What evidence is there to support the claim that the optimal scenario complexity of quicksort is O(n log n)?

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There are several pieces of evidence to support the claim that the optimal scenario complexity of quicksort is O(n log n):

1. Mathematical analysis: The average case time complexity of quicksort has been proven to be O(n log n) through mathematical analysis. This analysis takes into account the probability of each partition, and it has been shown that the expected number of comparisons required is proportional to n log n.

2. Empirical evidence: There have been numerous experiments and benchmarks conducted on quicksort that show its time complexity is O(n log n) in practice. These experiments have been conducted on a wide range of datasets and have consistently shown that quicksort performs well on average.

3. Comparison to other sorting algorithms: Quicksort is known to be one of the fastest general-purpose sorting algorithms. When compared to other sorting algorithms such as merge sort or heap sort, quicksort consistently performs better on average. Merge sort has a complexity of O(n log n), while heap sort has a complexity of O(n log n) in the worst-case scenario.

4. Practical usage: Quicksort is widely used in practice because of its fast average-case performance. It is used in popular programming languages such as C++, Java, and Python, and is often the default implementation for sorting algorithms in these languages.

Overall, the evidence suggests that quicksort has an optimal scenario complexity of O(n log n), making it one of the fastest sorting algorithms available.