5.5 Learning Objective: To demonstrate that the student understands the definition of Big O. To demonstrate that the student understands how to derive the function f(n) which computes how many times the key operation of an algo- rithm is performed as a function of the size of the problem. Instructions: This exercise is graded, so include your solution in your word processing document. Problem: Continuing with the previous exercise, derive a function f(n) which equates to the number of times the key operation is performed as a function of n, where n is the size of pList. State the worst case time complexity of split() in Big O notation. You do not need to formally prove it but explain your answer.

See explaination

Explanation:

Anytime we want to compute time complexity of an algorithm with input size n denoted as T(n).

And then check growth rate of that algorithm with any large number of input n.

We have to first obtain f(n) of that algorithm .

f(n) is total execution time of algorithm in terms of n.

we want write this function with growth rate notation.

we know there are basic three notations

1. big oh(O) notation

2.theta(Θ) notation

3. big omega(Ω) notation

we want explain big oh(O) notation

Here is the formal mathematical definition of Big O.

also called asymptotic upper bound

Let T(n) and f(n) are two positive functions. We write T(n) ∊ O(f(n)), if there are positive constants c and n₀ such that

T(n) ≤ c·f(n) for all n ≥ n₀.

This graph shows a situation where all of the conditions in the definition are exist. (see attachment for graph)

c.fin) T(n) cost no

we can say

T(n) ∊ O(f(n)) means that growth rate of T(n) is not faster than f(n).

example

T(n) = n -1. Using Big O notation this can be written as T(n) ∊ O(n).

let c = 1 and n₀ = 1,

then T(n) = n - 1 ≤ 1·n when n ≥ 1.