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Order NowIn this assignment you will write a Java program that finds all solutions to the n-Queens problem, for
1 n 13. Begin by reading the Wikipedia article on the Eight Queens puzzle at:
http://en.wikipedia.org/wiki/Eight_queens_puzzle
In the game of Chess a queen can move any number of spaces in any linear direction: horizontally,
vertically, or along a diagonal.
The Eight Queens puzzle is to find a placement of 8 queens on an otherwise empty
88
chessboard in such
a way that no two queens confront each other. One solution to this problem is pictured below.
The n-Queens problem is the natural generalization of placing n queens on an
nn
chessboard so that no
two queens lie on the same row, column or diagonal. There are many ways of solving this problem. Our
approach will be to start with a solution to the n-rooks problem (i.e. place n Rooks on an
nn
chessboard
so that no two rooks attack each other) then check if that arrangement is also a solution to n-Queens. The
rook move in chess is similar to the queen’s move except that it cannot move diagonally.
2
Solutions to the n-Rooks problem are easy to find since one need only position the rooks so that no two are
on the same row and no two are on the same column. Since there are n rows and n columns to choose from,
solutions abound.
A natural way to encode solutions to n-Rooks is as permutations of the integers {1, 2, 3, …, n}. A
permutation of a set is an ordered arrangement of the elements in that set. If we number the rows and
columns of the
nn
chessboard using the integers 1 through n, each square is then labeled by a unique pair
of coordinates (i, j) indicating the square in row i and column j. The permutation
( , , , , ) a1 a2 a3 an
corresponds to the placement of a piece on the square with coordinates
(a , j)
j
for
1 j n
. For instance,
the permutations
( 2, 7, 8, 5,1, 4, 6, 3)
and
( 2, 4, 6, 8, 3,1, 7, 5)
correspond to two the 8-Rooks solutions
pictured below.
Observe that the solution on the right is also a solution to 8-Queens, while the one on the left is not, since
certain pairs of rooks lie on the same diagonal (such as the rooks on (7, 2) and (5, 4).) In fact the solution
on the right is the 8-Queens solution pictured on the previous page. In general, any solution to the n-Queens
problem is also a solution to n-Rooks, but the converse is false. Not every solution to n-Rooks is a solution
to n-Queens.
Your program will generate all solutions to n-Rooks by systematically producing all permutations of the set
{1, 2, 3, …, n}. It will check each n-Rooks solution for diagonal attacks to see if the given permutation
8 R
7 R
6 R
5 R
4 R
3 R
2 R
1 R
1 2 3 4 5 6 7 8
8 R
7 R
6 R
5 R
4 R
3 R
2 R
1 R
1 2 3 4 5 6 7 8
3
also solves n-Queens. Whenever an n-Queens solution is found, your program will print out the
permutation, then move on to the next permutation. Thus when
n 8, ( 2, 4, 6, 8, 3,1, 7, 5)
would be printed
while
( 2, 7, 8, 5,1, 4, 6, 3)
would not. Two major problems must therefore be solved to complete the
project: (1) how can you produce all permutations of the set {1, 2, 3, …, n}, and (2) given one such
permutation how can you determine if two pieces lie on the same diagonal.
Permutations
There are
n!
permutations of a finite set containing n elements. To see this, observe that there are n ways
to choose the first element in the arrangement,
n 1
ways to choose the second element,
n 2
ways to
choose the third, … , 2 ways to choose the
th (n 1)
element, and finally 1 way to choose the
th
n
and last
element in the ordered arrangement. The number of ways of making all these choices in succession is
therefore
n(n 1)(n 2)3 21 n!
. For instance there are
3! 6
permutations of the set {1, 2, 3}: 123,
132, 213, 231, 312, 321. The permutations of a finite set {1, 2, 3, …, n} have a natural ordering called the
lexicographic order, or alphabetic order. The
4! 24
permutations of {1, 2, 3, 4} are listed in order as
follows.
1234 2134 3124 4123
1243 2143 3142 4132
1324 2314 3214 4213
1342 2341 3241 4231
1423 2413 3412 4312
1432 2431 3421 4321
As an exercise list the
5!120
permutations of {1, 2, 3, 4, 5} in lexicographic order. After finishing this
long exercise, you will see the need for an algorithm that systematically produces all permutations of a
finite set. We will represent a permutation
( , , , , ) a1 a2 a3 an
by an array
A[ ]
of length
n 1
, where
aj A[ j]
for
1 j n
, and the element
A[0]
is simply not used. Your program will include a function
with the following heading.
static void nextPermutation(int[] A){. . .}
This method will alter its argument A by advancing
( A[1], A[2], A[3],, A[n] )
to the next permutation in
the lexicographic ordering. If
( A[1], A[2], A[3],, A[n] )
is already at the end of the sequence, the function
will reset A to the initial permutation
(1, 2, 3 ,, n)
in the lexicographic order. The pseudo-code below
gives an outline for the body of nextPermutation().
scan the array from right-to-left
if the current element is less than its right-hand neighbor
call the current element the pivot
stop scanning
if the left end was reached without finding a pivot
reverse the array (permutation was lexicographically last, so start over)
return
scan the array from right-to-left again
if the current element is larger than the pivot
call the current element the successor
stop scanning
swap the pivot and the successor
reverse the portion of the array to the right of where the pivot was found
return
4
Run the above procedure by hand on the initial permutation (1, 2, 3, 4) and see that it does indeed produce
all 24 permutations of the set {1, 2, 3, 4} in lexicographic order. Also check that (4, 3, 2, 1) which is the
final permutation in lexicographic order, is returned to the initial state (1, 2, 3, 4).
Finding Diagonal Attacks
Your program will also include another function with the following heading.
static boolean isSolution(int[] A){. . .}
This method will return true if the permutation represented by
( A[1], A[2], A[3],, A[n] )
places no two
queens on the same diagonal, and will return false otherwise. To check if two queens at
( A[i], i)
and
( A[ j], j)
lie on the same diagonal, it is sufficient to check whether their horizontal distance apart is the
same as their vertical distance apart, as illustrated in the diagram below.
Function isSolution() should compare each pair of queens at most once. If a pair is found on the same
diagonal, do no further comparisons and return false. If all
n(n 1)/ 2
comparisons are performed without
finding a diagonal attack, return true. (Question: why is the number of 2-element subsets of an n element
set exactly
n(n 1)/ 2
?)
Program Operation
Your program for this project will be called Queens.java. You will include a Makefile that creates an
executable Jar file called Queens, allowing one to run the program by typing Queens at the command line.
Your program will read an integer n from the command line indicating the size of the Queens problem to
solve. The program will operate in two modes: normal and verbose (which is indicated by the command
line option “-v”). In normal mode, the program prints only the number of solutions to n-Queens. In verbose
mode, all permutations representing solutions to n-Queens will be printed in lexicographic order, followed
by the number of such solutions. Thus to find the number of solutions to 8-Queens you will type:
% Queens 8
To print all 92 unique solutions to 8-Queens type:
8
7 Q
6
5
4
3 Q
2
1
1 2 3 4 5 6 7 8
vertical distance
73 4
horizontal distance
62 4
5
% Queens –v 8
If the user types anything on the command line other than the option –v and a number n, the program will
print a usage message to stderr and quit. A sample session is included below.
% Queens
Usage: Queens [-v] number
Option: -v verbose output, print all solutions
% Queens x
Usage: Queens [-v] number
Option: -v verbose output, print all solutions
% Queens 5
5-Queens has 10 solutions
% Queens -v 5
(1, 3, 5, 2, 4)
(1, 4, 2, 5, 3)
(2, 4, 1, 3, 5)
(2, 5, 3, 1, 4)
(3, 1, 4, 2, 5)
(3, 5, 2, 4, 1)
(4, 1, 3, 5, 2)
(4, 2, 5, 3, 1)
(5, 2, 4, 1, 3)
(5, 3, 1, 4, 2)
5-Queens has 10 solutions
% Queens -v 6
(2, 4, 6, 1, 3, 5)
(3, 6, 2, 5, 1, 4)
(4, 1, 5, 2, 6, 3)
(5, 3, 1, 6, 4, 2)
6-Queens has 4 solutions
% Queens -v 4
(2, 4, 1, 3)
(3, 1, 4, 2)
4-Queens has 2 solutions
% Queens -v 3
3-Queens has 0 solutions
%
It is recommended that you write helper functions to perform basic subtasks such as: print the usage
message and quit, calculate the factorial of n, print out a formatted array as above, swap two elements in an
array, and reverse the elements in a subarray. Some of these methods have already been posted on the
website as examples.
There are more efficient procedures for solving n-queens that use programming techniques beyond the
scope of this course (such as recursion). Even if you know how to use these techniques, you are required
to solve the problem as indicated in this project description. Your program should work very quickly on
problem sizes up to 12 or 13. Beyond that, you should expect your program to slow down considerably.
What to turn in
Write a Makefile for this project that creates an executable Jar file called Queens, and that includes a clean
target (as in lab4). Submit the files Makefile and Queens.java to the assignment name pa5. As always start
early and ask questions of myself, the TAs and on Piazza.
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