
if board[row-1][col+1] == board[i+1][j+1] == '21':
legal = False
if legal:
b = [l[:] for l in board]
b[i][j+1] = '00'
b[row][col+1] = '00'
b[i][j] = '21'
b[row][col] = '21'
key = toKey(b)
ret.append(key)
elif board[i][j+1] == board[row][col+1] == '22':
b = [l[:] for l in board]
b[i][j+2] = '00'
b[row][col+2] = '00'
b[i][j] = '22'
b[row][col] = '22'
key = toKey(b)
ret.append(key)
# 1x2 blank
i = row
j = col + 1
if j < width and board[i][j] == '00':
if i-1 >= 0:
if board[i-1][j] == board[row-1][col] == '12':
legal = True
if col-1 >= 0 and j+1 < width:
if board[row-1][col-1] == board[i-1][j+1] == '12':
legal = False
if legal:
b = [l[:] for l in board]
b[i-1][j] = '00'
b[row-1][col] = '00'
b[i][j] = '12'
b[row][col] = '12'
key = toKey(b)
ret.append(key)
elif board[i-1][j] == board[row-1][col] == '22':
b = [l[:] for l in board]
b[i-2][j] = '00'
b[row-2][col] = '00'
b[i][j] = '22'
b[row][col] = '22'
key = toKey(b)
ret.append(key)
if i+1 < height:
if board[i+1][j] == board[row+1][col] == '12':
legal = True
if col-1 >= 0 and j+1 < width:
if board[row+1][col-1] == board[i+1][j+1] == '12':
legal = False
if legal:
b = [l[:] for l in board]
b[i+1][j] = '00'
b[row+1][col] = '00'
b[i][j] = '12'
b[row][col] = '12'
key = toKey(b)
ret.append(key)
elif board[i+1][j] == board[row+1][col] == '22':
b = [l[:] for l in board]
b[i+2][j] = '00'
b[row+2][col] = '00'
b[i][j] = '22'
b[row][col] = '22'
key = toKey(b)
ret.append(key)
return ret
# board to hashkey
def toKey(board):
return ''.join([''.join(l) for l in board])
# hashKey to board
def toBoard(key):
return [[key[2*(4*row+col):2*(4*row+col+1)] for col in range(width)] for row in range(height)]
# check if solved
def solved(board):
if board[4][1] == board[4][2] == '22':
return True
else:
return False
hashTable = {toKey(init): [toKey(init)]}
# draw board
def drawBoard(board):
for row in board:
for elem in row:
if elem == '22':
c='曹'
elif elem == '21':
c='丨'
elif elem == '12':
c='一'
elif elem == '11':
c='兵'
else:
c=' '
print(c, end='')
print('', end='\n')
print()
# draw moves
def drawMoves(key):
for (i, k) in enumerate(hashTable[key]):
print(i)
drawBoard(toBoard(k))
loop = True
buf1 = [(toKey(init), [toKey(init)])]
for i in range(200):
if loop:
print(i, len(hashTable))
buf = []
for (key, val) in buf1:
nxt = nextState(toBoard(key))
for newKey in nxt:
if newKey in hashTable:
pass
else:
if not newKey in map(operator.itemgetter(0), buf):
buf.append((newKey, hashTable[key] + [newKey]))
for (k, v) in buf:
hashTable[k] = v
if solved(toBoard(k)):
print("final state:", len(hashTable[k]) - 1, k)
loop = False
buf1 = buf[:]
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