backtracking 8 Queen.pptx

Backtracking methodology
• Many problems which deal with searching for a set of
solutions or for a optimal solution satisfying some constraints can
be solved using the backtracking formulation.
• In Backtracking method, the desired solution is expressed as
an n-tuple vector : (x1…………… xi ……………….xn ) where xi is
chosen from some finite set Si.
• The problem is to find a vector, which maximizes or
minimizes a criterion function P(x1…………… xi ……………….xn ).
• The major advantage of this method is, once we know that a
partial vector(x1,…xi) will not lead to an optimal solution then
the generation of remaining components i.e.,(xi+1………..xn) are
ignored entirely.

1
34
18
50
2
8 13 19
3
5 7 12
10 15 17 23
21 26 28 33
31 37 39 44
42 47 9 55
53 58 60 65
63
4 6 11
9 14 16 22
20 25 27 32
30 36 38 43
41 46 48 54
52 57 59 64
62
29 35 40
24 51 56 61
45
X1=1
X1=2 X1=3
X1=4
2 3 4 1 3 4
1 2 4 1 2 3
3 4 2 4 2 3 3 4 1 4 1 3 2 4 1 4 1 2 2 3 1 3 1 2
4 3 4 2 3 2 4 3 4 1 3 1 4 2 4 1 2 1 3 2 3 1 2 1
x2=
x3
=
x4
=
Tree organization of the 4-queens solution space. Nodes are
numbered as in depth first search.
State space
Tree.

State Space Tree Terminology
1. Criterion function: It is a function P(X1, X2… X n)
which needs to be maximized or minimized for a
given problem.
2. Solution Space: All tuples that satisfy the explicit
constraints define a possible solution space for a
particular instance ‘I’ of the problem
3. Problem state: Each node in the tree organization
defines a problem state.
4. Solution States: These are those problem states S for
which the path form the root to S define a tuple in
the solution space.
5. State space tree: If we represent solution space in the
form of a tree then the tree is referred as the state
space tree.
6. Answer States: These solution states S for which the
path from the root to S defines a tuple which is a
member of the set of solution (i.e. it satisfies the
implicit constraints) of the problem.

State Space Tree Terminology..contd
7. Live node: A node which has been generated and
all of whose children have not yet been
generated is live node.
8. E-node: The live nodes whose children are
currently being generated is called E-node (node
being expanded)
9. Dead node: It is a generated node that is either
not to be expanded further or one for which all of
its children has been generated.
10. Bounding function: It will be used to kill live
nodes without generating all their children
11. Branch and bound: It is a method in which E-
node remains E-node until it is dead

Applications of Backtracking
1. Producing all permutations of a set of values
2. Parsing languages
3. Games: anagrams, crosswords, word jumbles,
8 queens
4. Combinatory and logic programming
Example Problems
1. N queen’s problem
2. Sum of subsets problem.
3. Graph colorig problem
4. 0/1 knapsck problem
5. Hamiltonian cycle problem
In summary, backtracking consists of
Doing a depth-first search of a state
space tree by applying boundary
conditions.

4-Queens Problem
Failure (Diagonal Attack)
So The solver has to backtrack

4-Queens Problem
No Row Attack, No Column Attack
& No Diagonal Attack
Hence It is Solution 1

n– queens : Defining the problem:-
The problem is to place n queens on an n x n chessboard
so that
no two “attack” that is no two queens on the same row,
same
column, or same diagonal.
 Assume rows and columns of chessboard are
numbered 1 through n. Queens also be numbered 1
through n.
 Condition to ensure - No two queens on same Row:
Each queen must be on a different row ,hence queen i
is to be placed on row i. Therefore all solutions to the
n-queens problem can be represented as n-tuples (
x1,x2,…..xn), where xi is the column on which queen i is
placed.
 Condition to ensure - No two queens on same
Column:
Select a distinct value for each xi

If two queens are placed at positions ( i, j ) and ( k, l ).
They are on the same diagonal then following
conditions hold good.
1) Every element on the same diagonal that runs from
the upper left to the lower right has the same row –
column value. i – j = k – l -----(1)
2) Similarly, every element on the on the same
diagonal that goes from the upper right to the
lower left has the same row + column value.
i + j = k + l -----(2)
• First equation implies : j - l = i – k
Second equation implies : j – l = k – i
• Therefore, two queens lie on the same diagonal if
and only if
j – l = i – k
 Condition to ensure - No two queens on
same Diagonal

Algorithm place( k, i )
// It returns true if a queen can be placed in kth row and ith
// column . Otherwise it returns false. x[] is a goal array whose
// first( k-1) values have been set. Abs(r) returns the absolute
value of r.
{
for j = 1 to k-1 do
{ // Two in the same column or in the same
diagonal
if ( ( x [ i ] = i ) or ( Abs( x[ j ] - i ) = Abs( j - k )
) ) then
return false;
}
return true ;
}

Algorithm Nqueen(k, n)
// Using backtracking, this procedure prints all
possible placements
// of n queens on an n×n chessboard so that they
are nonattacking.
{
for i = 1 to n do // check place of
column for queen k
{
if place( k, i ) then
{
x[ k ] = i;
if( k = n ) then write ( x[1:n]
);
else NQueens( k+1, n);
}
}

1
18
2
8 13
19
3
15 31
11
9 14 16 30
29
24
x1=1 x1=2
x2=2 x2=3 x2=4 x2=4
x3=1
x4=3
Portion of the tree that is generated during
backtracking( n=4 ).
B
B B
B
B
B B

backtracking 8 Queen.pptx

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backtracking 8 Queen.pptx