Posted: May 24th, 2021
1.chose the true statments. (Your score for this questions will be proportional to the number of statements you mark correctly plus the number you correctly leave unmarked minus the number you should have marked but didn’t minus the number you shouldn’t have marked but did.) (a) lg n n = O(lg n!) (b) n (n+1) mod 2 = Ω(n n mod 2) for n ∈ N (c) n (3n) mod 2 = O(n n mod 2) for n ∈ N (d) If T(n) = 7T(n/3) + n 4 then T(n) = Ω(n 4/ log n). (e) If T(n) = 8T(n/2) + 8n 3 then T(n) = Θ(n 3 log n). 2. Explain how you can sort a sequence of n integers from a range of size n lg lg n in O(n log log n) time (assuming each integer fits in a constant number of machine words). 3. An in-place algorithm uses a constant number of machine words of memory on top of the memory initially occupied by its input and eventually occupied by its output. Give code or pseudo-code for an in-place version of QuickSort. Continued on Next Page! 1 4. You’ve probably seen in previous courses how to build a min-heap on n elements in O(n) time and how to extract the minimum value from one in O(log n) time. Do you think we can easily extract the minimum value in o(log n) time while still leaving a heap on the remaining elements? Why or why not? 5. Suppose you have an algorithm that given a sequence of n integers that can be partitioned into d non-decreasing subsequences but not fewer does so in O(n log d) time. Explain how you can also sort such a sequence in O(n log d) time. Requirements: 5 questions | .doc file
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