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Exam1ReviewQsSpr07 - Practice Questions 1 Which of the...

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Practice Questions 1. Which of the following are true? (a) 3n = O(n 2 ) (b) 3n 2 = O(nlgn) (c) nlgn = Θ (20000nlgn) (d) 2 n = Ω (n 1000 ) Answers: (a), (c), (d) (a) n 2 clearly grows larger than 3n as n gets large. (c) 20000 is a constant in front of nlgn (d) All exponential functions grow faster than all polynomial functions. (b) is false because 3n 2 grows faster than nlgn. 2. What is the run-time of the following segment of code in terms of n? Give an upper bound and justify it. Only a tight upper bound will be accepted as a correct answer. int i=1; while (i <= n) { int j = i; while (j > 0) j = j/2; i++; } The outer loop runs n times. The inner loop will run log i times at most since there is repeated halving. Since i never exceeds n, it is safe to say the inner loop runs at most log n times. Hence an upper bound on the run time is O(nlgn).
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3. Determine the following sum: - = 1 0 3 n i i This is a geometric sum with a first term of 1, with a common ratio of 3, with n terms. Here's the sum: 2 1 3 3 1 3 1 3 1 0 - = - - = - = n n n i i .
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