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Section I. Definitions of Time and Space Complexity
Definition of Big O: Let f (n) and g(n) be functions mapping non-negative integers to real numbers. We say that f (n) ∈ O(g(n)) if there is a real number c > 0 and a fixed integer n 0 ≥ 1 such that f (n) ≤ cg(n) for every integer n ≥ n 0. Definition of Big Ω: Let f (n) and g(n) be functions mapping non-negative integers to real numbers. We say that f (n) ∈ Ω(g(n)) if there is a real number c > 0 and a fixed integer n 0 ≥ 1 such that f (n) ≥ cg(n) for every integer n ≥ n 0. Definition of Big Θ: Let f (n) and g(n) be functions mapping non-negative integers to real numbers. We say that f (n) ∈ Θ(g(n)) if there are real numbers c, d > 0 and a fixed integer n 0 ≥ 1 such that dg(n) ≤ f (n) ≤ cg(n) for every integer n ≥ n 0. Note: For questions in this section, consider an algorithm that operates on n data items. It takes 1000 operations to initialize, regardless of the number of data items, then takes 3 n^2 operations to complete.
- (2 points) Which of the following equations correctly describes the runtime of this algorithm? A. f (n) = 1000 B. f (n) = n^2 C. f (n) = 3n^2 D. f (n) = n^2 + 1000 E. f (n) = 3n^2 + 1000
- (2 points) Which of the following equivalence classes would be valid to describe its Big O runtime? A. n B. n^2 C. n^3 D. A and B E. B and C
- (2 points) Which of the following equivalence classes would be best to describe its Big O runtime? A. n B. n^2 C. n^3 D. A and B E. B and C
- (2 points) Which of the following equivalence classes would be valid to describe its Big Ω runtime? A. n B. n^2 C. n^3 D. A and B E. B and C
- (2 points) Which of the following equivalence classes would be best to describe its Big Ω runtime? A. n B. n^2 C. n^3 D. A and B E. B and C
- (2 points) Which of the following equivalence classes would be valid to describe its Big Θ runtime? A. n B. n^2 C. n^3 D. A and B E. B and C
Section III. Applications of Time and Space Complexity
Consider the following pseudocode (noting that % is modulus division, which returns the remainder after integer division): Algorithm RecursiveFunction (a) // a is a non-negative integer if (a = 0) return 0 else return (a % 2) + (^10) * RecursiveFunction(a / 2) endif
- (2 points) What is the time complexity of the RecursiveFunction pseudocode shown above? A. Θ(a) B. Θ(log a) C. Θ(1) D. Θ(a/2) E. Θ(a%2)
- (2 points) What is the space complexity of the RecursiveFunction pseudocode shown above? A. Θ(a) B. Θ(log a) C. Θ(1) D. Θ(a/2) E. Θ(a%2)
Consider the following pseudocode: Algorithm IterativeFunction (a) // a is a non-negative integer b ← 0 while (a > 0) b ← 10 * b + (a % 2) a ← a / 2 endwhile return b
- (2 points) What is the time complexity of the IterativeFunction pseudocode shown above? A. Θ(a) B. Θ(log a) C. Θ(1) D. Θ(a/2) E. Θ(a%2)
- (2 points) What is the space complexity of the IterativeFunction pseudocode shown above? A. Θ(a) B. Θ(log a) C. Θ(1) D. Θ(a/2) E. Θ(a%2)
Consider the following pseudocode: Algorithm RecursiveFunction2 (A) // A is an array of n items (1 or more) if (A.size = 1) return A[0] else B ← A[1 : A.size-1] // creates new array B of n-1 items return max(A[0], RecursiveFunction2 (B)) endif
- (2 points) What is the time complexity of the RecursiveFunction2 pseudocode shown above?
A. Θ(n) B. Θ(log 2 n) C. Θ(1) D. Θ(n^2 ) E. Θ(n log 2 n)
- (2 points) What is the space complexity of the RecursiveFunction2 pseudocode shown above?
A. Θ(n) B. Θ(log 2 n) C. Θ(1) D. Θ(n^2 ) E. Θ(n log 2 n)
Consider the following pseudocode: Algorithm IterativeFunction2 (A) // A is an array of n items (1 or more) B ← A // creates new array B of n items m ← B[0] while (B.size > 1) B ← B[1 : B.size-1] // replaces B with array of one fewer items m ← max(m, B[0]) endwhile return m
- (2 points) What is the time complexity of the IterativeFunction2 pseudocode shown above?
A. Θ(n) B. Θ(log 2 n) C. Θ(1) D. Θ(n^2 ) E. Θ(n log 2 n)
- (2 points) What is the space complexity of the IterativeFunction2 pseudocode shown above?
A. Θ(n) B. Θ(log 2 n) C. Θ(1) D. Θ(n^2 ) E. Θ(n log 2 n)
- (2 points) Which of the following are true?
A. Bubble Sort ∈ Θ(n^2 ) B. Bubble Sort ∈ Θ(n log n) C. Bubble Sort ∈ Θ(n) D. All of the above E. None of the above
- (2 points) Which of the following are true?
A. Quick Sort ∈ O(n^2 ) B. Quick Sort ∈ O(n log n) C. Quick Sort ∈ O(n) D. B and C E. None of the above
- (2 points) Which of the following are true?
A. Quick Sort ∈ Ω(n^2 ) B. Quick Sort ∈ Ω(n log n) C. Quick Sort ∈ Ω(n) D. B and C E. None of the above
- (2 points) Which of the following are true?
A. Quick Sort ∈ Θ(n^2 ) B. Quick Sort ∈ Θ(n log n) C. Quick Sort ∈ Θ(n) D. B and C E. None of the above
- (2 points) Which of the following algorithms would be best to sort a randomly ordered array of data?
A. Straight Insertion Sort B. Bubble Sort C. Quick Sort D. All of the above E. None of the above
- (2 points) Which of the following algorithms would be best to sort a nearly sorted array of data?
A. Straight Insertion Sort B. Bubble Sort C. Quick Sort D. All of the above E. None of the above
- (2 points) Which of the following is an advantage of Quick Sort over Merge Sort?
A. Better average runtime equivalence class B. Stability with respect to previously sorted results C. Sorts within existing array D. All of the above E. None of the above
- (2 points) Which of the following is an advantage of Merge Sort over Quick Sort?
A. Better average runtime equivalence class B. Stability with respect to previously sorted results C. Sorts within existing array D. All of the above E. None of the above
Short Answer Question 1: Quicksort (30 points)
Algorithm Quicksort (A, left, right) if (left < right) pivotPoint ← b(left+right)/2c // note central pivot i ← left - 1 j ← right + 1 do do i ← i + 1 while (i < A.size) and (A[i] ≤ A[pivotPoint]) do j ← j - 1 while (j ≥ i) and (A[j] ≥ A[pivotPoint]) if (i < j) then swap (A[i], A[j]) while (i < j) if (i < pivotPoint) then j ← i swap (A[pivotPoint], A[j]) Quicksort (A, left, j-1) Quicksort (A, j+1, right) endif
Show the steps followed by the Quicksort algorithm given above in pseudocode when sorting the following array. Draw one figure for each call to Quicksort. You may omit calls where left ≮ right.
value 26 40 48 61 86 94 57 16 25 11
index 0 1 2 3 4 5 6 7 8 9
