Order Notation and the Mathematics for Analysis of Algorithms
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1 Elementary Data Structures and Algorithms Order Notation and the Mathematics for Analysis of Algorithms Name: Code: Always choose the best or most general answer, unless otherwise instructed. Concept: mathematics notation 1. log 2 n is equal to: (A log 2 n log 2 10 (B log 10 n log 2 10 (C log 2 n log 10 2 (D log 10 n log log(nm is equal to: (A n log m (B m log n (C (log n m (D log n + log m 3. The number of permutations of a list of n items is: (A 2 n (B n log n (C log n (D n (E n! Concept: relative growth rates 4. What is the correct ordering of growth rates for the following functions: f(n = n(log n 2 g(n = n log 2 n h(n = n log(log n (A f > h > g (B h > f > g (C h > g > f (D g > f > h (E g > h > f (F f > g > h Concept: order notation 5. T or F: All algorithms are θ(1. 6. T or F: There exist algorithms that are O(1. Concept: comparing algorithms using order notation The phrase by a stopwatch means the actual amount of time needed for the algorithm to run to completion, as measured by a stopwatch. 7. T or F: If f = ω(g, then algorithm f always runs slower than g (by a stopwatch, in all cases. 8. T or F: If f = ω(g and the input causes worst-case behaviors, then algorithm f always runs slower than g (by a stopwatch, regardless of input size.
2 9. T or F: If f = ω(g and the input causes worst-case behaviors, then algorithm f always runs slower than g (by a stopwatch, above a certain input size. 10. T or F: If f = ω(g, then algorithm f always runs slower than g (by a stopwatch, above a certain input size. 11. T or F: If f = θ(g, then algorithm f always takes the same time as g (by a stopwatch, in all cases. 12. T or F: If f = θ(g and the input causes worst-case behaviors, then algorithm f always takes the same time as g (by a stopwatch, regardless of input size. 13. T or F: If f = θ(g and the input causes worst-case behaviors, then algorithm f always takes the same time as g (by a stopwatch, above a certain input size. 14. T or F: If f = θ(g, then algorithm f always takes the same time as g (by a stopwatch, above a certain input size. 15. T or F: If f = θ(g, then algorithm f always takes the same time as g (within a constant factor, in all cases. 16. T or F: If f = θ(g and the input causes worst-case behaviors, then algorithm f always takes the same time as g (within a constant factor, regardless of input size. 17. T or F: If f = θ(g and the input causes worst-case behaviors, then algorithm f always takes the same time as g (within a constant factor, above a certain input size. 18. T or F: If f = θ(g, then algorithm f always takes the same time as g (within a constant factor, above a certain input size. 19. T or F: If f = ω(g, then f and g can be the same algorithm. 20. T or F: If f = Ω(g, then f and g can be the same algorithm. 21. T or F: If f = o(g, then f and g can be the same algorithm. 22. T or F: If f = O(g, then f and g can be the same algorithm. 23. T or F: Suppose algorithm f = θ(g. f and g can be the same algorithm. 24. T or F: If f = Ω(g and f = O(g, then f = Θ(g. 25. T or F: If f = Ω(g and f = O(g, then f and g must be the same algorithm. Concept: analyzing code In the pseudocode, the lower limit of a for loop is inclusive, while the upper limit is exclusive. The additive step, if not specified, is one. Tracing recursive functions When asked about the number of recursive calls, do not include the original call. 26. How many recursive calls are made if n = 5? Assume the initial value of i is zero. f(i+1,n; (A 4 (B the function recurs infinitely (C 3 (D 6 (E 5 (F none of the other answers are correct 2
3 27. How many recursive calls are made if n = 64. Assume the initial value of i is one. f(i*sqrt(n,n; (A none of the other answers are correct (B 4 (C 9 (D the function recurs infinitely (E 7 (F 2 (G 1 (H 8 Time complexity, recursive functions, single recursion 28. What is the time complexity of this function? Assume the initial value of i is zero. f(i+1,n; (A θ(n (C θ( n (D θ(n 2 (E θ(log n (F θ(log n 29. What is the time complexity of this function? Assume positive, integral input and integer division. function f(n if (n > 0 for (var i from 0 until n println(n; f(n/2; (A θ(log n (C θ(n log n (D θ(n n (E θ(n (F θ(n 2 3
4 Time complexity, recursive functions, double recursion Space complexity, recursive functions, single recursion 30. What is the space complexity of this function? Assume the initial value of i is zero. f(i+2,n; (A θ( n (B θ(n log n (C θ(n (D θ(1 (E θ(n 2 (F θ(log n 31. What is the space complexity of this function? Assume the initial value of i is one. f(i*sqrt(n,n; (A θ(1 (B θ(n n (C θ( n (D θ(n (E θ(n n (F θ(n n n Space complexity, recursive functions, double recursion Time complexity, iterative loops, single loops 32. What is the time complexity of this code fragment? (A θ((log n 2 (B θ(n n (C θ(n 2 (D θ(n log n (E θ(log n 33. What is the time complexity of this code fragment? i = i * sqrt(n; (A θ(n n (C θ(n n n (D θ( n (E θ(n (F θ(n n 4
5 Time complexity, iterative loops, double loops 34. What is the time complexity of this code fragment? for (j from 0 until i by 1 println(i,j; (A θ(n (C θ(log 2 n (D θ(n 2 (E θ(n n (F θ(n log n 35. What is the time complexity of this code fragment? for (j from 0 until n by 2 println(j; (A θ(n n (B θ(n (C θ(1 (D θ(n log n (E θ(log 2 n (F θ(n 2 Space complexity, iterative loops, single loops 36. What is the space complexity of this code fragment? for (i from 0 until n by 1 (A θ(n 2 (B θ(n (C θ(1 (D θ( n (E θ(n log n (F θ(log n 37. What is the space complexity of this code fragment? (A θ(log n (C θ(n 2 (D θ( n (E θ(n log n 5
6 Space complexity, iterative loops, double loops 38. What is the space complexity of this code fragment? for (i from 0 until n by 1 for (j from 0 until n by 1 println(j; (A θ(n (B θ(n 2 (C θ(n log n (D θ(log n (E θ(1 (F θ( n 39. What is the space complexity of this code fragment? for (i from 0 until n by 1 for (j from 0 until i by 1 println(i,j; (A θ(log n (B θ( n (C θ(n 2 (D θ(n log n (E θ(n (F θ(1 Concept: analysis of classic, simple algorithms 40. Which of the following describes the classic recursive fibonacci s space complexity? (A θ(1 (B θ( n n (C θ(n n (D θ( Φ n (E θ( n 41. Which of the following describes iterative fibonacci s space complexity? (A θ(1 (B θ( Φ n (C θ( n (D θ(n n (E θ( n n 6
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