The n Queen Problem

The n-queen
problem
Prepared by::
SUSHANT GOEL
(b090010291)
SUKRIT GUPTA
(b090010285)

Introduction
• N-Queens dates back to the 19th century
(studied by Gauss)
• Classical combinatorial problem, widely used
as a benchmark because of its simple and regular structure
• Problem involves placing N queens on an N  N chessboard
such that no queen can attack any other
• Benchmark code versions include finding the first solution and
finding all solutions

The n-queen problem
• What is its input??
• Ans. Positive integer n
• What is its task??
• Ans. Place n queens on an n by n chessboard so that no two
queens attack each other (on same row, column, diagonal), or
report that this is impossible.
• Solving particular problem for the different values of
n=1,2,3,4.
• Pure brute force search
• 16 squares on a 4 by 4 chessboard
• Leads to 16 * 15 * 14 * 13 = 43,680 possible solutions

For the larger values of n
n=8
• At most one queen per row: 88 = 16,777,216
• Early backtracking: 2057 nodes total
• Time to find first solution: 114 nodes
n=12
• At most one queen per row: 1212
• Early backtracking: 856,189 nodes total
• Time to find first solution: 262 nodes

N-Queens Solutions
• Various approaches to the problem
• Brute force
• Local search algorithms
• Backtracking
• Divide and conquer approach
• Permutation generation
• Mathematical solutions
• Graph theory concepts

Example:-
1 1
. . 2
1
2
. . . .
1
2
. 3
1
2
3
. . 4
1
. . . 2
11
2
3
. . . .

The backtracking method
• A given problem has a set of constraints and possibly an
objective function
• The solution optimizes an objective function, and/or is
feasible.
• We can represent the solution space for the problem
using a
• The root of the tree represents 0 choices,
• Nodes at depth 1 represent first choice
• Nodes at depth 2 represent the second choice, etc.
• In this tree a path from a root to a leaf represents a
candidate solution

Backtracking approach
• One of the only approaches that guarantees a solution,
though it can be slow
• Can be seen as a form of intelligent depth-first search
• Complexity of backtracking typically rises exponentially
with problem size
• Good test case for performance analysis of RC systems,
as the problem is complex even for small data size
• Traditional processors provide a suboptimal platform for
this iterative application due to serial nature of their
processing pipelines
• Tremendous speedups achieved by adding parallelism at
the logic level via RC

Algorithm
• AlgorithmNQueens(k,n)
• //Using backtracking,this procedure prints all possible
//placements of n queens on an n*n
//chessboard so that they are non attacking
{
for(i=1 to n ) do
{
if Place(k,i) then
{
x[k]=I
If(k=n) then write (x[1:n]);
else Nqueens(k+1,n);
}
}
}

• Algorythm Place(k,i)
{
For j=1 to k-1 do
if((x[j]=i) //same colum or (Abs(x[j]-i)=Abs(j-k)))//same
diagonal
then return false;
Return true;
}

The n Queen Problem

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The n Queen Problem