Algorithm Design and Structure Chapter 3 Dynamic Planning Assignments

True or False

1-2
The root node of the optimal binary search tree must store the keyword with the highest search probability.

T F

1-3 When
using dynamic programming instead of recursive methods to solve problems, the key is to save the calculation results of the sub-problems so that each different sub-problem only needs to be calculated once. The solution of the subproblem can be stored in an array or hash table.

T F

Multiple choice

2-1
In dynamic programming, we need to derive the recurrence relationship between the solution of one sub-problem and the solutions of other sub-problems. To convert this relationship into a bottom-up dynamic programming algorithm, we need to fill in the sub-problem solution form in the correct order, so that when solving any sub-problem, all the sub-problems it needs have been solved. Which one of the following relations cannot be calculated?

AA (i, j) = min (A (i − 1, j), A (i, j − 1), A (i − 1, j − 1))
BA (i, j) = F (A (min {i, j} −1, min {i, j} −1), A (max {i, j} −1, max {i, j} −1))
CA (i, j) = F (A ( i, j − 1), A (i − 1, j − 1), A (i − 1, j + 1))
D.A(i,j)=F(A(i−2,j−2),A(i+2,j+2))

Draw a matrix and you will know it, and it can only be derived from the previously derived elements, namely the upper left and right elements;

2-2
Given the recurrence equation f ​i,j,k​​ =f ​i,j+1,k​​ + min​0≤l≤k​​ (f ​i−1,j,l​ +w​j,l​​ }. To solve this equation in a loop, we must not use which of the following methods to fill in the table?

A.for k in 0 to n: for i in 0 to n: for j in n to 0
B.for i in 0 to n: for j in 0 to n: for k in 0 to n
C.for i in 0 to n: for j in n to 0: for k in n to 0
D.for i in 0 to n: for j in n to 0: for k in 0 to n

2-3
Log cutting problem: Given a log with a length of N meters; there is also a segmented price list, giving the price P​L​​ corresponding to the length L=1,2,...,M. You are required to find the maximum profit R​N that can be obtained by appropriately cutting logs and selling them in sections. For example, according to the price list given below, if you want to sell a log of 8 meters long, the optimal solution is to cut it into two segments of 2 meters and 6 meters, so that you can get the maximum profit R​8​​ =P​ 2+P6=5+17=22. If you want to sell a 3 meter long log, the best solution is not to cut it at all and sell it directly.

Length L	1	2	3	4	5	6	7	8	9	10
Price PL	1	5	8	9	10	17	17	20	23	28

Which of the following statements is wrong?

A. This problem can be solved by dynamic programming
B. If N≤M, then R​N​ =max{P​N​​ ,max​1≤i<N​ {R​i​​ +R​N− i​​ }}
C. If N>M, then RN​​ =max1≤i<N​​ {R​i​​ +R​N−M}
D. The time complexity of the algorithm is O(N​2​​)