Ai lecture 10(unit02)
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The document explains the n-queens problem, which involves placing n queens on an n×n chessboard such that no two queens can attack each other. It details a backtracking approach to solving the 4-queens problem on a 4x4 chessboard, illustrating steps and backtracking processes when placements are invalid. The document emphasizes safety checks for placements in rows, columns, and diagonals.
![N-Queens Problem Solution Using Backtracking
Example :4 x 4 Chessboard To place 4-
Queens
Step:-1
r=0,c=0
Check, board[0][0] =safe
Now ,Increment column By 1(Place Queen)
Step:2
r=0,c=1
Check, board[0][1] ≠ safe
Now ,Increment Row By 1
Step:3
r=1,c=1
Check, board[1][1] ≠ safe
Now ,Increment Row By 1
Q
Q
Q](https://image.slidesharecdn.com/ailecture-10unit02-210506092545/85/Ai-lecture-10-unit02-3-320.jpg)
![N-Queens Problem Solution Using Backtracking
Example :4 x 4 Chessboard To place 4-
Queens
Step:-4
r=2,c=1
Check, board[2][1] =safe
Now ,Increment column By 1(Place Queen)
Step:5
r=0,c=2
Check, board[0][2] ≠ safe
Now ,Increment Row By 1
Step:6
r=1,c=2
Check, board[1][2] ≠ safe
Now ,Increment Row By 1
Q
Q
Q
Q
Q
Q](https://image.slidesharecdn.com/ailecture-10unit02-210506092545/85/Ai-lecture-10-unit02-4-320.jpg)
![N-Queens Problem Solution Using Backtracking
Example :4 x 4 Chessboard To place 4-Queens
Step:-7
r=2,c=2
Check, board[2][2] ≠ safe
Now ,Increment Row By 1
Step:8
r=3,c=2
Check, board[3][2] ≠ safe
Now ,Increment Row By 1
Now after this if we try to increment Row by 1 it
becomes 4..which is invalid, as we have 4*4
chessboard here, so here we have reach to END of
Board.. and we could not able to placed 4 queens so,
here we have to BACKTRACKKKKK…
Q
Q
Q
Q](https://image.slidesharecdn.com/ailecture-10unit02-210506092545/85/Ai-lecture-10-unit02-5-320.jpg)
![N-Queens Problem Solution Using Backtracking
Example :4 x 4 Chessboard To place 4-Queens
Note: When u backtrack we have to remove last
placed queen and increment ROW BY 1
So ,here at position Board[2][1] we have lastly
placed the queen so ,we remove it increment ROW
BY 1…So we get next step as follows…
Step:9
r=3,c=1
Check, board[3][1] =safe
Now ,Increment Column By 1(placed Queen)
Step:10
r=0,c=2
Check, board[0][2] ≠ safe
Now ,Increment Row By 1
Q
Q
Q
Q](https://image.slidesharecdn.com/ailecture-10unit02-210506092545/85/Ai-lecture-10-unit02-6-320.jpg)
![N-Queens Problem Solution Using Backtracking
Step:11
r=1,c=2
Check, board[1][2] =safe
Now ,Increment Column By 1(placed queen)
Step:12
r=0,c=3
Check, board[0][3] ≠ safe
Now ,Increment Row By 1
Step:13
r=1,c=3
Check, board[1][3] ≠ safe
Now ,Increment Row By 1
Q
Q
Q
Q
Q
Q
Q
Q
Q](https://image.slidesharecdn.com/ailecture-10unit02-210506092545/85/Ai-lecture-10-unit02-7-320.jpg)
![N-Queens Problem Solution Using Backtracking
Step:14
r=2,c=3
Check, board[2][3] ≠ safe
Now ,Increment Row By 1
Step:15
r=3,c=3
Check, board[3][3] ≠ safe
Now ,Increment Row By 1
Now after this if we try to increment Row by 1 it
becomes 4..which is invalid, as we have 4*4
chessboard here, so here we have reach to END of
Board.. and we could not able to placed 4 queens
so, here we have to BACKTRACKKKKK…
Q
Q
Q
Q
Q
Q](https://image.slidesharecdn.com/ailecture-10unit02-210506092545/85/Ai-lecture-10-unit02-8-320.jpg)
![N-Queens Problem Solution Using Backtracking
Note: When u backtrack we have to remove last
placed queen and increment ROW BY 1
So ,here at position board[1][2] we have lastly
placed the queen so ,we remove it increment ROW
BY 1…So we get next step as follows…
Step:16
r=2,c=2
Check, board[2][2] ≠ safe
Now ,Increment Row By 1
Step:17
r=3,c=2
Check, board[3][2] ≠ safe
Now ,Increment Row By 1
Q
Q
Q
Q](https://image.slidesharecdn.com/ailecture-10unit02-210506092545/85/Ai-lecture-10-unit02-9-320.jpg)
![N-Queens Problem Solution Using Backtracking
Now after this if we try to increment Row by 1 it becomes 4..which is
invalid, as we have 4*4 chessboard here, so here we have reach to END
of Board.. and we could not able to placed 4 queens so, here we have to
BACKTRACKKKKK…
Note: When u backtrack we have to remove last placed queen and
increment ROW BY 1
So ,here at position board[3][1] we have lastly placed the queen so ,we
remove it increment ROW BY 1
Now after this if we try to increment Row by 1 it becomes 4..which is
invalid, as we have 4*4 chessboard here, so here we have reach to END
of Board.. and we could not able to placed 4 queens so, here we have to
BACKTRACKKKKK…](https://image.slidesharecdn.com/ailecture-10unit02-210506092545/85/Ai-lecture-10-unit02-10-320.jpg)
![N-Queens Problem Solution Using Backtracking
Note: When u backtrack we have to remove last
placed queen and increment ROW BY 1
So ,here at position board[0][0] we have lastly
placed the queen so ,we remove it increment ROW
BY 1…So we get next step as follows…
Step:18
r=1,c=0
Check, board[1][0] = safe
Now ,Increment column By 1
Step:19
r=0,c=1
Check, board[0][1] ≠ safe
Now ,Increment Row By 1
Q
Q](https://image.slidesharecdn.com/ailecture-10unit02-210506092545/85/Ai-lecture-10-unit02-11-320.jpg)
![N-Queens Problem Solution Using Backtracking
Step:20
r=1,c=1
Check, board[1][1] ≠ safe
Now ,Increment Row By 1
Step:21
r=2,c=1
Check, board[2][1] ≠ safe
Now ,Increment Row By 1
Step:22
r=3,c=1
Check, board[3][1] = safe
Now ,Increment Column By 1
Q
Q
Q
Q](https://image.slidesharecdn.com/ailecture-10unit02-210506092545/85/Ai-lecture-10-unit02-12-320.jpg)
![N-Queens Problem Solution Using Backtracking
Step:23
r=0,c=2
Check, board[0][2] =safe
Now ,Increment Column By 1
Step:24
r=0,c=3
Check, board[3][0] ≠ safe
Now ,Increment Row By 1
Step:25
r=1,c=3
Check, board[1][3] ≠ safe
Now ,Increment Row By 1
Q
Q
Q
Q
Q
Q
Q
Q
Q](https://image.slidesharecdn.com/ailecture-10unit02-210506092545/85/Ai-lecture-10-unit02-13-320.jpg)
![N-Queens Problem Solution Using Backtracking
Step:26
r=2,c=3
Check, board[2][3] =safe
Now ,Increment Column By 1
So, finally we have placed all 4 queen
Q
Q
Q
Q](https://image.slidesharecdn.com/ailecture-10unit02-210506092545/85/Ai-lecture-10-unit02-14-320.jpg)
Ai lecture 10(unit02)
- 1. Topic To Be Covered: Example of CSP:N-Queens Problem Solution Using Backtracking Jagdamba Education Society's SND College of Engineering & Research Centre Department of Computer Engineering SUBJECT: Artificial Intelligence & Robotics Lecture No-10(UNIT-02) Prof.Dhakane Vikas N
- 2. N-Queens Problem Solution Using Backtracking What is N-Queen Problem? This problem is to find an arrangement of N queens on a chess board, such that no queen can attack any other queens on the board. The N Queen is the problem of placing N chess queens on an N×N chessboard so that no two queens attack each other. Queen can attack each other if they are in same column, row & diagonal. Problem Statement: We have to place 4-Queens on 4 x 4 chessboard such that no queen will attack each other(For that we have to check no two queens are placed in same Row, Column & Diagonal
- 3. N-Queens Problem Solution Using Backtracking Example :4 x 4 Chessboard To place 4- Queens Step:-1 r=0,c=0 Check, board[0][0] =safe Now ,Increment column By 1(Place Queen) Step:2 r=0,c=1 Check, board[0][1] ≠ safe Now ,Increment Row By 1 Step:3 r=1,c=1 Check, board[1][1] ≠ safe Now ,Increment Row By 1 Q Q Q
- 4. N-Queens Problem Solution Using Backtracking Example :4 x 4 Chessboard To place 4- Queens Step:-4 r=2,c=1 Check, board[2][1] =safe Now ,Increment column By 1(Place Queen) Step:5 r=0,c=2 Check, board[0][2] ≠ safe Now ,Increment Row By 1 Step:6 r=1,c=2 Check, board[1][2] ≠ safe Now ,Increment Row By 1 Q Q Q Q Q Q
- 5. N-Queens Problem Solution Using Backtracking Example :4 x 4 Chessboard To place 4-Queens Step:-7 r=2,c=2 Check, board[2][2] ≠ safe Now ,Increment Row By 1 Step:8 r=3,c=2 Check, board[3][2] ≠ safe Now ,Increment Row By 1 Now after this if we try to increment Row by 1 it becomes 4..which is invalid, as we have 4*4 chessboard here, so here we have reach to END of Board.. and we could not able to placed 4 queens so, here we have to BACKTRACKKKKK… Q Q Q Q
- 6. N-Queens Problem Solution Using Backtracking Example :4 x 4 Chessboard To place 4-Queens Note: When u backtrack we have to remove last placed queen and increment ROW BY 1 So ,here at position Board[2][1] we have lastly placed the queen so ,we remove it increment ROW BY 1…So we get next step as follows… Step:9 r=3,c=1 Check, board[3][1] =safe Now ,Increment Column By 1(placed Queen) Step:10 r=0,c=2 Check, board[0][2] ≠ safe Now ,Increment Row By 1 Q Q Q Q
- 7. N-Queens Problem Solution Using Backtracking Step:11 r=1,c=2 Check, board[1][2] =safe Now ,Increment Column By 1(placed queen) Step:12 r=0,c=3 Check, board[0][3] ≠ safe Now ,Increment Row By 1 Step:13 r=1,c=3 Check, board[1][3] ≠ safe Now ,Increment Row By 1 Q Q Q Q Q Q Q Q Q
- 8. N-Queens Problem Solution Using Backtracking Step:14 r=2,c=3 Check, board[2][3] ≠ safe Now ,Increment Row By 1 Step:15 r=3,c=3 Check, board[3][3] ≠ safe Now ,Increment Row By 1 Now after this if we try to increment Row by 1 it becomes 4..which is invalid, as we have 4*4 chessboard here, so here we have reach to END of Board.. and we could not able to placed 4 queens so, here we have to BACKTRACKKKKK… Q Q Q Q Q Q
- 9. N-Queens Problem Solution Using Backtracking Note: When u backtrack we have to remove last placed queen and increment ROW BY 1 So ,here at position board[1][2] we have lastly placed the queen so ,we remove it increment ROW BY 1…So we get next step as follows… Step:16 r=2,c=2 Check, board[2][2] ≠ safe Now ,Increment Row By 1 Step:17 r=3,c=2 Check, board[3][2] ≠ safe Now ,Increment Row By 1 Q Q Q Q
- 10. N-Queens Problem Solution Using Backtracking Now after this if we try to increment Row by 1 it becomes 4..which is invalid, as we have 4*4 chessboard here, so here we have reach to END of Board.. and we could not able to placed 4 queens so, here we have to BACKTRACKKKKK… Note: When u backtrack we have to remove last placed queen and increment ROW BY 1 So ,here at position board[3][1] we have lastly placed the queen so ,we remove it increment ROW BY 1 Now after this if we try to increment Row by 1 it becomes 4..which is invalid, as we have 4*4 chessboard here, so here we have reach to END of Board.. and we could not able to placed 4 queens so, here we have to BACKTRACKKKKK…
- 11. N-Queens Problem Solution Using Backtracking Note: When u backtrack we have to remove last placed queen and increment ROW BY 1 So ,here at position board[0][0] we have lastly placed the queen so ,we remove it increment ROW BY 1…So we get next step as follows… Step:18 r=1,c=0 Check, board[1][0] = safe Now ,Increment column By 1 Step:19 r=0,c=1 Check, board[0][1] ≠ safe Now ,Increment Row By 1 Q Q
- 12. N-Queens Problem Solution Using Backtracking Step:20 r=1,c=1 Check, board[1][1] ≠ safe Now ,Increment Row By 1 Step:21 r=2,c=1 Check, board[2][1] ≠ safe Now ,Increment Row By 1 Step:22 r=3,c=1 Check, board[3][1] = safe Now ,Increment Column By 1 Q Q Q Q
- 13. N-Queens Problem Solution Using Backtracking Step:23 r=0,c=2 Check, board[0][2] =safe Now ,Increment Column By 1 Step:24 r=0,c=3 Check, board[3][0] ≠ safe Now ,Increment Row By 1 Step:25 r=1,c=3 Check, board[1][3] ≠ safe Now ,Increment Row By 1 Q Q Q Q Q Q Q Q Q
- 14. N-Queens Problem Solution Using Backtracking Step:26 r=2,c=3 Check, board[2][3] =safe Now ,Increment Column By 1 So, finally we have placed all 4 queen Q Q Q Q