Algorithms and Data Structures

N.Wirth. Algorithms and Data Structures. Oberon version 120
marrying individuals and testing stability after each proposed marriage, x and y are needed even before all
their components are defined. In order to keep track of defined components, we may introduce Boolean
arrays
singlem, singlew: ARRAY n OF BOOLEAN
with the meaning that singlem m implies that x m is undefined, and singlew w implies that y w is undefined. An
inspection of the proposed algorithm, however, quickly reveals that the marital status of a man k is
determined by the value mthrough the relation
~singlem[k] = k < m
This suggests that the array singlem be omitted; accordingly, we will simplify the name singlew to single.
These conventions lead to the refinement shown by the following procedure Try. The predicate
acceptable can be refined into the conjunction of single and stable, where stable is a function to be still
further elaborated.
PROCEDURE Try (m: man);
VAR r: rank; w: woman;
BEGIN
IF m < n THEN
FOR r := 0 TO n-1 DO
w := wmr[m,r];
IF single[w] & stable THEN
x[m] := w; y[w] := m; single[w] := FALSE;
Try(m+1);
single[w] := TRUE
END
END
ELSE
record the solution
END
END Try
At this point, the strong similarity of this solution with procedure AllQueens is still noticeable. The crucial
task is now the refinement of the algorithm to determine stability. Unfortunately, it is not possible to
represent stability by such a simple expression as the safety of a queen's position. The first detail that
should be kept in mind is that stability follows by definition from comparisons of ranks. The ranks of men
or women, however, are nowhere explicitly available in our collection of data established so far. Surely, the
rank of woman w in the mind of man m can be computed, but only by a costly search of w in wmr m . Since
the computation of stability is a very frequent operation, it is advisable to make this information more
directly accessible. To this end, we introduce the two matrices
rmw: ARRAY man, woman OF rank;
rwm: ARRAY woman, man OF rank
such that rmw m,w denotes woman w's rank in the preference list of man m, and rwm w,m denotes the rank
of man m in the list of w. It is plain that the values of these auxiliary arrays are constant and can initially be
determined from the values of wmr and mwr.
The process of determining the predicate stable now proceeds strictly according to its original
definition. Recall that we are trying the feasibility of marrying m and w, where w = wmr m,r , i.e., w is man
m's r-th choice. Being optimistic, we first presume that stability still prevails, and then we set out to find
possible sources of trouble. Where could they be hidden? There are two symmetrical possibilities:
1. There might be a women pw, preferred to w by m, who herself prefers m over her husband.