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算法帖——用舞蹈链算法（Dancing Links）求解俄罗斯方块覆盖问题
阅读量：7289 次
发布时间：2019-06-30

本文共 13387 字，大约阅读时间需要 44 分钟。

问题的提出：如下图，用13块俄罗斯方块覆盖8*8的正方形。如何用计算机求解？

 

 

解决这类问题的方法不一而足，然而核心思想都是穷举法，不同的方法仅仅是对穷举法进行了优化

 

用13块不同形状的俄罗斯方块（每个方块只能使用一次）覆盖住棋盘，很容易就想到这是“精确覆盖问题”（13个俄罗斯方块完全覆盖住8*8的正方形）。而舞蹈链算法（Dancing Links）是比较好求解“精确覆盖问题”的算法，因为该算法在穷举的过程中，不再额外增加空间负担，状态的回溯也比较方便，能快捷的排除无效的穷举过程。有关舞蹈链算法（Dancing Links），在这里不再赘述，详情参看“”

 

用舞蹈链算法（Dancing Links）解决问题的核心是把问题转换为问题矩阵

 

很直观的，这样的矩阵一共有77列，其中第1-64列表示8*8正方形的每一个单元格，第65-77列代表方块的编号

这样求解出来的解就是正方形的每一个单元格都有方块填充，每个方块都被使用了一次

 

以上图为例，我把左下角的深绿色的方块定义为方块1，而这个深绿色方块又占用了第49、57、58、59、60单元格

那么这个深绿色的方块所构造的数据行就是如下表示

{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0}

 

为了方便描述，我们把上面的行矩阵记作｛49，57、58、59、60、65｝

 

而我们要做的就是，构造出所有的数据行

 

先把如下图方块1的所有能在的位置做成数据行

则一共能有7行*5列=35种可能

 

同时，巧妙利用中心旋转的算法，分别得出旋转90度、180度、270度的位置可能

如下所示

旋转90度的图

旋转180度的图

旋转270度的图

 

这样一来，只需要遍历最先图的形状位置即可，其余旋转的形状的可以依次推导。

上面的形状还有一个如下图的，需要遍历

 

这样一来，这个形状1的所有位置就遍历完成了。

 

依次遍历13个形状，这样就生成了问题矩阵的所有行

代码如下：

 
Public 
Class 
clsTetris
        
Implements 
I_Question
   
Private _Shapes
As 
List(
Of 
clsTetrisShape)
   
Private _Index()
As 
Integer
   
Public 
ReadOnly 
Property Cols
As 
Integer 
Implements 
I_Question.Cols
       
Get
           
Return 77
       
End 
Get
   
End 
Property
   
Public 
Function ConvertFromDance(Answer()
As 
Integer)
As 
Object 
Implements 
I_Question.ConvertFromDance
       
Debug.Print(Answer.Length)
       
Dim tBmp
As 
New 
Bitmap(320, 320)
       
Dim tG
As 
Graphics =
Graphics.FromImage(tBmp)
        tG.Clear(
Color.White)
       
Dim I
As 
Integer
       
For I = 0
To Answer.Length - 1
            _Shapes(_Index(Answer(I) - 1)).DrawShape(tG)
       
Next
       
Return tBmp
   
End 
Function
   
Public 
ReadOnly 
Property ExtraCols
As 
Integer 
Implements 
I_Question.ExtraCols
       
Get
           
Return 77
       
End 
Get
   
End 
Property
   
Public 
Sub ConvertToDance(Dance
As 
clsDancingLinksImproveNoRecursive)
Implements 
I_Question.ConvertToDance
        _Shapes =
New 
List(
Of 
clsTetrisShape)
       
Dim I
As 
Integer, J
As 
Integer
       
Dim tShape
As 
clsTetrisShape, tRotateShape
As 
clsTetrisShape
       
Dim S
As 
Integer
       
'Shape 1
       
For I = 0
To 6
           
For J = 0
To 4
                S = I * 8 + J
                tShape =
New 
clsTetrisShape(1, S, S + 1, S + 2, S + 3, S + 8)
                AppendAllShapes(Dance, tShape)
           
Next
       
Next
       
For I = 0
To 6
           
For J = 0
To 4
                S = I * 8 + J
                tShape =
New 
clsTetrisShape(1, S, S + 8, S + 9, S + 10, S + 11)
                AppendAllShapes(Dance, tShape)
           
Next
       
Next
       
'Shape 2
       
For I = 0
To 5
           
For J = 0
To 5
                S = I * 8 + J
                tShape =
New 
clsTetrisShape(2, S, S + 1, S + 9, S + 10, S + 18)
                AppendAllShapes(Dance, tShape)
           
Next
       
Next
       
'Shape3
       
For I = 0
To 5
           
For J = 0
To 5
                S = I * 8 + J
                tShape =
New 
clsTetrisShape(3, S, S + 1, S + 9, S + 10, S + 17)
                AppendAllShapes(Dance, tShape)
           
Next
       
Next
       
For I = 0
To 5
           
For J = 1
To 6
                S = I * 8 + J
                tShape =
New 
clsTetrisShape(3, S, S + 1, S + 7, S + 8, S + 16)
                AppendAllShapes(Dance, tShape)
           
Next
       
Next
       
'Shape 4
       
For I = 0
To 5
           
For J = 0
To 5
                S = I * 8 + J
                tShape =
New 
clsTetrisShape(4, S, S + 1, S + 2, S + 8, S + 16)
                AppendAllShapes(Dance, tShape)
           
Next
       
Next
       
'Shape5
       
For I = 0
To 6
           
For J = 0
To 4
                S = I * 8 + J
                tShape =
New 
clsTetrisShape(5, S, S + 1, S + 2, S + 10, S + 11)
                AppendAllShapes(Dance, tShape)
           
Next
       
Next
       
For I = 0
To 6
           
For J = 1
To 5
                S = I * 8 + J
                tShape =
New 
clsTetrisShape(5, S, S + 1, S + 2, S + 7, S + 8)
                AppendAllShapes(Dance, tShape)
           
Next
       
Next
       
'Shape6
       
For I = 0
To 5
           
For J = 0
To 5
                S = I * 8 + J
                tShape =
New 
clsTetrisShape(6, S, S + 8, S + 9, S + 10, S + 18)
                _Shapes.Add(tShape)
                tRotateShape = tShape.Rotate90
                _Shapes.Add(tRotateShape)
           
Next
       
Next
       
For I = 0
To 5
           
For J = 2
To 7
                S = I * 8 + J
                tShape =
New 
clsTetrisShape(6, S, S + 6, S + 7, S + 8, S + 14)
             
                _Shapes.Add(tShape)
                tRotateShape = tShape.Rotate90
                _Shapes.Add(tRotateShape)
           
Next
       
Next
       
'Shape 7
       
For I = 0
To 5
           
For J = 0
To 5
                S = I * 8 + J
                tShape =
New 
clsTetrisShape(7, S, S + 1, S + 2, S + 9, S + 17)
                AppendAllShapes(Dance, tShape)
           
Next
       
Next
       
'Shape 8
       
For I = 0
To 6
           
For J = 0
To 5
                S = I * 8 + J
                tShape =
New 
clsTetrisShape(8, S, S + 1, S + 2, S + 8, S + 9)
                AppendAllShapes(Dance, tShape)
           
Next
       
Next
       
For I = 0
To 6
           
For J = 0
To 5
                S = I * 8 + J
                tShape =
New 
clsTetrisShape(8, S, S + 1, S + 2, S + 9, S + 10)
                AppendAllShapes(Dance, tShape)
           
Next
       
Next
       
'Shape 9
       
For I = 0
To 6
           
For J = 0
To 4
                S = I * 8 + J
                tShape =
New 
clsTetrisShape(9, S, S + 1, S + 2, S + 3, S + 9)
                AppendAllShapes(Dance, tShape)
           
Next
       
Next
       
For I = 0
To 6
           
For J = 0
To 4
                S = I * 8 + J
                tShape =
New 
clsTetrisShape(9, S, S + 1, S + 2, S + 3, S + 10)
                AppendAllShapes(Dance, tShape)
           
Next
       
Next
       
'Shape 10
       
For I = 0
To 6
           
For J = 0
To 6
                S = I * 8 + J
                tShape =
New 
clsTetrisShape(10, S, S + 1, S + 8, S + 9)
                _Shapes.Add(tShape)
           
Next
       
Next
       
'Shape 11
       
For I = 0
To 5
           
For J = 1
To 6
                S = I * 8 + J
                tShape =
New 
clsTetrisShape(11, S, S + 7, S + 8, S + 9, S + 16)
                _Shapes.Add(tShape)
           
Next
       
Next
       
'Shape12
       
For I = 0
To 7
           
For J = 0
To 3
                S = I * 8 + J
                tShape =
New 
clsTetrisShape(12, S, S + 1, S + 2, S + 3, S + 4)
                _Shapes.Add(tShape)
                tRotateShape = tShape.Rotate90
                _Shapes.Add(tRotateShape)
           
Next
       
Next
       
'Shape 13
       
For I = 0
To 6
           
For J = 0
To 5
                S = I * 8 + J
                tShape =
New 
clsTetrisShape(13, S, S + 1, S + 2, S + 8, S + 10)
                AppendAllShapes(Dance, tShape)
           
Next
       
Next
       
ReDim _Index(_Shapes.Count - 1)
       
For I = 0
To _Shapes.Count - 1
            _Index(I) = I
       
Next
       
Dim R
As 
New 
Random, tSwap
As 
Integer
       
For I = _Shapes.Count - 1
To Int(_Shapes.Count / 3)
Step -1
            J = R.Next(I)
            tSwap = _Index(J)
            _Index(J) = _Index(I)
            _Index(I) = tSwap
       
Next
       
For I = 0
To _Shapes.Count - 1
            Dance.AppendLine(_Shapes(_Index(I)).GetLineValue)
       
Next
   
End 
Sub
   
Private 
Sub AppendAllShapes(Dance
As 
clsDancingLinksImproveNoRecursive, tShape
As 
clsTetrisShape)
       
Dim tRotateShape
As 
clsTetrisShape
        _Shapes.Add(tShape)
        tRotateShape = tShape.Rotate90
        _Shapes.Add(tRotateShape)
        tRotateShape = tShape.Rotate180
        _Shapes.Add(tRotateShape)
        tRotateShape = tShape.Rotate270
        _Shapes.Add(tRotateShape)
   
End 
Sub
   
Public 
ReadOnly 
Property IsRandomSolution
As 
Boolean 
Implements 
I_Question.IsRandomSolution
       
Get
           
Return 
False
       
End 
Get
   
End 
Property
End 
Class

 

 

上面这个类实现了I_Question接口，代码如下：

 

 
Public 
Interface 
I_Question
   
ReadOnly 
Property Cols
As 
Integer
   
ReadOnly 
Property ExtraCols
As 
Integer
   
ReadOnly 
Property IsRandomSolution
As 
Boolean
   
Sub ConvertToDance(Dance
As 
clsDancingLinksImproveNoRecursive)
   
Function ConvertFromDance(Answer()
As 
Integer)
As 
Object
End 
Interface

 

几个参数解释一下

Cols：问题矩阵的数据列数

ExtraCols：问题矩阵必须覆盖的列数。大多数的情况下，和Cols相等，也就是所有列完全覆盖

IsRandomSolution：一个开关，指示求解过程中，是按照最少列优先求解（为False的时候）还是随机选择列求解（为True的时候），在列数比较少的情况下，可以为True，否则不建议使用True，为True的时候，如果存在多个解，每次求解有可能得出不同的解。

ConvertToDance：将数据转换为问题矩阵，并输入到指定的Dance类

ConvertFromDance：Dance类计算得出结果后，将结果返回给实现接口的类，让该类对结果进行相应的处理。

 

 

类clsTetris还内置了clsTetrisShape类，定义每个形状的编号、位置、并最终将每个形状绘制到指定的图上，如下：

 

 
Public 
Class 
clsTetrisShape
   
Private Poi()
As 
Integer
   
Private ShapeType
As 
Integer
   
Public 
Sub 
New(ShapeType
As 
Integer,
ParamArray Poi()
As 
Integer)
       
Me.ShapeType = ShapeType
       
Dim I
As 
Integer
       
ReDim 
Me.Poi(Poi.Length - 1)
       
For I = 0
To Poi.Length - 1
           
Me.Poi(I) = Poi(I)
       
Next
   
End 
Sub
   
Public 
Function GetLineValue()
As 
Integer()
       
Dim Value(76)
As 
Integer
       
Dim I
As 
Integer
       
For I = 0
To 76
            Value(I) = 0
       
Next
       
For I = 0
To Poi.Length - 1
            Value(Poi(I)) = 1
       
Next
        Value(63 + ShapeType) = 1
       
Return Value
   
End 
Function
   
Public 
Function Rotate90()
As 
clsTetrisShape
       
Dim NewPoi(Poi.Length - 1)
As 
Integer
       
Dim I
As 
Integer, X
As 
Integer, Y
As 
Integer
       
For I = 0
To Poi.Length - 1
            X = Int(Poi(I) / 8)
            Y = Poi(I)
Mod 8
            NewPoi(I) = Y * 8 + 7 - X
       
Next
       
Return 
New 
clsTetrisShape(ShapeType, NewPoi)
   
End 
Function
   
Public 
Function Rotate180()
As 
clsTetrisShape
       
Dim NewPoi(Poi.Length - 1)
As 
Integer
       
Dim I
As 
Integer
       
For I = 0
To Poi.Length - 1
            NewPoi(I) = 63 - Poi(I)
       
Next
       
Return 
New 
clsTetrisShape(ShapeType, NewPoi)
   
End 
Function
   
Public 
Function Rotate270()
As 
clsTetrisShape
       
Dim NewPoi(Poi.Length - 1)
As 
Integer
       
Dim I
As 
Integer, X
As 
Integer, Y
As 
Integer
       
For I = 0
To Poi.Length - 1
            X = Int(Poi(I) / 8)
            Y = Poi(I)
Mod 8
            NewPoi(I) = (7 - Y) * 8 + X
       
Next
       
Return 
New 
clsTetrisShape(ShapeType, NewPoi)
   
End 
Function
   
Public 
Sub DrawShape(G
As 
Graphics)
       
Dim tBrush
As 
SolidBrush
       
Select 
Case ShapeType
           
Case 1
                tBrush =
New 
SolidBrush(
Color.FromArgb(84, 130, 53))
           
Case 2
                tBrush =
New 
SolidBrush(
Color.FromArgb(112, 48, 160))
           
Case 3
                tBrush =
New 
SolidBrush(
Color.FromArgb(166, 166, 166))
           
Case 4
                tBrush =
New 
SolidBrush(
Color.FromArgb(0, 176, 240))
           
Case 5
                tBrush =
New 
SolidBrush(
Color.FromArgb(0, 32, 96))
           
Case 6
                tBrush =
New 
SolidBrush(
Color.FromArgb(0, 0, 0))
           
Case 7
                tBrush =
New 
SolidBrush(
Color.FromArgb(192, 0, 0))
           
Case 8
                tBrush =
New 
SolidBrush(
Color.FromArgb(255, 217, 102))
           
Case 9
                tBrush =
New 
SolidBrush(
Color.FromArgb(0, 112, 192))
           
Case 10
                tBrush =
New 
SolidBrush(
Color.FromArgb(0, 176, 80))
           
Case 11
                tBrush =
New 
SolidBrush(
Color.FromArgb(255, 255, 0))
           
Case 12
                tBrush =
New 
SolidBrush(
Color.FromArgb(198, 89, 17))
           
Case 13
                tBrush =
New 
SolidBrush(
Color.FromArgb(146, 208, 80))
           
Case Else
                tBrush =
New 
SolidBrush(
Color.FromArgb(146, 208, 80))
       
End 
Select
       
Dim I
As 
Integer, X
As 
Integer, Y
As 
Integer
       
For I = 0
To Poi.Length - 1
            X = Int(Poi(I) / 8)
            Y = Poi(I)
Mod 8
            G.FillRectangle(tBrush,
New 
Rectangle(Y * 40, X * 40, 40, 40))
       
Next
   
End 
Sub
End 
Class

 

 

 

然后是贴出求解类

 

 
Public 
Class 
clsDancingCentre
   
Public 
Shared 
Function Dancing(Question
As 
I_Question)
As 
Object
       
Dim _Dance
As 
New 
clsDancingLinksImproveNoRecursive(Question.Cols, Question.ExtraCols)
        Question.ConvertToDance(_Dance)
       
Return Question.ConvertFromDance(_Dance.Dance(Question.IsRandomSolution))
   
End 
Function
End 
Class

 

该类只有一个核心方法，定义一个舞蹈链算法（Dancing Links）类，并对该类和I_Question接口搭桥求解问题

 

在clsTetris类中，原本如果设置IsRandomSolution为True的话，那么求解过程非常缓慢（曾经1小时没有求出一个解出来），但如果设置为False的时候，每次求解是秒破，但是每次求解都是同一个结果。后来想到，交换问题矩阵的行，会影响求解的顺序，但不影响求解的结果。如果求解的结果是唯一的，那么矩阵的行交不交换都一样，但是如果求解的问题不是唯一的，那么改变问题矩阵的行，那么每次求解出来的解就有可能不同。故在clsTetris中，在最后把数据添加到Dance类的时候，是改变了添加顺序的，这样每次求解都是秒破，并且得出的结果也不一样。求解100个解，不到30秒。

 

最后贴出Dancing类，这才是舞蹈链算法（Dancing Links）的核心

 

 
Public 
Class 
clsDancingLinksImproveNoRecursive
   
Private Left()
As 
Integer, Right()
As 
Integer, Up()
As 
Integer, Down()
As 
Integer
   
Private Row()
As 
Integer, Col()
As 
Integer
   
Private _Head
As 
Integer
   
Private _Rows
As 
Integer, _Cols
As 
Integer, _NodeCount
As 
Integer
   
Private Count()
As 
Integer
   
Private Ans()
As 
Integer
   
Public 
Sub 
New(
ByVal Cols
As 
Integer)
       
Me.New(Cols, Cols)
   
End 
Sub
   
Public 
Sub 
New(
ByVal Cols
As 
Integer, ExactCols
As 
Integer)
       
ReDim Left(Cols), Right(Cols), Up(Cols), Down(Cols), Row(Cols), Col(Cols), Ans(Cols)
       
ReDim Count(Cols)
       
Dim I
As 
Integer
        Up(0) = 0
        Down(0) = 0
        Right(0) = 1
        Left(0) = Cols
       
For I = 1
To Cols
            Up(I) = I
            Down(I) = I
            Left(I) = I - 1
            Right(I) = I + 1
            Col(I) = I
            Row(I) = 0
            Count(I) = 0
       
Next
        Right(Cols) = 0
        _Rows = 0
        _Cols = Cols
        _NodeCount = Cols
        _Head = 0
       
Dim N
As 
Integer = Right(ExactCols)
        Right(ExactCols) = _Head
        Left(_Head) = ExactCols
        Left(N) = _Cols
        Right(_Cols) = N
   
End 
Sub
   
Public 
Sub AppendLine(
ByVal 
ParamArray Value()
As 
Integer)
       
Dim V
As 
New 
List(
Of 
Integer)
       
Dim I
As 
Integer
       
For I = 0
To Value.Length - 1
           
If Value(I) <> 0
Then V.Add(I + 1)
       
Next
        AppendLineByIndex(V.ToArray)
   
End 
Sub
   
Public 
Sub AppendLine(Line
As 
String)
       
Dim V
As 
New 
List(
Of 
Integer)
       
Dim I
As 
Integer
       
For I = 0
To Line.Length - 1
           
If Line.Substring(I, 1) <>
"0" 
Then V.Add(I + 1)
       
Next
        AppendLineByIndex(V.ToArray)
   
End 
Sub
   
Public 
Sub AppendLineByIndex(
ByVal 
ParamArray Index()
As 
Integer)
       
If Index.Length = 0
Then 
Exit Sub
        _Rows += 1
       
Dim I
As 
Integer, K
As 
Integer = 0
       
ReDim 
Preserve Left(_NodeCount + Index.Length)
       
ReDim 
Preserve Right(_NodeCount + Index.Length)
       
ReDim 
Preserve Up(_NodeCount + Index.Length)
       
ReDim 
Preserve Down(_NodeCount + Index.Length)
       
ReDim 
Preserve Row(_NodeCount + Index.Length)
       
ReDim 
Preserve Col(_NodeCount + Index.Length)
       
ReDim 
Preserve Ans(_Rows)
       
For I = 0
To Index.Length - 1
            _NodeCount += 1
           
If I = 0
Then
                Left(_NodeCount) = _NodeCount
                Right(_NodeCount) = _NodeCount
           
Else
                Left(_NodeCount) = _NodeCount - 1
                Right(_NodeCount) = Right(_NodeCount - 1)
                Left(Right(_NodeCount - 1)) = _NodeCount
                Right(_NodeCount - 1) = _NodeCount
           
End 
If
            Down(_NodeCount) = Index(I)
            Up(_NodeCount) = Up(Index(I))
            Down(Up(Index(I))) = _NodeCount
            Up(Index(I)) = _NodeCount
            Row(_NodeCount) = _Rows
            Col(_NodeCount) = Index(I)
            Count(Index(I)) += 1
       
Next
   
End 
Sub
   
Public 
Function Dance(
Optional Random
As 
Boolean =
False)
As 
Integer()
       
Dim P
As 
Integer, C1
As 
Integer
       
Dim I
As 
Integer, J
As 
Integer
       
Dim K
As 
Integer = 0
       
Dim R
As 
New 
Random
       
Do
           
If (Right(_Head) = _Head)
Then
               
ReDim 
Preserve Ans(K - 1)
               
For I = 0
To Ans.Length - 1
                    Ans(I) = Row(Ans(I))
               
Next
               
Return Ans
           
End 
If
            P = Right(_Head)
            C1 = P
           
If Random =
False 
Then
               
Do 
While P <> _Head
                   
If Count(P) < Count(C1)
Then C1 = P
                    P = Right(P)
               
Loop
           
Else
                I = R.Next(_Cols)
               
For J = 1
To I
                    P = Right(P)
               
Next
               
If P = _Head
Then P = Right(_Head)
                C1 = P
           
End 
If
            RemoveCol(C1)
            I = Down(C1)
           
Do 
While I = C1
                ResumeCol(C1)
                K -= 1
               
If K < 0
Then 
Return 
Nothing
                C1 = Col(Ans(K))
                I = Ans(K)
                J = Left(I)
               
Do 
While J <> I
                    ResumeCol(Col(J))
                    J = Left(J)
               
Loop
                I = Down(I)
           
Loop
            Ans(K) = I
            J = Right(I)
           
Do 
While J <> I
                RemoveCol(Col(J))
                J = Right(J)
           
Loop
            K += 1
       
Loop
   
End 
Function
   
Private 
Sub RemoveCol(
ByVal ColIndex
As 
Integer)
        Left(Right(ColIndex)) = Left(ColIndex)
        Right(Left(ColIndex)) = Right(ColIndex)
       
Dim I
As 
Integer, J
As 
Integer
        I = Down(ColIndex)
       
Do 
While I <> ColIndex
            J = Right(I)
           
Do 
While J <> I
                Up(Down(J)) = Up(J)
                Down(Up(J)) = Down(J)
                Count(Col(J)) -= 1
                J = Right(J)
           
Loop
            I = Down(I)
       
Loop
   
End 
Sub
   
Private 
Sub ResumeCol(
ByVal ColIndex
As 
Integer)
        Left(Right(ColIndex)) = ColIndex
        Right(Left(ColIndex)) = ColIndex
       
Dim I
As 
Integer, J
As 
Integer
        I = Up(ColIndex)
       
Do 
While (I <> ColIndex)
            J = Right(I)
           
Do 
While J <> I
                Up(Down(J)) = J
                Down(Up(J)) = J
                Count(Col(J)) += 1
                J = Right(J)
           
Loop
            I = Up(I)
       
Loop
   
End 
Sub
End 
Class

 

注：

求解了1000个解，发现很有趣的一个现象，就是长条（1*5的那个），几乎都在边上，在当中的解少之又少

下面贴几个解

 

 

 

 

 

 

 

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