Starter Activity (5 mins)
Organizing Chaos...
Example Answers:
- 1. You might sort them by suit (hearts, diamonds, clubs, spades) and then by rank (Ace, 2, 3... King) within each suit. Or just by rank.
- 2. Sorted data is useful because it makes searching for specific items much faster (e.g., using Binary SearchA searching algorithm that divides the search space in half each time until it finds the target. Requires the array to be sorted.), it's easier to identify duplicates, find minimum/maximum values, or see patterns.
- 3. Most people would break a large task into smaller, more manageable parts. This makes the overall task less daunting, easier to plan, and progress can be tracked more effectively. This is a key idea in DecompositionBreaking a problem down into smaller steps in order to solve it. A CT skill. and also in the Divide and ConquerAn algorithmic paradigm where a problem is recursively broken down into two or more sub-problems of the same or related type, until these become simple enough to be solved directly. The solutions to the sub-problems are then combined to give a solution to the original problem. strategy used by Merge Sort!
Lesson Outcomes
By the end of this lesson, you should be able to:
- Describe the Merge SortA divide and conquer sorting algorithm that recursively divides a list into smaller sub-lists, sorts them, and then merges them back together. algorithm.
- Explain the "Divide and Conquer" strategy used by Merge Sort.
- Trace the execution of a Merge Sort algorithm on a given list of data.
- Understand the process of splitting lists and merging sorted sub-listsA smaller list created by dividing a larger list during the merge sort process..
- Identify the advantages and disadvantages of Merge Sort.
- Conceptually understand its time complexityA measure of how the runtime of an algorithm scales with the input size. Merge Sort has an average and worst-case time complexity of O(n log n). (O(n log n)) and space complexityA measure of the amount of memory an algorithm uses. Merge Sort typically requires O(n) auxiliary space..
Introduction to Merge Sort
Merge SortA divide and conquer sorting algorithm that recursively divides a list into smaller sub-lists, sorts them, and then merges them back together. is an efficient, comparison-based sorting algorithm. It's a classic example of the Divide and ConquerAn algorithmic paradigm where a problem is recursively broken down into two or more sub-problems of the same or related type, until these become simple enough to be solved directly. The solutions to the sub-problems are then combined to give a solution to the original problem. strategy.
The basic idea is to:
- Divide: If the list has more than one element, split it into two (approximately) equal halves.
- Conquer: Recursively sort each half. If a half has only one element, it's considered sorted.
- Combine (Merge): MergeThe process of combining two sorted sub-lists into a single sorted list. the two sorted halves back into a single, sorted list.
This process continues until the entire list is sorted. Merge Sort is known for its consistent performance. Hover over keywordsThis is an example tooltip! for definitions.
Merge Sort in Action
Interactive Simulation
Enter a list of numbers (comma-separated, e.g., 5,2,8,1,9,4) or use the default. Then, step through the Merge Sort process.
Task 1: Tracing Merge Sort (Max 4 points)
Consider the list: [6, 1, 7, 3]. Describe the main steps Merge Sort would take. Focus on the splitting and the first two merge operations.
Example Trace Steps:
- Split 1: [6, 1, 7, 3] is split into [6, 1] and [7, 3].
- Split 2 (Left): [6, 1] is split into [6] and [1]. (These are now sorted sub-lists of size 1).
- Split 2 (Right): [7, 3] is split into [7] and [3]. (These are now sorted sub-lists of size 1).
- Merge 1 (Left): Merge [6] and [1]. Compare 6 and 1. 1 is smaller. Result: [1, 6].
- Merge 1 (Right): Merge [7] and [3]. Compare 7 and 3. 3 is smaller. Result: [3, 7].
- Merge 2 (Final): Merge [1, 6] and [3, 7].
- Compare 1 and 3. 1 is smaller. Merged: [1]
- Compare 6 and 3. 3 is smaller. Merged: [1, 3]
- Compare 6 and 7. 6 is smaller. Merged: [1, 3, 6]
- Add remaining 7. Merged: [1, 3, 6, 7]
(For 4 marks, focus on the initial splits and the first two merge operations clearly showing comparisons and resulting sub-lists.)
Advantages & Disadvantages of Merge Sort
Weighing the Options
Advantages:
- Very efficientA measure of how well an algorithm performs in terms of time (speed) and space (memory usage). Binary search is generally more efficient than linear search for large, sorted lists. for large lists, with a consistent time complexityA measure of how the runtime of an algorithm scales with the input size. Merge Sort has an average and worst-case time complexity of O(n log n). of O(n log n).
- It's a stable sortA sorting algorithm where two objects with equal keys appear in the same order in the sorted output as they appear in the input array to be sorted. Merge sort is stable., meaning it preserves the relative order of equal elements.
- Performance is not significantly affected by the initial order of the list (unlike some other sorts like Bubble Sort or Insertion Sort which can be very fast on nearly sorted data).
Disadvantages:
- Requires extra space complexityA measure of the amount of memory an algorithm uses. Merge Sort typically requires O(n) auxiliary space. (typically O(n)) because it usually needs a temporary array or list to merge the sub-lists. This can be an issue for very large datasets if memory is limited.
- Can be slower than other algorithms (like Insertion Sort) for very small lists due to the overhead of recursionA process where a function calls itself directly or indirectly. Merge sort is often implemented recursively for the splitting phase. or iterative setup.
- The recursive implementation can lead to stack overflow errors for extremely large lists if not handled carefully (though iterative versions exist).
Real-World Context for Merge Sort
Merge Sort's efficiency and stability make it useful in various real-world applications:
- External Sorting: When data is too large to fit into RAM (e.g., sorting massive database files stored on disk), Merge Sort is effective because it can work with chunks of data, sort them in memory, and then merge the sorted chunks from disk.
- Standard Library Sort Functions: Many programming languages' built-in sort functions (or variations like Timsort, which combines Merge Sort and Insertion Sort) use Merge Sort due to its O(n log n) guarantee and stability. For example, Python's `list.sort()` and `sorted()` use Timsort.
- Counting Inversions in a List: Merge Sort can be adapted to count the number of inversions (pairs of elements that are out of order) in a list efficiently.
- Parallel Sorting: The divide-and-conquer nature of Merge Sort makes it suitable for parallelization, where different sub-lists can be sorted concurrently on multiple processors.
Merge Sort Myth Busters!
Myth 1: "Merge Sort is always the fastest sorting algorithm."
Reality: While Merge Sort has an excellent worst-case and average-case time complexityA measure of how the runtime of an algorithm scales with the input size. Merge Sort has an average and worst-case time complexity of O(n log n). of O(n log n), it's not always the absolute fastest for *all* situations. For very small lists, simpler algorithms like Insertion Sort can be faster due to less overhead. Also, algorithms like Quicksort can have a better average-case constant factor, making them faster in practice for many datasets (though Quicksort's worst-case is O(n2)).
Myth 2: "Merge Sort sorts the list 'in-place' without needing extra memory."
Reality: The standard implementation of Merge Sort is NOT in-place. It typically requires auxiliary (extra) memory space proportional to the size of the input list (O(n) space complexityA measure of the amount of memory an algorithm uses. Merge Sort typically requires O(n) auxiliary space.) to hold the sub-listsA smaller list created by dividing a larger list during the merge sort process. during the mergeThe process of combining two sorted sub-lists into a single sorted list. step. While in-place versions of Merge Sort exist, they are much more complex and often less efficient in practice.
Task 2: Exam Practice Questions
Apply your knowledge to these exam-style questions.
1. Describe the 'divide' step and the 'merge' step of the Merge Sort algorithm. [4 marks]
Mark Scheme:
- Divide Step: The list is repeatedly split/divided into two halves (1 mark). This process continues recursively until each sub-list contains only one element (which is considered sorted) (1 mark).
- Merge Step: Pairs of sorted sub-lists are combined/merged (1 mark). Elements are compared from the two sub-lists and placed into a new temporary list in sorted order, until all elements from both sub-lists are in the new merged list (1 mark).
2. State one advantage and one disadvantage of using Merge Sort. [2 marks]
Mark Scheme:
- Advantage: Efficient for large lists / O(n log n) time complexity / Stable sort. (1 mark for one valid advantage)
- Disadvantage: Requires extra memory space (O(n)) / Not in-place / Can be slower than other sorts for small lists due to overhead. (1 mark for one valid disadvantage)
Key Takeaways
- Merge SortA divide and conquer sorting algorithm that recursively divides a list into smaller sub-lists, sorts them, and then merges them back together. uses a Divide and ConquerAn algorithmic paradigm where a problem is recursively broken down into two or more sub-problems of the same or related type, until these become simple enough to be solved directly. The solutions to the sub-problems are then combined to give a solution to the original problem. strategy: recursively splitting the list, then merging sorted sub-listsA smaller list created by dividing a larger list during the merge sort process..
- The Divide step breaks the list into halves until sub-lists have one (or zero) elements.
- The Merge step combines two sorted sub-lists into a single larger sorted list by comparing elements.
- Advantages: Efficient (O(n log n) time complexity), stable.
- Disadvantages: Requires extra O(n) space, can be slower for very small lists.
What's Next?
You've explored the Merge Sort algorithm!
Continue learning about other sorting algorithms in J277 Topic 2.1.3:
- Bubble Sort
- Insertion Sort
Comparing these different sorting methods will help you understand their trade-offs.
Extension Activities
1. Iterative Merge Sort
Merge Sort is often explained and implemented using recursionA process where a function calls itself directly or indirectly. Merge sort is often implemented recursively for the splitting phase.. However, it can also be implemented iteratively (using loops).
Research Task: What is the general approach for an iterative (bottom-up) Merge Sort? How does it manage the merging of sub-lists without recursive calls?
2. Timsort
Timsort is a hybrid stable sorting algorithm, derived from Merge Sort and Insertion Sort, designed to perform well on many kinds of real-world data. It's used in Python and Java, among others.
Research Task: Why was Timsort developed? What are "runs" in Timsort, and how does it combine the strengths of Merge Sort and Insertion Sort?