lab01
The N-Queens Problem
 Suppose you have N chess queens…
 …and a N by N chess board
 Can the queens be placed on board without conflicts?
 Conflict: Any two queens on board attacking each other
is a conflict.
Attacking of queens
 Two queens are attacking each other if they are
 in the same row
 in the same column
 in the same diagonal
Denoting the positions of queens
 A 4 by 4 chess board. (4 rows, 4 columns)
row 3, column 1
row 2, column 3
row 1, column 1
 A 8 by 8 chess board. (8 rows, 8 columns)
a8
b4
Pseudocode of N-queens Problem
 Initialize a stack s where we can keep track of the
placement of queens
 Place the first queen, push its position onto s and set
filled to 0.
 int filled =0;
Continued…
 Repeat these steps:
 if there are no conflicts
increase filled by 1. If filled is now N, then algorithm is done.
Otherwise move to the next row and place a queen in the first column.
Push its position onto the stack.
 else if there is a conflict and there is room to shift the
current queen rightward
Move the current queen rightward, adjusting the record on top of stack
to indicate the new position.
 else if there is a conflict and there is no room to shift the
current queen rightward
Keep popping the stack, and decrease filled by 1 until a row is reached
where the queen can be shifted rightward.
Shift the queen rightward and adjust the record on top of stack to
indicate the new position.
Example:4-queues problem(4
queens on a 4 by 4 board)
 Initialize a stack s where we can keep track of the
placement of queens
top
Stack s
bottom
 Place the first queen, push its position onto s and set
filled to 0.
 int filled =0;
 there are no conflicts, increase filled by 1.
 move to the next row (row 2) and place a queen in the
first column. Push its position onto the stack.
 If there is a conflict (yellow line), then we shift the new
queen to the next column. (rightward ). Adjust the
record on top of stack to indicate the new position
(ROW 2, COL 2).
 If there is a conflict (yellow line), then we shift the new
queen to the next column. (rightward ). Adjust the
record on top of stack to indicate the new position
(ROW 2, COL 3).
ROW 2, COL 3
 there are no conflicts, increase filled by 1.
ROW 2, COL 3
 move to the next row and place a queen in the first
column. Push its position onto the stack.
ROW 3, COL 1
 there is a conflict and there is room to shift the current
queen rightward. Move the current queen rightward,
adjusting the record on top of stack to indicate the new
position. (ROW 2, COL 2)
 there is a conflict and there is room to shift the current
queen rightward. Move the current queen rightward,
adjusting the record on top of stack to indicate the new
position. (ROW 2, COL 3)
 there is a conflict and there is room to shift the current
queen rightward. Move the current queen rightward,
adjusting the record on top of stack to indicate the new
position. (ROW 2, COL 4)
 there is a conflict and there is no room to shift the current
queen rightward. Keep popping the stack, and decrease
filled by 1 until a row is reached where the queen can be
shifted rightward.
 Shift the queen rightward. Adjust the record on top of
stack to indicate the new position (ROW 2, COL 4).
ROW 2, COL 4
 there are no conflicts, increase filled by 1
 move to the next row and place a queen in the first
column. Push its position onto the stack (ROW2, COL
1).
 there is a conflict and there is room to shift the current
queen rightward. Move the current queen rightward,
adjusting the record on top of stack to indicate the new
position.
ROW 3, COL 1
ROW 3, COL 2
 there are no conflicts. Increase filled by 1.
 move to the next row and place a queen in the first
column. Push its position onto the stack.
ROW 4, COL 1
ROW 3, COL 2
ROW 3, COL 2
3
3
 there is a conflict and there is room to shift the current
queen rightward.
 Move the current queen rightward, adjusting the record
on top of stack to indicate the new position.
ROW 4, COL 1
ROW 4, COL 2
ROW 3, COL 2
ROW 3, COL 2
3
3
 there is a conflict and there is room to shift the current
queen rightward.
 Move the current queen rightward, adjusting the record
on top of stack to indicate the new position.
ROW 4, COL 2
ROW 4, COL 3
ROW 3, COL 2
ROW 3, COL 2
3
3
ROW 4, COL 3
ROW 4, COL 4
ROW 3, COL 2
ROW 3, COL 2
3
3
there is a conflict and there is no room to shift the current queen
rightward
Keep popping the stack, and decrease filled by 1 until a row is reached
where the queen can be shifted rightward. Shift the queen rightward.
Adjust the record on top of stack to indicate the new position (ROW ,
COL )
ROW 4, COL 4
ROW 3, COL 2
ROW 3, COL 3
3
2
ROW 3, COL 3
ROW 3, COL 4
2
2
there is a conflict and there is no room to shift the current queen
rightward
Keep popping the stack, and decrease filled by 1 until a row is reached
where the queen can be shifted rightward. Shift the queen rightward.
Adjust the record on top of stack to indicate the new position (ROW ,
COL )
ROW 2, COL 4
ROW 2, COL 4
2
ROW 1, COL
1
1=filled
Keep popping the stack, because current queen can not move rightward.
decrease filled by 1
Shift the queen rightward. Adjust the record on top of stack to indicate
the new position (ROW 1 , COL 2).
ROW 2, COL 4
ROW 1, COL
2
ROW 1, COL
1
1=filled
0=filled
Two steps are omitted here.
ROW 2, COL 1
ROW 2, COL 4
ROW 1, COL
2
ROW 1, COL
2
1=filled
2=filled
ROW 3, COL 1
ROW 2, COL 4
ROW 1, COL
2
3=filled
if there are no conflicts increase filled by 1.
If filled is now N(N=4), then algorithm is done.
ROW 4, COL 3
ROW 3, COL 1
ROW 2, COL 4
ROW 1, COL
2
4=filled