Algorithms and Data Structures

N.Wirth. Algorithms and Data Structures. Oberon version 126
Weight: 10 11 12 13 14 15 16 17 18 19
Value: 18 20 17 19 25 21 27 23 25 24
limw ↓
maxv
10 * 18
20 * 27
30 * * 52
40 * * * 70
50 * * * * 84
60 * * * * * 99
70 * * * * * 115
80 * * * * * * 130
90 * * * * * * 139
100 * * * * * * * 157
110 * * * * * * * * 172
120 * * * * * * * * 183
Table 3.5. Sample Output from Optimal Selection Program.
The asterisks mark the objects that form the optimal sets opts for the total weight limits ranging from 10
to 120.
This backtracking scheme with a limitation factor curtailing the growth of the potential search tree is also
known as branch and bound algorithm.
Exercises
3.1. (Towers of Hanoi). Given are three rods and n disks of different sizes. The disks can be stacked up
on the rods, thereby forming towers. Let the n disks initially be placed on rod A in the order of decreasing
size, as shown in Fig. 3.9 for n = 3. The task is to move the n disks from rod A to rod C such that they are
ordered in the original way. This has to be achieved under the constraints that
1. In each step exactly one disk is moved from one rod to another rod.
2. A disk may never be placed on top of a smaller disk.
3. Rod B may be used as an auxiliary store.
Find an algorithm that performs this task. Note that a tower may conveniently be considered as consisting
of the single disk at the top, and the tower consisting of the remaining disks. Describe the algorithm as a
recursive program.
0
1
2
Fig. 3.9. The towers of Hanoi
A B C
3.2. Write a procedure that generates all n! permutations of n elements a 0 , ..., a n-1 in situ, i.e., without
the aid of another array. Upon generating the next permutation, a parametric procedure Q is to be called
which may, for instance, output the generated permutation.
Hint: Consider the task of generating all permutations of the elements a 0 , ..., a m-1 as consisting of the
m subtasks of generating all permutations of a 0 , ..., a m-2 followed by a m-1 , where in the i-th subtask the
two elements a i and a m-1 had initially been interchanged.