Algorithms and Data Structures

N.Wirth. Algorithms and Data Structures. Oberon version 115
data representation which makes checking as simple as possible. The best recipe is to represent as directly
as possible that information which is truly relevant and most often used. In our case this is not the position
of the queens, but whether or not a queen has already been placed along each row and diagonals. (We
already know that exactly one is placed in each column k for 0 ≤ k < i.) This leads to the following choice
of variables:
VAR
where
x: ARRAY 8 OF INTEGER;
a: ARRAY 8 OF BOOLEAN;
b, c: ARRAY 15 OF BOOLEAN
x i denotes the position of the queen in the i-th column;
a j means "no queen lies in the j-th row";
b k means "no queen occupies the k-th /-diagonal;
c k means "no queen sits on the k-th \-diagonal.
We note that in a /-diagonal all fields have the same sums of their coordinates i and j, and that in a \-
diagonal the coordinate differences i-j are constant. The appropriate solution is shown in the following
program Queens. Given these data, the statement SetQueen is elaborated to
x[i] := j; a[j] := FALSE; b[i+j] := FALSE; c[i-j+7] := FALSE
the statement RemoveQueen is refined into
a[j] := TRUE; b[i+j] := TRUE; c[i-j+7] := TRUE
The field is safe if it lies in a row and in diagonals which are still free. Hence, it can be expressed
by the logical expression
a[j] & b[i+j] & c[i-j+7]
This allows one to formulate the procedures for enumeration of safe rows j for the i-th queen by analogy
with the preceding example.
This completes the development of this algorithm, that is shown in full below. The computed solution is x
= (1, 5, 8, 6, 3, 7, 2, 4) and is shown in Fig. 3.8.
0
1
2
3
4
5
6
7
0 1 2 3 4 5 6 7
Fig. 3.8. A solution to the Eight Queens problem