周末闲来无事，忽然想起大学时候学《数据结构》时候的期末上机考试，要求在两个小时内完成两道编程的题目。其中一道题目就是让编程解决“八皇后问题”。现在想起来还是挺有意思，就打算用 C# 再实现一次 :)
八皇后问题的大致意思是这样的：在一个 8*8 的矩形格子中排放 8 个皇后，要满足的条件包括：任意两个皇后都不能在同一行、同一列，也不能在同一条对角线（斜率等于1或-1）。
在考试的时候，我是用递归的方法解决的。现在想一想，还是用递归比较方便。但是需要注意的是，递归有个很大的毛病，就是递归的规模不太好控制，而且在递归很深的程序中，要调试也是一个很伤脑筋的问题。在有的公司，递归就是基本上禁止的。不过这里我就不去考虑非递归的解决方案了，反正是娱乐娱乐，何必自找麻烦呢:)
在使用递归解决此问题的时候，我采用的方法大致是这样：
- 将一个皇后向下移动一个位置；
- 如果没有成功移动（比如已经超出边界），失败；
- 如果成功移动，则判断当前位置是否可用（和其他的皇后时候冲突）？如果冲突，则重做 1；
- 继续给下一个皇后安排位置。
代码如下：
1
// AppEntry.cs
2using System;
3
4namespace Chenglin
5
{
6
class AppEntry
7
{
8 static void Main(string[] args)
9 {
10 int queenNumber = 8;
11 QueenRowCollection q = new QueenRowCollection(queenNumber);
12
13 bool flag;
14 DateTime timeStarted = DateTime.Now;
15 flag = q.PositionQueens();
16 TimeSpan ts = DateTime.Now.Subtract( timeStarted );
17
18
19 if( flag ) {
20 Console.WriteLine( q.ToString() );
21
}
22 else {
23 Console.WriteLine( "Failed
" );
24 }
25
26 Console.WriteLine( "{0} seconds has been elapsed.", ts.TotalSeconds );
27 }
28 }
29
}
// AppEntry.cs
2
using System;
3
4
namespace Chenglin
5
{
6
class AppEntry
7
{
8
static void Main(string[] args)
9
{
10
int queenNumber = 8;
11
QueenRowCollection q = new QueenRowCollection(queenNumber);
12
13
bool flag;
14
DateTime timeStarted = DateTime.Now;
15
flag = q.PositionQueens();
16
TimeSpan ts = DateTime.Now.Subtract( timeStarted );
17
18
19
if( flag ) {
20
Console.WriteLine( q.ToString() );
21
}
22
else {
23
Console.WriteLine( "Failed" );
24
}
25
26
Console.WriteLine( "{0} seconds has been elapsed.", ts.TotalSeconds );
27
}
28
}
29
}
1// QueenRowCollection.cs
2using System;
3using System.Text;
4
5namespace Chenglin
6{
7 public class QueenRowCollection
8 {
9
10 public QueenRowCollection( int numberOfQueens ){
11 this.numberOfQueens = numberOfQueens;
12 this.rows = new QueenRow[ numberOfQueens ];
13
14 for( int i=0;i<numberOfQueens;i++ ){
15 rows[i] = new QueenRow( numberOfQueens );
16 }
17 }
18
19 public bool PositionQueens()
20 {
21 return PositionQueen( 0 );
22 }
23
24 private bool PositionQueen( int row )
25 {
26 if( row>=this.numberOfQueens ) {
27 return true;
28 }
29
30 QueenRow q = rows[row];
31 while( q.MoveNext() )
32 {
33 if( PositionAvailable( row, q.CurrentPosition ) ) {
34 // An available position has been found for the current queen,
35 // and try to find a proper position for the next queen.
36 //
37 // If no available position can be found for the next queen,
38 // the current queen should move to the next position and try again.
39 //
40 if( PositionQueen( row+1 ) )
41 {
42 // Both the current queen and the next queen
43 // have found available positions.
44 //
45 return true;
46 }
47 }
48 }
49
50 // No position is available for the current queen,
51 //
52 return false;
53 }
54
55 private bool PositionAvailable( int row, int column ){
56 for( int i=row-1; i>=0; i-- )
57 {
58 if( rows[i].PositionOccupied( column ) )
59 return false;
60
61 if( rows[i].PositionOccupied( column-(i-row) ) )
62 return false;
63
64 if( rows[i].PositionOccupied( column+(i-row) ) )
65 return false;
66 }
67
68 return true;
69 }
70
71 public override string ToString()
72 {
73 StringBuilder s = new StringBuilder();
74
75 foreach( QueenRow q in rows ){
76 s.AppendFormat( "{0}{1}", q, Environment.NewLine );
77 }
78
79 return s.ToString();
80 }
81
82 private int numberOfQueens;
83 private QueenRow [] rows;
84 }
85}
// QueenRowCollection.cs
2
using System;
3
using System.Text;
4
5
namespace Chenglin
6
{
7
public class QueenRowCollection
8
{
9
10
public QueenRowCollection( int numberOfQueens ){
11
this.numberOfQueens = numberOfQueens;
12
this.rows = new QueenRow[ numberOfQueens ];
13
14
for( int i=0;i<numberOfQueens;i++ ){
15
rows[i] = new QueenRow( numberOfQueens );
16
}
17
}
18
19
public bool PositionQueens()
20
{
21
return PositionQueen( 0 );
22
}
23
24
private bool PositionQueen( int row )
25
{
26
if( row>=this.numberOfQueens ) {
27
return true;
28
}
29
30
QueenRow q = rows[row];
31
while( q.MoveNext() )
32
{
33
if( PositionAvailable( row, q.CurrentPosition ) ) {
34
// An available position has been found for the current queen,
35
// and try to find a proper position for the next queen.
36
//
37
// If no available position can be found for the next queen,
38
// the current queen should move to the next position and try again.
39
//
40
if( PositionQueen( row+1 ) )
41
{
42
// Both the current queen and the next queen
43
// have found available positions.
44
//
45
return true;
46
}
47
}
48
}
49
50
// No position is available for the current queen,
51
//
52
return false;
53
}
54
55
private bool PositionAvailable( int row, int column ){
56
for( int i=row-1; i>=0; i-- )
57
{
58
if( rows[i].PositionOccupied( column ) )
59
return false;
60
61
if( rows[i].PositionOccupied( column-(i-row) ) )
62
return false;
63
64
if( rows[i].PositionOccupied( column+(i-row) ) )
65
return false;
66
}
67
68
return true;
69
}
70
71
public override string ToString()
72
{
73
StringBuilder s = new StringBuilder();
74
75
foreach( QueenRow q in rows ){
76
s.AppendFormat( "{0}{1}", q, Environment.NewLine );
77
}
78
79
return s.ToString();
80
}
81
82
private int numberOfQueens;
83
private QueenRow [] rows;
84
}
85
}
1// QueenRow.cs
2using System;
3using System.Text;
4
5namespace Chenglin
6{
7 public class QueenRow
8 {
9 public QueenRow( int numberOfPositions )
10 {
11 this.numberOfPositions = numberOfPositions;
12 this.currentPosition = -1;
13 this.columns = new bool[ numberOfPositions ];
14 }
15
16 public bool MoveNext(){
17 if( currentPosition>=0 && currentPosition<this.numberOfPositions ){
18 columns[currentPosition] = false;
19 }
20
21 if( currentPosition<this.numberOfPositions-1){
22 currentPosition ++;
23 columns[currentPosition] = true;
24 return true;
25 }
26 else {
27 currentPosition = -1;
28 return false;
29 }
30 }
31
32 public bool PositionOccupied( int column ){
33 if( column<0 || column>=numberOfPositions ){
34 return false;
35 }
36
37 return columns[column];
38 }
39
40 public override string ToString()
41 {
42 StringBuilder s = new StringBuilder();
43
44 foreach( bool b in columns ){
45 s.AppendFormat( "{0} ", (b ? "*" : "-") );
46 }
47
48 return s.ToString();
49 }
50
51 public int CurrentPosition
52 {
53 get { return currentPosition; }
54 }
55
56 private int currentPosition;
57 private int numberOfPositions;
58 private bool [] columns;
59 }
60}
// QueenRow.cs
2
using System;
3
using System.Text;
4
5
namespace Chenglin
6
{
7
public class QueenRow
8
{
9
public QueenRow( int numberOfPositions )
10
{
11
this.numberOfPositions = numberOfPositions;
12
this.currentPosition = -1;
13
this.columns = new bool[ numberOfPositions ];
14
}
15
16
public bool MoveNext(){
17
if( currentPosition>=0 && currentPosition<this.numberOfPositions ){
18
columns[currentPosition] = false;
19
}
20
21
if( currentPosition<this.numberOfPositions-1){
22
currentPosition ++;
23
columns[currentPosition] = true;
24
return true;
25
}
26
else {
27
currentPosition = -1;
28
return false;
29
}
30
}
31
32
public bool PositionOccupied( int column ){
33
if( column<0 || column>=numberOfPositions ){
34
return false;
35
}
36
37
return columns[column];
38
}
39
40
public override string ToString()
41
{
42
StringBuilder s = new StringBuilder();
43
44
foreach( bool b in columns ){
45
s.AppendFormat( "{0} ", (b ? "*" : "-") );
46
}
47
48
return s.ToString();
49
}
50
51
public int CurrentPosition
52
{
53
get { return currentPosition; }
54
}
55
56
private int currentPosition;
57
private int numberOfPositions;
58
private bool [] columns;
59
}
60
}
程序运行的时候，当皇后个数增加的时候，运行的时间也会急剧增加！下面这副图表示了皇后个数与运行时间的大致关系： 
