How Many Comparisons Are in a Merge Sort?

When discussing the efficiency of sorting algorithms, one important factor to consider is the number of comparisons made during the sorting process. In the case of merge sort, the number of comparisons can be analyzed based on its recursive structure, which divides the array and sorts each half before merging them back together.

Recursive Division and Merging Process

Merge sort fundamentally relies on a divide-and-conquer strategy. The array is recursively divided into halves until each subarray contains a single element. The depth of recursion is (log_{2}n), where (n) is the number of elements in the array. This ensures that the algorithm efficiently processes small, manageable subarrays before working towards a final sorted output.

During the merge step, each pair of elements from the two sorted halves is compared. In the worst-case scenario, where the two halves need to be merged completely, up to (n - 1) comparisons may be necessary. This merging process is crucial for combining sorted subarrays into a larger sorted array.

Total Number of Comparisons

The total number of comparisons in a merge sort can be approximated by the recurrence relation:

[C_n C_{frac{n}{2}} C_{frac{n}{2}} n - 1]

This relation simplifies to:

[C_n approx n log_{2}n]

Conclusion and Efficiency

Therefore, the number of comparisons made by the merge sort algorithm is approximately (O(n log n)). This makes merge sort particularly efficient for large datasets where stability is required. The (O(n log n)) complexity ensures that the algorithm scales well even with large inputs, making it a preferred choice for certain applications.

Additional Insights

Interestingly, the number of comparisons in a merge sort is also equal to the number of inversions in the original array. The number of inversions is typically (O(n log n)), further supporting the efficiency of merge sort. Additionally, the merging process in merge sort involves breaking down the sequence into halves at each step, reducing the size of the sequence by 2 at every step. This ensures that the algorithm continues to efficiently handle smaller subarrays as it progresses.

For a more detailed breakdown of the total number of comparisons, consider the following:

When (N 2^k), the initial comparison value is simple: There are (frac{N}{2}) pairs that are sorted with 1 comparison each for (frac{N}{2}) comparisons. There are (frac{N}{4}) pairs of sets from 1 that are sorted with at most 3 comparisons each for (3 cdot frac{N}{4}) comparisons. This pattern continues with each subsequent level of merging, leading to a total of:

[sum_{i1}^{log N} (2^i - 1) cdot frac{N}{2^i} N log_2 N - N]

This formula provides a precise calculation of the total number of comparisons, emphasizing the (O(n log n)) complexity.

In conclusion, merge sort is not only efficient due to its divide-and-conquer nature but also due to the predictable and manageable number of comparisons it makes. This makes it a highly effective algorithm for sorting large datasets where both efficiency and stability are critical.