Algorithms and Data Structures

N.Wirth. Algorithms and Data Structures. Oberon version 123
husbands, and the sums of the ranks of all men in the preference lists of their wives. These are the values
rm = Sm: 0 ≤ m < n: rmw m,x[m]
rw = Sm: 0 ≤ m < n: rwm x[m],m
x0 x1 x2 x3 x4 x5 x6 x7 rm rw c
0 6 3 2 7 0 4 1 5 8 24 21
1 1 3 2 7 0 4 6 5 14 19 449
2 1 3 2 0 6 4 7 5 23 12 59
3 5 3 2 7 0 4 6 1 18 14 62
4 5 3 2 0 6 4 7 1 27 7 47
5 5 2 3 7 0 4 6 1 21 12 143
6 5 2 3 0 6 4 7 1 30 5 47
7 2 5 3 7 0 4 6 1 26 10 758
8 2 5 3 0 6 4 7 1 35 3 34
c = number of evaluations of stability.
Solution 0 = male optimal solution; solution 8 = female optimal solution.
Table 3.4. Result of the Stable Marriage Problem.
The solution with the least value rm is called the male-optimal stable solution; the one with the smallest
rw is the female-optimal stable solution. It lies in the nature of the chosen search strategy that good
solutions from the men's point of view are generated first and the good solutions from the women's
perspective appear toward the end. In this sense, the algorithm is based toward the male population. This
can quickly be changed by systematically interchanging the role of men and women, i.e., by merely
interchanging mwr with wmr and interchanging rmw with rwm.
We refrain from extending this program further and leave the incorporation of a search for an optimal
solution to the next and last example of a backtracking algorithm.
3.7 The Optimal Selection Problem
The last example of a backtracking algorithm is a logical extension of the previous two examples
represented by the general schema. First we were using the principle of backtracking to find a single
solution to a given problem. This was exemplified by the knight's tour and the eight queens. Then we
tackled the goal of finding all solutions to a given problem; the examples were those of the eight queens
and the stable marriages. Now we wish to find an optimal solution.
To this end, it is necessary to generate all possible solutions, and in the course of generating them to
retain the one that is optimal in some specific sense. Assuming that optimality is defined in terms of some
positive valued function f(s), the algorithm is derived from the general schema of Try by replacing the
statement print solution by the statement
IF f(solution) > f(optimum) THEN optimum := solution END
The variable optimum records the best solution so far encountered. Naturally, it has to be properly
initialized; morever, it is customary to record to value f(optimum) by another variable in order to avoid its
frequent recomputation.
An example of the general problem of finding an optimal solution to a given problem follows: We choose
the important and frequently encountered problem of finding an optimal selection out of a given set of
objects subject to constraints. Selections that constitute acceptable solutions are gradually built up by
investigating individual objects from the base set. A procedure Try describes the process of investigating
the suitability of one individual object, and it is called recursively (to investigate the next object) until all