This document presents an overview of the N-Queens problem and the backtracking algorithm used to solve it. The N-Queens problem aims to place N queens on an N×N chessboard so that no two queens attack each other. The document outlines the constraints of the problem, explores the solution space using a state space tree representation, provides pseudocode for the backtracking algorithm, and analyzes the time complexity which is O(N!). Examples are given for solving the 4-Queens problem using backtracking.

1. The N-Queens
problem
Presented By:
Anoop Chaudhary
A Seminar on

2.  Introduction to N-Queens Problem
 Constraints and Solution Space
 The Backtracking Approach
 Solution to 4-Queens Problem using backtracking
 Algorithm for N-Queens Problems
 Time complexity of N-Queens algorithm
 Advantages of using backtracking
Contents:
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3. What is a queen ?
 A queen is one of six different chess pieces.
 It can move any distance available vertically, horizontally
or diagonally on the 8 x 8 grid of chessboard.
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Figure 1: Movement of a Queen on a Chessboard

4. Introduction to N-Queens Problem
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Place n queens on an n x n chessboard such that no two
queens attack each other.
One solution to each n-queens problem for n=4,6 and 8
4-queens 6-queens 8-queens

5. History
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 1848: Problem proposed by Max Bezzel.
 1850: Friedrich Gauss claims Q(8)=92
 1874: Dr. Glaisher prove Gauss’s claim
 1874: Paul proves that Q(n)>0 for all n>=4
 20th Century: Problem became a benchmark for
combinatorial optimization
 N-Queen problem is generalized form of 8-Queens problem

6.  Assumption: Queen i is to be placed on row i. Hence, all solutions
to the N-queens problem can be represented as n-tuples (x1,
…, xn), where xi is the column on which queen i is placed.
 Explicit Constraints:
 xi are chosen from set Si ={1, …, n} and 1<=i<=n.
So the Solution space consists of nn n-tuples.
 Implicit Constraints:
 No two xi ’s can be same
Now, the solution space reduced to n!
 No two queens can be on the same diagonal.
Constraints of N-Queens Problem
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(2, 4, 1, 3)

7. The Solution Space of N-Queens Problem
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 The solution space for the N-Queens problem can be
represented using a State Space Tree –
 The root of the tree represents 0 choices.
 Nodes at depth 1 represent first choice.
 Nodes at depth 2 represents the second choice, and so on till
depth n.
 Edges from depth i to i+1 are labeled with the values of xi.
 In this tree a path from a root to a leaf represents a candidate
solution and All the candidate solutions define the solution
space.

8. Example: 4-Queens Problem
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State Space Tree:

9. The Backtracking Approach
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 Backtracking is a standard method to find solutions for a particular
kind of problems known as “Constraint-Satisfaction-Problems”.
 Backtracking is finding the solution of a problem whereby the solution
depends on the previous steps taken. Thus the general steps of
backtracking are :
 Start with a sub-solution (partial candidate)
 Check if partial candidate will lead to the solution or not.
 If not, then come back and change the partial candidate and
continue again.
 Backtracking can be seen as a form of intelligent depth-first search
(DFS).

10. Example: 4-Queens Problem
a b c d
e f g h
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11. State Space tree, generated during backtracking
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* The B denotes the dead node.

12. Algorithm for N-Queens Problems
1. Place (k, i)
2. {
3. for j ← 1 to k - 1 do
4. // Check if two queens in same column or in same diagonal
5. if ( x[j] == i ) or ( Abs x[j]) - i) == (Abs (j - k) )
6. then return false;
7. return true;
8. }
 Returns true if a queen can be placed in kth row and ith column. Otherwise returns false.
 x[] is a global array whose first (k-1) values have been set.
 Abs(r) returns the absolute value of r.
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13. Algorithm for N-Queens Problems
1. NQueens(k, n)
2. {
3. for i ← 1 to n do
4. {
5. if Place (k, i) then
6. {
7. x[k] ← i;
8. if (k ==n) then write ( x[1 : n] );
9. else NQueens(k + 1, n);
10. }
11. }
12. }
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14. Complexity
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 Time complexity of N queens algorithm : For finding a
single solution where the first queen Q has been assigned
the first column and can be put on N positions, the
second queen has been assigned the second column and
would choose from N-1 possible positions and so on; the
time complexity is O ( N ) * ( N - 1 ) * ( N - 2 ) * … 1 ). i.e
The worst-case time complexity is O(N!) .
 Thus, for finding all the solutions to the N Queens
problem the time complexity runs in polynomial time.

15. Advantages of using backtracking
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 Ability to find and count all the possible solutions rather
than just one.
 Parallel version of algorithm can increase speed several
times.

16. References
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 “Computer Algorithms” by Ellis Horowitz, Sartaz Sahni and
Rajashekharan printed by Computer Science Press.
 The n-Queens Problem, Article in “The American Mathematical
Monthly” 101(7)- August 1994, DOI:10.2307/2974691
 http://mathcenter.oxford.emory.edu/site/cs171/nQueensProblemAnd
Backtracking/