20. Assume P ≠NP. Which of the following is true?
(A)
(B)NP-complete = NP
(C) NP-hard -NP
(D) P=NP-complete

21. Let S be an NP-complete problem and Q and R be two other problems not known to be in NP. Q is polynomial-time reducible to S and S is polynomial-time reducible to R. Which of the following statements is true? (A)R is NP-complete. (B) Ris NP-hard. (C) Ois NP-complete. (D) Q is NP-hard.

22. Suppose you can take any arbitrary portion of cach item. What is the highest total value you can get by filling your knapsack? (A) 184 (B) 100(C)90 (D) 84

23. Suppose you can only choose to take or leave cach entire item. Let V[i]U] denote the highest total value you can obtain with items 1~i and a knapsack of weight capacity j. Which of the following recurrence relations is correct? (A) V[i][j] = max(V[i - 1][j]，V[i - 1][j -1] +v[i]) (B) V[i][j]=max(V[i - 1][j]，V[i - 1][j -w[i])+v[i]) (C) V[i][j] = max(V[i][j - 1]，V[i -1] [j-1] +v[i]) (D)V[i][j]= max(V[i][j-w[i]]，V[i -1][j-w[i]]+  v[1])

24. Suppose you can only choose to take or leave each entire item. What is the highest total value you can get by filling your knapsack? (A) 184 (B)90 (C) 84 (D) 80 

25. The recurrence relation in the previous problem solves the 0-1 knapsack problem in a dynamic programming manner by filling the table of V[i][j]. Which of the following statements is correct about the running time of this algorithm? (A)It is a linear-t -time algorithm. (B) It is a polynomial-time algorithm, but not a lincar-time algorithr. (C) It is an exponential-time algorithm, but not a polynomial-time algorithr. (D) Nobody knows yet whether or not it is a polynomial-time algorithm.

26. Consider the following five recursive functions: (A) (B) (C) (D) (E)Which recursion yields the highest asymptotic complexity for T(n)?

27. For an AVL tree with 2021 nodes, let x be the minimum possible height, y be the maximum height, then (y-x) mod 5 is (A)0 (B)1(C)2(D)3(E)4

28. For a 2-3-4 tree of height 5, let x be the minimum possible size, y be the maximum possible size, then (xy) mod 5 is (A) 0 (B) 1 (C) 2 (D) 3 (E)4

29. For a binary heap of height 2021, let x be the minimumn possible size, y be the maximum possible size, then (3xty) mod 5 (A) 0 (B) 1 (C)2 (D) (E) 4

30. Consider a binary heap that results from successively inserting keys 6, 3, 7, 8, 5, 1, 2, 4, 9 into an initially empty heap. Then the depth of key 5 is (A)0 (B)1 (C) 2 (D) 3(E) None of above

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