Question

Suppose that \(L\) is a list of 10,000 elements. Find the average number of comparisons made by quick sort and merge sort to sort \(L\)

Short answer

Quicksort averages about 184,200 comparisons, while merge sort averages about 132,900 comparisons.

Step-by-step solution

Understanding Quicksort and Comparisons

Quicksort is a divide-and-conquer algorithm that works by selecting a 'pivot' element and partitioning the other elements into two sub-arrays according to whether they are less than or greater than the pivot. The average number of comparisons made by quicksort is proportional to the complexity of the algorithm, which is \(O(n \log n)\).

Calculating Comparisons for Quicksort

The average number of comparisons for quicksort can be approximated by \(2n \ln n\) where \(n\) is the number of elements. Substitute \(n = 10000\) to get \( \approx 2 \times 10000 \times \ln 10000 \approx 2 \times 10000 \times 9.21 \approx 184200\) comparisons.

Understanding Merge Sort and Comparisons

Merge sort is another divide-and-conquer algorithm that divides the list into halves, recursively sorts each half, and then merges the sorted halves to produce a sorted list. The algorithm is \(O(n \log n)\), similar to quicksort, but the number of comparisons differs due to the nature of the divide and merge steps.

Calculating Comparisons for Merge Sort

The number of comparisons in merge sort can be expressed by the average-case formula \(n \log n\), where \(n = 10000\). Substitute into the formula: \(10000 \times \log_2 10000 \). Since \(\log_2 10000 \approx 13.29\), the number of comparisons is approximately \(10000 \times 13.29 = 132900\).

Key concepts

Quicksort

Quicksort is a popular sorting algorithm known for its efficiency and simplicity. Being a divide-and-conquer algorithm, it works by selecting a 'pivot' element from the array. Subsequent elements are divided into two parts: those less than the pivot and those greater than it. This division is referred to as partitioning. After partitioning, the sub-arrays (smaller parts of the list) are sorted recursively, which gradually sorts the entire array. This method is often faster in practice compared to other sorting algorithms, such as bubble sort or insertion sort, even if both have the same average case complexity of \(O(n \log n)\).
Quicksort is not only elegant but also has an in-place sorting nature, meaning it requires small additional space. However, its performance can degrade to \(O(n^2)\) time complexity in the worst-case scenario. By strategically choosing pivots, such as using the median-of-three method, this drawback can often be mitigated.

Merge Sort

Merge sort is another efficient, stable, and straightforward sorting algorithm. It also follows the divide-and-conquer paradigm, splitting the unsorted list into halves until it divides into single-element sublists. Each sublist is sorted, and the sorted sublists are merged back into a sorted list. The key operation here is the merging step, which guarantees that the resulting list is sorted.
Unlike quicksort, merge sort has a consistent time complexity of \(O(n \log n)\) in all cases, making it predictable in terms of performance, though often slower in practice than quicksort due to higher constant factors and overhead in merging. Merge sort is particularly useful for sorting linked lists and large datasets because it is stable and works well with data that doesn’t fit in memory since it adapts more easily to data in external storage systems.

Average Case Analysis

Understanding the average case allows us to predict the expected performance of algorithms under normal conditions. In the context of sorting, it involves measuring the average number of comparisons and permutations required. For quicksort, the average case performance is often calculated using the approximate expression of \(2n \ln n\). This gives an indication of the number of times elements are compared during the execution of the algorithm.
For merge sort, although it has a consistent time complexity of \(n \log n\), the exact number of operations depends on how elements are split and merged. Average case analysis helps to provide insights into an algorithm's behavior under expected circumstances and serves as a practical measure that balances between best and worst case scenarios.

Divide-and-Conquer

The divide-and-conquer approach is a strategic method in computer science for solving problems. It involves breaking down a problem into smaller sub-problems that are easier to manage and solve. Each sub-problem is solved independently, and these solutions are then combined to form the solution to the original problem.
Both quicksort and merge sort employ this strategy efficiently. Dividing the data allows both algorithms to break large, complex tasks into simpler ones, which can be processed faster. For example, in quicksort, the division is based on a pivot, while in merge sort, it involves dividing the list into equal halves. This approach not only simplifies problem-solving but also enhances the speed and efficiency of algorithms.

Complexity Analysis

Complexity analysis is essential for understanding the efficiency of algorithms. It provides a theoretical estimate of resources like time and space that an algorithm consumes. These resources are generally measured in terms of input size \(n\).
In quicksort, complexity varies: the best-case is \(O(n \log n)\) when partitions are balanced, but it can degrade to \(O(n^2)\) when they are skewed. In contrast, merge sort consistently has a complexity of \(O(n \log n)\), as it always splits lists evenly and merges sorted lists in linear time.
While complexity analysis provides a worst-case scenario, it often overlooks factors like constant factors and lower load time that can significantly impact the actual runtime, making empirical testing crucial to truly gauge an algorithm's performance.