8 Queens Backtrack

7/28/2019 8 Queens Backtrack

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The 8 Queens problem

By

R. Kishore Kumar


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Objective of 8 Queen problem

Considering an n*n chessboard and try to find

all ways to place n nonattacking queens.


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Understanding 4 Queen problem

Here (X1,..,Xn) represent a solution in which

Xi is the column of the ith row where ith queen

is placed.

The Xis will all be distinct since no two

queens can be placed in the same column.


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Example of a backtracking solution to

4-queen problem

1 1

. . 2

1

2

. . . .

1

2

3

. . . .

1

2

3

1 1

. . . 2

1

2

3

. . 4


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8-Queen problem

Chessboards squares being numbered as the

indices of two dimensional array a[1:n][1:n].

We can observe that every element on the

same diagonal that runs from upper left to

lower right has the same row-column value.


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Example of 8 Queen problem

Consider the queen at a[4][2]

The square that are diagonal to this queen are

a[3][1], a[5][3], a[6][4], a[7][5] and a[8][6].

All these squares have row-column value of 2.

Every element on the same diagonal that goes

from the upper right to lower left has the samerow + column value.


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Two queens are placed at position (i,j) and (k,l)

then by the above they are on same diagonal

only if

i - j = k - l or i + j = k + l

The first equation implies j - l = i - k

Two queens lie on the same diagonal if and

only if |j-l| = |i-k|

Example of 8 Queen problem cont.


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1

2

3

4

5

(8,5,4,3,2) = 1649

1

2

3 4

5

6

(8,5,3,1,2,1) = 769

Solution of 8-Queen


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1

2

3

4

6 5

(8,6,4,2,1,1,1) = 1401

1

2

3

4

5

(8,6,4,3,2) = 1977

1

2

3

4

5

6

7

8

(8,5,3,2,2,1,1,1) = 2329

Solution of 8-Queen cont.


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Algorithm for NQueen

Place(k,i) returns a boolean value that is true

if the kth queen can be placed in column i.

It test both whether i is distinct from all

previous values X[1],,X[k-1] and also

whether there is no other queen on the same

diagonal.


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Bool Place(int k, int i)

//Returns true if a queen can be placed in kth row

//and ith column. Otherwise it returns false. x[]

//is a global array whose first (k-1) values have//been set. Abs(r) returns the absolute value of r

{

for(int j=1; j


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void NQueens(int k,int n)

//Using backtracking,this procedure prints all

//possible placements of n queens on an nxn

//chessboard so that they are nonattacking

{

for(int i=1;i


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Algorithm explanation

For an 8*8 chessboard there are (648) possible

ways to place 8 pieces.

Placement of each queen on the chessboard

was chosen randomly.

With each choice we keep track of number of

columns a queen could legitimately be placed

on.


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The vector represents the value that function

estimate would produce from these sizes.

The average of these five trials is 1625.

The total number of nodes in the 8-queens

state space tree is

69,281 1 + [7=0 (8)

= ] Its computational time is O(k-1).

Algorithm explanation cont.


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THANK YOU!!!

8 Queens Backtrack

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