Algorithms and Data Structures

N.Wirth. Algorithms and Data Structures. Oberon version 117
of the algorithm corresponds to a search of the candidate tree in a systematic fashion in which every node
is visited exactly once. It allows — once a solution is found and duly recorded — merely to proceed to
the next candidate delivered by the systematic selection process. The modification is formally accomplished
by carrying the procedure function CanBeDone from the loop's guard into its body and substituting the
procedure body in place of its call. To return a logical value is no longer necessary. The general schema is
as follows:
PROCEDURE Try;
BEGIN
IF solution incomplete THEN
initialize selection of candidate moves and select the first one;
WHILE ~(no more moves) DO
record move;
Try;
erase move;
select next move
END
ELSE
print solution
END
END Try
It comes as a surprise that the search for all possible solutions is realized by a simpler program than the
search for a single solution.
In the eight queenn problem another simplification is posssible. Indeed, the somewhat cumbersome
mechanism of enumeration of safe positions which consists of the two procedures First and Next, was
used to disentangle the linear search of the next safe position (the loop over j within Next) and the linear
search of the first j which yields a complete solution. Now, thanks to the simplification of the latter loop,
such a disentanglement is no longer necessary as the simplest loop over j will suffice, with the safe j filtered
by an IF embedded within the loop, without invoking the additional procedures.
The extended algorithm to determine all 92 solutions of the eight queens problem is shown in the
following program. Actually, there are only 12 significantly differing solutions; our program does not
recognize symmetries. The 12 solutions generated first are listed in Table 3.2. The numbers n to the right
indicate the frequency of execution of the test for safe fields. Its average over all 92 solutions is 161.
PROCEDURE write; (* ADenS35_Queens *)
VAR k: INTEGER;
BEGIN
FOR k := 0 TO 7 DO Texts.WriteInt(W, x[k], 4) END;
Texts.WriteLn(W)
END write;
PROCEDURE Try (i: INTEGER);
VAR j: INTEGER;
BEGIN
IF i < 8 THEN
FOR j := 0 TO 7 DO
IF a[j] & b[i+j] & c[i-j+7] THEN
x[i] := j; a[j] := FALSE; b[i+j] := FALSE; c[i-j+7] := FALSE;
Try(i + 1);
x[i] := -1; a[j] := TRUE; b[i+j] := TRUE; c[i-j+7] := TRUE
END