数独小解
本文简述了一种关于数独游戏的程序解法
数独游戏相信大家都有所耳闻,规则上非常简明:
在 9 x 9 的数格中(由 9 个 3 x 3 的九宫格组成)会预先给定不少于(>=)17个数字,你需要在满足以下条件的情况下补完其余数格中的数字:
- 每一行的数字不重复
- 每一列的数字不重复
- 每个 3 x 3 九宫格中的数字不重复
这里有个示例:
使用程序求解数独的一种简单方法便是穷举(搜索),较粗糙的一种方式就是枚举 9 x 9 数格中所有的数字组合(为每个数格指定 1 ~ 9 之间的某个数字),只是这种搜索方式的解空间较大,实际运用中需要进行剪枝,剪枝的思路其实也很朴素,依据数独中相关数字不可重复的3条规则即可:
按照空白数格顺序进行深度优先搜索,对于每个空白数格生成候选数字列表(即满足规则下可能填入该空白数格的数字列表),然后依次填入可能的候选数字,并在下一个空白数格处继续进行相同的深度优先搜索,如果成功则找到答案,否则返回失败.
(可以使用启发式算法优化,譬如按候选数字列表最短的空白数格顺序进行搜索(实际测试中因为工程原因其实比顺序搜索方法更慢))
下面是使用 Lua 实现的示例程序:
function get_sudoku_row_and_col(data)
local row = #data
local col = data[1] and #data[1] or 0
return row, col
end
function print_sudoku(data)
local row, col = get_sudoku_row_and_col(data)
for i = 1, row do
local str_buffer = {}
for j = 1, col do
table.insert(str_buffer, data[i][j])
end
print(table.concat(str_buffer, ", "))
end
print()
end
function check_sudoku_row(data, row_index)
local row, col = get_sudoku_row_and_col(data)
local sum = 1
for i = 1, col do
sum = sum | (1 << (data[row_index][i]))
end
return sum == 0x3FF
end
function check_sudoku_col(data, col_index)
local row, col = get_sudoku_row_and_col(data)
local sum = 1
for i = 1, row do
sum = sum | (1 << (data[i][col_index]))
end
return sum == 0x3FF
end
function check_sudoku_square(data, row_index, col_index)
local row, col = get_sudoku_row_and_col(data)
local sum = 1
for i = row_index, row_index + 2 do
for j = col_index, col_index + 2 do
sum = sum | (1 << (data[i][j]))
end
end
return sum == 0x3FF
end
function check_sudoku(data)
local row, col = get_sudoku_row_and_col(data)
-- check row
for i = 1, row do
if not check_sudoku_row(data, i) then
return false
end
end
-- check col
for i = 1, col do
if not check_sudoku_col(data, i) then
return false
end
end
-- check square
for i = 1, row // 3 do
local row_index = (i - 1) * 3 + 1
for j = 1, col // 3 do
local col_inex = (j - 1) * 3 + 1
if not check_sudoku_square(data, row_index, col_inex) then
return false
end
end
end
return true
end
function apply_sudoku(data, delta)
local row, col = delta.row, delta.col
if data[row] and data[row][col] then
data[row][col] = delta.val
end
end
function revert_sudoku(data, delta)
local row, col = delta.row, delta.col
if data[row] and data[row][col] then
data[row][col] = 0
end
end
function get_pending_list(data, row, col)
if data[row] and data[row][col] then
if data[row][col] == 0 then
-- only empty pos has pending list
local pending_list = { true, true, true, true, true, true, true, true, true }
local data_row, data_col = get_sudoku_row_and_col(data)
-- check row
for i = 1, data_col do
pending_list[data[row][i]] = false
end
-- check col
for i = 1, data_row do
pending_list[data[i][col]] = false
end
-- check square
local square_row = (row - 1) // 3 * 3 + 1 -- or math.ceil(row / 3) * 3 - 2
local square_col = (col - 1) // 3 * 3 + 1 -- or math.ceil(col / 3) * 3 - 2
for i = square_row, square_row + 2 do
for j = square_col, square_col + 2 do
pending_list[data[i][j]] = false
end
end
-- gen pending list number
local pending_list_num = {}
for i = 1, #pending_list do
if pending_list[i] then
table.insert(pending_list_num, i)
end
end
return pending_list_num
end
end
end
function get_next_search_pos(data, row, col)
-- now we just search in order
local data_row, data_col = get_sudoku_row_and_col(data)
-- top right part
for j = col, data_col do
if data[row][j] == 0 then
return row, j
end
end
-- rest part
for i = row + 1, data_row do
for j = 1, data_col do
if data[i][j] == 0 then
return i, j
end
end
end
end
function search_sudoku_recur(data, row, col)
row, col = get_next_search_pos(data, row, col)
if row and col then
local pending_list = get_pending_list(data, row, col)
if pending_list and #pending_list > 0 then
for i = 1, #pending_list do
apply_sudoku(data, { row = row, col = col, val = pending_list[i] })
local val = search_sudoku_recur(data, row, col)
if val then
return true
else
revert_sudoku(data, { row = row, col = col, val = pending_list[i] })
end
end
-- all pending list is not valid
return false
else
-- no valid pending list
return false
end
else
-- all data filled
if check_sudoku(data) then
-- valid data
return true
else
-- invalid data
return false
end
end
end
function search_sudoku(data)
return search_sudoku_recur(data, 1, 1)
end
示例程序并不简短,但是理解起来并不困难,要点如下:
- 数独的数格表示使用了"二维数组"
- 函数 check_sudoku 用以检查数格中的数字是否满足数独条件
- 函数 search_sudoku 实现了上述的深度优先搜索方法
下图是号称世界上迄今难度最大的数独游戏,有兴趣的朋友可以试下: