CS 350 Midterm Algorithms and Complexity

Transcription

1 It is recommended that you read through the exam before you begin. Answer all questions in the space provided. Name: Answer whether the following statements are true or false and briefly explain your answer in the space provided. 1. [TRUE / FALSE] If an in-place sorting algorithm is given a sorted-array, it will always output [5 pts] an unchanged array. 2. [TRUE / FALSE] If f(n) O(g(n)) and g(n) O(h(n)), then h(n) Ω(f(n)). [5 pts] CS 350 Midterm Page 1 of 8

2 3. [TRUE / FALSE] There exists a comparison sort of 5 numbers that uses at most 6 comparisons [5 pts] in the worst case. 4. [TRUE / FALSE] If f(n) O(g(n)) and g(n) O(f(n)), then f(n) = g(n). [5 pts] 5. [TRUE / FALSE] If f(n) O(g(n)) and g(n) Ω(f(n)), then f(n) Θ(g(n)). [5 pts] CS 350 Midterm Page 2 of 8

3 6. Prove that f(n) = n(100n + 200n 3 ) + n 3 is an element of O(n 4 ). 7. Prove that f(n) = n lg(n 4 ) + n is an element of O(n log n). CS 350 Midterm Page 3 of 8

4 8. Describe the circumstances in which Quicksort exhibits worst-case performance. 9. Answer the following questions regarding insertion sort. (a) Describe an input for this algorithm for which it will exhibit worst case performance. [3 pts] (b) Describe an input for this algorithm for which it will exhibit best case performance. [3 pts] CS 350 Midterm Page 4 of 8

5 10. The following algorithm finds the sum of all elements in an array using a divide and conquer strategy. func sumarray(a, low, high) if low > high return 0 if low = high return A[low] mid <- (high + low) / 2 leftsum <- sumarray(a, low, mid) rightsum <- sumarray(a, mid+1, high) return leftsum + rightsum (a) Give a recurrence relation for the worst case performance of this algorithm. (b) Find the asymptotic complexity of your recurrence relation. CS 350 Midterm Page 5 of 8

6 11. The following recursive algorithm finds the k-th smallest element. It is assumed that the list contains at least k elements. (Hint: This algorithm is similar to quicksort) func select(list, left, right, k) if left = right return list[left] pivotindex <-... // select a pivotindex between left and right pivotindex <- partition(list, left, right, pivotindex) // The pivot is now in its final sorted position if k = pivotindex return list[k] else if k < pivotindex return select(list, left, pivotindex - 1, k) else return select(list, pivotindex + 1, right, k) (a) What is the best case complexity for this algorithm? would result in that performance. Describe the circumstances that [3 pts] (b) Give a recurrence relation for the worst case performance of this algorithm. asymptotic complexity of your recurrence relation. Find the [8 pts] CS 350 Midterm Page 6 of 8

7 12. Find the asymptotic complexity of the following recurrence relation. ( n ) T (n) = 8T + n 2 T (1) = Find a closed form for the following recurrence relation. T (n) = T (n 2) + n T (1) = 0 T (0) = 0 CS 350 Midterm Page 7 of 8

8 14. The following algorithm is a brute-force solution for a problem called MaxSubarraySum. It finds the contiguous subarray with the largest sum and returns that sum. For example: if the input array is [ 6, 1, 6, 1, 4, 1, 5, 3] the algorithm would output 7. func MaxSubArraySum(list) maxsum <- -infinity for i < list.size() - 1 runningsum <- 0 for j <- i... list.size() - 1 runningsum <- runningsum + list[j] if runningsum > maxsum maxsum <- runningsum return maxsum (a) What parameter should be used to measure the size of the input? [2 pts] (b) Give a summation that represents the number additions that are performed. [7 pts] (c) Find a closed form for your summation and give its order of growth. [7 pts] CS 350 Midterm Page 8 of 8