Algorithms Analysis & Design - Lecture 9
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The document discusses two sorting algorithms: Merge Sort and Quick Sort. Merge Sort uses a divide-and-conquer approach and is stable, but not in-place, with a worst-case complexity of O(n log n), making it suitable for large datasets. Quick Sort also employs a divide-and-conquer strategy, is in-place and generally faster, though it has a worst-case complexity of O(n^2) and is not stable.
Algorithms Analysis & Design - Lecture 9
- 1. Algorithms By Mohamed Gamal © Mohamed Gamal 2024
- 4. Merge Sort Algorithm – Merge Sort uses the divide-and-conquer approach to break the problem into smaller sub-problems, solve each sub-problem recursively, and then combine the results. – Merge Sort is a comparison-based sorting algorithm that repeatedly compares pairs of elements to determine their order. – Merge Sort is typically implemented using recursion, although it can also be implemented iteratively.
- 5. Merge Sort Algorithm 1) Divide the array into two halves. 2) Recursively apply Merge Sort to each half. 3) Merge the two sorted halves back together: a) Compare the elements of the two halves and arrange them in order. b) Continue merging elements until all elements from both halves are merged into a single sorted array. 4) Repeat the process until the entire array is sorted. 2 8 5 3 9 4 1 7
- 10. Complexity: ▪ Worst Case: 𝑂(𝑛 log 𝑛) ▪ Average Case: 𝑂(𝑛 log 𝑛) ▪ Best Case: 𝑂(𝑛 log 𝑛) Advantages: ▪ Merge Sort is well-suited for sorting large datasets that do not fit into memory (external sorting) because it accesses data sequentially and has good performance with large datasets. ▪ Merge Sort is a stable sort, meaning that it preserves the relative order of equal elements in the sorted array. Disadvantages: ▪ Merge Sort is not in-place sorting algorithm since it requires additional space proportional to the size of the array being sorted.
- 11. Let 𝐶(𝑛) be the number of comparisons made by the merge-sort algorithm. Recurrence relation equation: 𝐶 𝑛 = 2𝐶 𝑛 2 + 𝑛 and 𝐶 1 = 0 • Using backward substitution: 𝐶(𝑛) = 2𝐶 𝑛 2 + 𝑛 = 2 2𝐶 𝑛 4 + 𝑛 2 + 𝑛 = 4𝐶 𝑛 4 + 2𝑛 = 4 2𝐶 𝑛 8 + 𝑛 4 + 2𝑛 = 8𝐶 𝑛 8 + 3𝑛 𝐶 𝑛 = 2𝑖 𝐶 𝑛 2𝑖 + 𝑖 𝑛 = 2log2 𝑛 𝐶 𝑛 2log2 𝑛 + log2 𝑛 𝑛 = 𝑛 log2 𝑛 𝑛 2𝑖 = 1 2𝑖 = 𝑛 𝑖 = log2 𝑛 Basic Operation n represents the number of comparisons required to merge the two sorted halves
- 12. Divide N Conquer
- 13. Quick Sort Algorithm – Quick Sort uses the divide-and-conquer approach to break the problem into smaller sub-problems, sort each sub-problem recursively, and then combine the results. – Quick Sort is typically implemented using recursion, although it can also be implemented iteratively. – The choice of the pivot element is crucial for the performance of Quick Sort. Good pivot selection strategies can significantly improve its efficiency.
- 15. Note: Pivots shown in bold.
- 17. Complexity: ▪ Worst Case: 𝑂(𝑛2) – sorted array! ▪ Average Case: 𝑂(𝑛 log 𝑛) – random arrays ▪ Best Case: 𝑂 𝑛 log 𝑛 – split in the middle Advantages: ▪ Quick Sort is in-place sorting algorithm, meaning it requires only a small, constant amount of additional memory space. ▪ Despite its worst-case scenario, Quick Sort is often faster in practice compared to other 𝑂 𝑛 log 𝑛 algorithms like Merge Sort due to its good cache performance and low overhead. Disadvantages: ▪ The worst-case time complexity is 𝑂(𝑛2), which occurs when the pivot selection consistently results in highly unbalanced partitions. ▪ Quick Sort is not stable, which means it may not preserve the relative order of equal elements.
- 18. Which algorithm would Obama pick? Source: http://www.youtube.com/watch?v=k4RRi_ntQc8
- 19. ThankYou!