Determine ways to find the complexity of an algorithm? What is the relation between the time and space complexities of an algorithm? Justify your answer with an example.
Answers
Answer:
Time complexity of an algorithm quantifies the amount of time taken by an algorithm to run as a function of the length of the input. Similarly, Space complexity of an algorithm quantifies the amount of space or memory taken by an algorithm to run as a function of the length of the input.
The different ways to find the complexity of an algorithm are:
- The amount of time and/or space needed by an algorithm to process an input of a specific size is known as the computational complexity, or simply complexity, of an algorithm.
- Asymptotic Notations are used to calculate an algorithm's time complexity.
The most used Asymptotic Notations to find the time complexity of an algorithm is:
- Average case in Theta Notation: The function from above and below is enclosed in theta notation. It is used to examine the average case complexity of an algorithm because it represents the upper and lower bounds of the running time of an algorithm.
- Best case in Omega Notation: The lower bound of an algorithm's running time is shown in omega notation. As a result, it offers an algorithm's best case complexity. Omega Ω(f(n)) provides the minimal amount of time that the algorithm needs for any value of n.
- The worst case in Big-O Notation: Big-O notation is used to express an algorithm's maximum allowable running time. As a result, it provides an algorithm's worst-case complexity. As we are constantly interested in the worst-case scenario, it is frequently used to analyze an algorithm.
The term "space complexity" refers to how much memory the program requires to run completely. The only factor affecting an algorithm's time complexity is the size of the input; as a result, it is a function of the input size 'n'. Time complexity is the term used to describe how long it takes an algorithm to run through to completion.
For example:
- Let's say we are tasked with determining whether an integer x is present in the array A.
- The entire array A can be traversed as a simple solution to this issue to see if any element is equal to x. Each operation should take c clock cycles.
- The value of x determines how many lines of code are run. Analyzing the algorithm, we will primarily take the worst-case scenario into account, which is the case when x is absent from array A.
- In the worst case, where n is the length of the array A, the if condition will execute n times. Therefore, in the worst case, the return statement will take, c seconds and the if condition will take n×c seconds to complete.
- Hence, total running time will be (n×c + c)
For more information on Time complexities and Space complexities, refer to these answers:
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