Algorithms and Data Structures

N.Wirth. Algorithms and Data Structures. Oberon version 113
23 10 15 4 25
16 5 24 9 14
11 22 1 18 3
6 17 20 13 8
21 12 7 2 19
23 4 9 14 25
10 15 24 1 8
5 22 3 18 13
16 11 20 7 2
21 6 17 12 19
1 16 7 26 11 14
34 25 12 15 6 27
17 2 33 8 13 10
32 35 24 21 28 5
23 18 3 30 9 20
36 31 22 19 4 29
Table 3.1. Three Knights' Tours.
What abstractions can now be made from this example? Which pattern does it exhibit that is typical for
this kind of problem-solving algorithms? What does it teach us? The characteristic feature is that steps
toward the total solution are attempted and recorded that may later be taken back and erased in the
recordings when it is discovered that the step does not possibly lead to the total solution, that the step leads
into a dead-end street. This action is called backtracking. The general pattern below is derived from
TryNextMove, assuming that the number of potential candidates in each step is finite.
PROCEDURE Try;
BEGIN
IF solution incomplete THEN
initialize selection of candidates and choose the first one;
WHILE ~(no more candidates) & ~CanBeDone(candidate) DO
select next
END
END
END Try;
PROCEDURE CanBeDone (move): BOOLEAN;
BEGIN
record move;
Try;
IF not successful THEN
cancel recording
END;
RETURN successful
END CanBeDone
Actual programs may, of course, assume various derivative forms of this schema. In particular, the way
input is passed to TryNextMove may vary depending on the problem specifics. Indeed, this procedure
accesses global variables in order to record the solution, and those variables contain, in principle, a
complete information about the current state of the construction process. For instance, in the knight's tour
problem TryNextMove needs to know the knight's last position on the board, which can be found by a
search in the matrix h. However, this information is explicitly available when the procedure is called, and it
is simpler to pass it via parameters. In subsequent examples we will see variations on this theme.
Note that the search condition in the while loop is modeled as a procedure-function CanBeDone for a
maximal clarification of the logic of the algorithm while keeping the program easily comprehensible.