Algorithms and Data Structures

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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.
N.Wirth. Algorithms and Data Structures. Oberon version 127 3.3. Deduce the recursion scheme of Fig. 3.10 which is a superposition of the four curves W 1 , W 2 , W 3 , W 4 . The structure is similar to that of the Sierpinski curves in Fig. 3.6. From the recursion pattern, derive a recursive program that draws these curves. Fig. 3.10. Curves W 1 – W 4 . 3.4. Only 12 of the 92 solutions computed by the Eight Queens algorithm are essentially different. The other ones can be derived by reflections about axes or the center point. Devise a program that determines the 12 principal solutions. Note that, for example, the search in column 1 may be restricted to positions 1- 4. 3.5. Change the Stable Marriage program so that it determines the optimal solution (male or female). It therefore becomes a branch and bound program of the type represented by the program Selection. 3.6. A certain railway company serves n stations S 0 , ... , S n-1 . It intends to improve its customer information service by computerized information terminals. A customer types in his departure station SA and his destination SD, and he is supposed to be (immediately) given the schedule of the train connections with minimum total time of the journey. Devise a program to compute the desired information. Assume that the timetable (which is your data bank) is provided in a suitable data structure containing departure (= arrival) times of all available trains. Naturally, not all stations are connected by direct lines (see also Exercise 1.6). 3.7. The Ackermann Function A is defined for all non-negative integer arguments mand n as follows: A(0, n) = n + 1 A(m, 0) = A(m-1, 1) (m > 0) A(m, n) = A(m-1, A(m, n-1)) (m, n > 0) Design a program that computes A(m,n) without the use of recursion. As a guideline, use the procedure NonRecursiveQuickSort from sec. 2.3.3. Devise a set of rules for the transformation of recursive into iterative programs in general. References [3.1] D.G. McVitie and L.B. Wilson. The Stable Marriage Problem. Comm. ACM, 14, No. 7 (1971),
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