电子科技大学研究生算法设计与分析拟考题及答案评分细则 (2)

一、Please answer T or F for each of the following statements to indicate whether the

statement is true or false

1. An algorithm is an instance, or concrete representation, for a computer program

in some programming language. ( F )

2. The following problem is a Decision Problem: What is the value of a best

possible solution? ( F )

3. The dynamic programming method can not solve a problem in polynomial time.

( F)

4. Assume that there is a polynomial reduction from problem A to problem B. If

we can prove that A is NP-hard, then we know that B is NP-hard. ( F )

5. If one can give a polynomial-time algorithm for a problem in NP, then all the

problems NP can be solved in polynomial time. ( F )

6. In an undirected graph, the minimum cut between any two vertices a and b is

unique. ( F)

7. Linear programming can be solved in polynomial time, but integer linear

programming can not be solved in polynomial time. ( T )

8. We can solve the maximum independent set problem in a graph with at most

100 vertices in polynomial time. ( T ) 结论

9. If an algorithm solves a problem of size n by dividing it into two subproblems of

size n/2, recursively solving each subproblems, and then combine the solutions in linear time. Then the algorithm runs in O(n log n) time. ( T )

10. Neural Computation, Fuzzy Computation and Evolution Computing are the

three research fields of Computational Intelligence. ( T )

二、Given the following seven functions f1(n) = n5+ 10n4, f2(n) = n2+ 3n

, f3(n) =

f4(n) = log n + (2log n)3, f5(n) = 2n+n!+ 5e n, f6(n) = 3log(2n) + 5log n, f7(n) = 2n log n+log n n. Please answer the questions:

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(a) Give the tight asymptotic growth rate (asymptotic expression with θ) to each

of them; (7分)

(b) Arrange them in ascending order of asymptotic growth rate。(3分) 参考答案和评分标准：

a)

(1) n5 + 10n4 = θ (n5) （1分，非最简表达式或写成O或Ω不符合题意，不给分）

(2) n2 +3n = θ (3n) （1分，标准同上）

(3) 210000

= θ (n0.75) （1分，标准同上）

(4) log n + (2log n)3 =θ ( (log n)3) （1分，标准同上）

(5) 2n+n!+ 5e n =θ (n!) （1分，标准同上）

(6) 3log2n + 5log n =θ (n) （1分，标准同上）

(7) 2n log n+log n n. =θ (n n) （1分，标准同上）

b)

f4 f3 f6 f1 f2 f5 f7 (3分，每个错误位置扣0.5分，扣完为止）

三、Please answer the following questions:

(a)。

四、In the interval scheduling problem, we are given n jobs each of which has a

starting time s and a finishing time f, and the goal is to find a maximum set of mutually compatible jobs (two jobs are compatible if they don’t overlap). Please answer the following questions:

(a) Design a greedy algorithm for the interval scheduling problem and prove the

correctness of it.

(b) Assume that we are given 8 jobs with starting time and finishing time (s, t)

being (0,2), (1,3), (8,9), (3,7), (7,8), (2,4）,(6,9), (4,5). Use your algorithm to find

a solution to this instance.

参考答案及评分标准：

a）

将所有工作（jobs）按其完成时间的先后进行排序；

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