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Programming Assignment 5 Solution
In 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 such a way that no two queens confront each other. One solution to this problem is pictured
The n-Queens problem is the natural generalization of placing n queens on an n n chessboard so that no two queens confront one another. 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 n n 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.
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
n n
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
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.
8
R
8
R
7
R
7
R
6
R
6
R
5
R
5
R
4
R
4
R
3
R
3
R
2
R
2
R
1
R
1
R
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
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 pieces lie on the same diagonal. 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 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, printed while ( 2, 7, 8, 5, 1, 4, 6, 3) would not. Two major problems must therefore be
the project: (1) how can you produce all permutations of the set {1, 2, 3, …, n}, and permutation how can you determine if two pieces lie on the same diagonal.
3, 1, 7, 5 ) would be solved to complete (2) given one such
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 (n 1)th element, and finally 1 way to choose the n th 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 2 1 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
A[ j] 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
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 pieces on the same diagonal, and will return false otherwise.
( A[1], A[2], A[3], , A[n] ) To check if two queens at
places no two
( 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.
horizontal distance
8
7
6
5
4
Q
1 2 3 4 5
6 2 4
Q
6 7 8
vertical distance
7 3 4
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, 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:
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.
Note that there are much more efficient procedures for solving this problem that use programming methods outside the scope of this course. Even if you know how to use such methods, 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 that the program will 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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