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Kenneth Rosen Edition 7 Exercise 8.1 Question 31 (Page No. 512)
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Use the recurrence relation developed in Example $5$ to determine $C_{5},$ the number of ways to parenthesize the product of six numbers so as to determine the order of multiplication.
Check your result with the closed formula for $C_{5}$ mentioned in the solution of Example $5.$
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Kenneth Rosen Edition 7 Exercise 8.1 Question 57 (Page No. 512)
Dynamic programming can be used to develop an algorithm for solving the matrix-chain multiplication problem introduced in Section $3.3.$ This is the problem of determining how the product $A_{1}A_{2} \dots A_{n}$ can be computed ... algorithm from part $(D)$ has $O(n^{3})$ worst-case complexity in terms of multiplications of integers.
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Kenneth Rosen Edition 7 Exercise 8.1 Question 56 (Page No. 512)
In this question, we will develop a dynamic programming algorithm for finding the maximum sum of consecutive terms of a sequence of real numbers. That is, given a sequence of real numbers ... worst-case complexity in terms of the number of additions and comparisons of your algorithm from part $(C)$ is linear.
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Kenneth Rosen Edition 7 Exercise 8.1 Question 55 (Page No. 512)
For each part of question $54,$ use your algorithm from question $53$ to find the optimal schedule for talks so that the total number of attendees is maximized. $20, 10, 50, 30, 15, 25, 40.$ $100, 5, 10, 20, 25, 40, 30. $ $2, 3, 8, 5, 4, 7, 10. $ $10, 8, 7, 25, 20, 30, 5.$
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Kenneth Rosen Edition 7 Exercise 8.1 Question 54 (Page No. 512)
Use Algorithm $1$ to determine the maximum number of total attendees in the talks in Example $6$ if $w_{i},$ the number of attendees of talk $i, i = 1, 2,\dots, 7,$ is $20, 10, 50, 30, 15, 25, 40.$ $100, 5, 10, 20, 25, 40, 30. $ $2, 3, 8, 5, 4, 7, 10. $ $10, 8, 7, 25, 20, 30, 5.$
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