[Solved] CMPT280Assignment 2-timing analysis and ADT specification

The questions on this assignment are about timing analysis and ADT specification. There is no programming on this assignment.

Question 1 ():

For each of the following functions, give the tightest upper bound chosen from among the usual simple functions listed in Section 3.5 of the course readings. Answers should be expressed in big-O notation.

⁴ log₂₈₀ n + ²₄₂ⁿ (a) (1 point) f₁(n) = n log₂ n + n

• (1 point) f₃(n) = 4n⁴ + n² log n² + log₂(2ⁿ)
• (1 point) f₂(n) = 4n⁷ + 29n log₂ n + 280

Question 2 ():

Suppose the exact time required for an algorithm A in both the best and worst cases is given by the function

T_A n

• (2 points) For each of the following statements, indicate whether the statement istrue or false.
  1. Algorithm A is O(log n)
  2. Algoirthm A is O(n²)
  3. Algoirthm A is O(n³)
  4. Algoirthm A is O(2ⁿ)
• (1 point) What is the time complexity of algorithm A in big- notation.

Question 3 ():

If possible, simplify the following expressions. Hint: See slide 11 of topic 4 of the lecture slides!

• (1 point) O(n²)+ O(log n)+ O(n log n)
• (1 point) O(2ⁿ) O(n²)
• (1 point) 42O(n log n)+ 18O(n³)
• (1 point) O(n² log₂ n²)+ O(m) (yes, thats an m, not a typo; note that m is independent of n)

Question 4 ():

Consider the following Java code fragment:

• (3 points) Use the statement counting approach to determine the exact number of statements that are executed when we run this code fragment as a function of n. Show all of your calculations.
• (1 point) Express the answer you obtained in part (a) in big- notation.

Question 5 ():

Consider the following pseudocode:

• (6 points) Use the statement counting approach to determine the exact number of statements that are executed by this pseudocode as a function of n. Show all of your calculations..
• (1 point) Express the answer you obtained in part a) in big- notation.

Question 6 (3 points):

Using the active operation approach, determine the time complexity of the pseudocode in question 5. Show all your work and express your final answer in Big- notation.

Question 7 (6 points):

Consider the following pseudocode.

Using the active operation approach to timing analysis determine the time complexity of this pseudocode in the worst case. Assume that the list of arrays contains n arrays and that each array has exactly m items in it. Show all your work and express your final answer in Big-O notation.

Question 8 (17 points):

A priority queue is a queue where a numeric priority is associated with each element. Access to elements that have been inserted into the queue is limited to inspection and removal of the elements with smallest and largest priority only. A priority queue may have multiple items that are of equal priority.

Give the ADT specification for a bounded priority queue using the specification method described in Topic 7 of the lecture notes. By bounded, it is meant that the priority queue has a maximum capacity specified when it is created, and it can never contain more than that number of items.

Your specification must specify the following operations:

newPriorityQueue: make a new queue insert: inserts an element with a certain priority isEmpty: test if the queue is empty isFull: test if the queue is full maxItem: obtain the item in the queue with the highest priority minItem: obtain the item in the queue with the lowest priority deleteMax: remove from the queue the item with the highest priority deleteAllMax: remove from the queue all items that are tied for the highest priority deleteMin: remove from the queue the item with the lowest priority

frequency: obtain the number of times a certain item occurs in the queue (with any priority)