What does it mean when we say that an algorithm X is asymptotically more efficient than Y?a)X will be a better choice for all inputsb)X will be a better choice for all inputs except small inputsc)X will be a better choice for all inputs except large inputsd)Y will be a better choice for small inputsCorrect answer is option 'B'. Can you explain this answer?

Question

Answer

In asymptotic analysis we consider growth of algorithm in terms of input size. An algorithm X is said to be asymptotically better than Y if X takes smaller time than y for all input sizes n larger than a value n0 where n0 > 0.

Answer

Explanation:

In asymptotic analysis, we consider growth of algorithm in terms of input size.

An algorithm X is said to be asymptotically better than Y if X takes smaller time than y for all input sizes n larger than a value n0 where n0 > 0.

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Answer

Asymptotic Efficiency of Algorithms:

- In computer science, the efficiency of an algorithm is measured by its time complexity and space complexity.
- The time complexity defines the amount of time required by an algorithm to run, while space complexity defines the amount of memory required by an algorithm.
- When we say that an algorithm X is asymptotically more efficient than Y, it means that X has a better time or space complexity compared to Y as the input size grows larger.

Option B Explanation:

- Option B states that X will be a better choice for all inputs except small inputs. This means that as the input size grows larger, the time or space complexity of X will be better than Y, but for small input sizes, Y may be a better choice.
- This is because for small input sizes, the difference in efficiency between X and Y may not be significant enough to make a noticeable impact, and Y may have other advantages such as simpler implementation or lower memory usage.
- However, as the input size grows larger, the efficiency of X will become increasingly superior to Y, making it a better choice overall.

Example:

- Suppose we have two algorithms X and Y for sorting a list of integers. X is a quicksort algorithm with a time complexity of O(n log n), while Y is a bubble sort algorithm with a time complexity of O(n^2).
- For small lists (e.g. less than 10 items), Y may be a better choice as it has a simpler implementation and may be faster for small input sizes.
- However, as the list size grows larger (e.g. 1000 items), X will become increasingly more efficient than Y, and will be a better choice overall.

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