Select the correct answerWhat is the average running time of a quick sort algorithm?OptionsO(N^2)O(N log N)O(log N)O(N)

Break Down the Problem

1. Identify the algorithm in question: Quick sort.
2. Understand the nature of the quick sort algorithm and its average case running time.

Relevant Concepts

1. The average case time complexity for the quick sort algorithm is crucial in algorithm analysis.
2. The general complexity levels to compare are:
   • $O(N^2)$ for worst case,
   • $O(N \log N)$ for average case,
   • $O(log N)$ for logarithmic operations,
   • $O(N)$ for linear operations.

Analysis and Detail

1. Quick sort uses a divide-and-conquer approach:

   • On average, it partitions the data into two halves, leading to logarithmic depth.
   • Each partitioning step requires linear time to complete.

2. The average case running time can be formulated as: $T(n) = 2T\left(\frac{n}{2}\right) + O(n)$ This results from the recursive nature of sorting along with linear partitioning time.

3. By applying the Master theorem or looking at related recurrence relations, we can conclude: $T(n) = O(N \log N)$

Verify and Summarize

1. The time complexity of quick sort varies based on pivot choice, but in average scenarios, it often runs efficiently in comparison to other sorting algorithms.
2. In practical implementations, its average case performance is consistently $O(N \log N)$.

Final Answer

The average running time of a quick sort algorithm is $O(N \log N)$.