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Welcome everyone! Today, we will explore Merge Sort, one of the most efficient sorting algorithms. Who can tell me what efficient means in this context?
Efficient means it works faster than other sorting methods, right?
Exactly! Merge Sort operates in O(n log n) time. Can anyone tell me what O(n log n) means?
It means that as our input size grows, the time taken to sort increases in relation to both n and log n.
Correct! Remember that merge sort is based on a merging technique that combines two sorted lists. Let's look deeper into that process.
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Now, let's analyze the merge function. If we have two lists, A with m elements and B with n elements, how do we combine them?
We take one element from each list and put them into a new sorted list, right?
Exactly! Each time we merge, we perform some fixed operations: comparisons and assignments. Therefore, the time complexity is O(m + n). Can anyone describe what 'linear' means in this context?
Linear means the time taken grows directly with the size of the elements we're merging.
Yes! Now let's link this back to our entire merge sort process.
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Merge Sort divides the list into half until we reach sublists of size one. How do we express this in terms of time?
It sounds like a recurrence relation! T(n) = 2T(n/2) + O(n)?
Great! And what does that mean when we solve for T(n)?
It means we can unwind it. After a few steps, we find that T(n) is O(n log n)!
Exactly, well done! Now, letβs apply this understanding to practical scenarios with the merge function.
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Apart from sorting, the merge function can help perform unions, intersections, and more. Can anyone give me an example of a union using merge?
If we have lists [1, 2, 3] and [2, 3, 4], we can merge them into [1, 2, 3, 4].
Exactly! Now, what about intersection? How would that function?
We would only keep the common elements, right? Like merging [1, 2, 6] with [2, 6, 8] results in [2, 6].
Perfect! Merging functions showcase their versatility in various applications. But what are some challenges of merge sort?
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Given the strengths of merge sort, what are some limitations we've encountered?
Merge sort requires additional memory space due to creating new lists, right?
Absolutely! This extra memory can be a drawback. What about the recursive calls?
They can slow down performance, too, since we need to allocate and manage call stacks.
Great observations! Despite these challenges, merge sort remains a key algorithm to understand in computer science.
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This section delves into merge sort's recursive nature, exploring the efficiency of its merge function and presenting a thorough analysis of its time complexity, concluding that merge sort operates in O(n log n) time. The section also discusses possible applications of the merge function beyond sorting.
Merge sort is a sorting algorithm based on the divide-and-conquer paradigm. The section begins by contrasting merge sort's efficiency against algorithms like insertion sort and selection sort, emphasizing its O(n log n) performance. Understanding merge sort necessitates analysis of its core function, merge:
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In order to analyze merge sort, the first thing we need to do is to give an analysis of the merge function itself. How much time does merge take in terms of the input sizes of the two lists, A and B. So, suppose A has m elements and B has n elements and we want to put all these elements together into a single sorted list in C. Remember that we had an iteration where in each step of the iteration we looked at the first element in A and B and move the smaller of the two to C. So clearly C grows by one element with each iteration and since we have to move all m plus n elements from A and B to C, the size of C is m plus n. What do we do in each iteration? Well we do a comparison and then, we do an assignment and then, we increment some indices. So, this is a fixed number of operations. So, it is a constant. So, the total amount of work is proportional to m plus n.
The merge operation is a crucial part of merge sort. It takes two sorted lists A and B, with m and n elements respectively, and combines them into a new sorted list C. The merge function works by comparing elements from A and B one at a time and adding the smaller element to C. For each element that we need to merge, we perform a constant amount of work, specifically a comparison and an assignment. Since we need to process m + n elements to complete the merge, we conclude that the time complexity for the merge function is O(m + n). This means that the time taken is linearly dependent on the total number of elements in the two lists.
Imagine you are preparing a sorted seating arrangement for a group of people from two different lines (A and B). You look at the first person in line A and the first person in line B. You decide who should sit first based on their names being in alphabetical order. Each time you make a decision, you move one person to the sorted seating chart (C). Eventually, everyone from both lines is placed into the seating chart, and you end up with a neatly organized list of names. The time taken depends on the total number of people you had initially.
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Now, having analyzed merge, let us look at merge sort. So, merge sort says that if a list is small, has zero or one elements, nothing is to be done. Otherwise you have to solve the problem for two half lists and then, merge them. As with any recursive function, we have to describe the time taken by such a function in terms of a recurrence. So, let us assume for now since we are going to keep dividing by 2 that we can keep dividing by 2 without getting an odd number in between. Let us assume that the input n is some perfect power of 2. It is n is 2 to the k. When we break up merge sort into two lists, we have two lists of size n by 2. The time taken for n elements is two times time taken for two lists of n by 2 and this is the merge component.
The merge sort algorithm uses a recursive approach that breaks the original list into smaller sub-lists until it reaches a point where each sub-list contains one or zero elements. At this point, the recursion starts to build back up by merging the sub-lists. The process of merging two halves of a list that each contain n/2 elements takes linear time because of the earlier analysis of the merge function β it takes O(n). To formalize this, we set up a recurrence relation where the time T(n) taken to sort a list of size n is T(n/2) for the left half, T(n/2) for the right half, plus the O(n) time for merging.
Think of sorting a messy drawer of utensils. First, you take out all utensils and start sorting them into smaller groups. Once they are sorted into smaller containers (each having one or two utensils), you then start combining these small groups back into one neatly organized drawer. Each time you combine groups, it takes some time to arrange them correctly. Just like in merge sort, where you break down a large problem into manageable pieces, this method allows you to sort efficiently.
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If we expand this out, we read substitute for T n by 2 and we get two times T n by 4 plus n by 2 because that if we take this as a new input, this expands using the same definition and if we rewrite this. So, we write two times 2 as 2 square and we write this 4 as two squared. We will find that this is equivalent to writing it in this form 2 into 2, 2 squared T n by 2 square and now notice that you have two times n by 2 over here. This 2 and this 2 will cancel. So, we have one factor n and the other factor of n. The important thing is that you have 2 here in the exponent and you have 2 here before then.
As we develop the recurrence for merge sort, we start from the base case, where sorting a list of size 1 requires 0 work. The recurrence relation T(n) = 2T(n/2) + O(n) can be expanded by substituting T(n/2) with the same recurrence formula recursively. This process reveals that the number of levels in the recursion is logarithmic in relation to n, leading us to realize that after several expansions, we derive a linear component multiplied by log(n) due to the nature of the binary divisions in the list. Ultimately, this analysis leads us to the conclusion that merge sort runs with a time complexity of O(n log n).
Envision a project where you need to group your work into stages. Each stage requires doing two essential tasks before you can move to the next stage. The number of stages grows logarithmically with the amount of work, but every stage involves handling all your tasks linearly. By the end, while the number of stages (log n) is fewer, the amount of time spent on each stage (n) sums up, leading you to understand the overall time taken for the project is O(n log n).
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Merge sort is clearly superior to insertion sort and selection sort because it is order n log n, it can handle lists as we saw of size 100,000 as opposed to a few thousand, but it does not mean that merge sort does not have limitations. One of the limitations of merge sort is that we are forced to create a new array, every time we merge two lists. There is no obvious way to efficiently merge two lists without creating a new list. So, there is a penalty in terms of extra storage. We have to double this space that we use when we start with lists and we want to sort it within merge sort.
While merge sort is efficient, especially for large datasets, it has some drawbacks. One primary limitation is that it requires additional space for a temporary array to hold the sorted elements during the merging process. This means that if you are sorting a very large list, you might need twice the storage, which can be significant. Additionally, merge sort is inherently recursive. Each recursive call requires extra overhead from the system to manage those calls (like maintaining the state of local variables), which can be costly in terms of time and system resources.
Consider a factory that splits raw materials into smaller batches for easier processing. While this makes handling each batch efficient, they must have extra storage space to keep all the batches organized, which can be a hindrance if their space is limited. Similarly, merge sort needs that extra space to manage all the elements being sorted, which can be a drawback in memory-constrained environments.
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Key Concepts
Divide-and-Conquer: A strategy that divides a problem into smaller subproblems to solve them more efficiently.
Linear Time Complexity: An algorithm is considered to run in linear time if its time complexity can be expressed as O(n).
Auxiliary Space: The extra space required by an algorithm beyond the main input data.
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To merge the sorted lists [1, 3, 5] and [2, 4, 6], the merge function would produce [1, 2, 3, 4, 5, 6].
The merge function can also be used to find the intersection of two lists, e.g., merging [1, 2, 2, 3] with [2, 3, 4] results in [2, 3].
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When sorting with merge, split and mix, O(n log n) does the fix!
Imagine you have two baskets of fruits, one with apples and another with oranges. To prepare a fruit salad, you split them by type and then combine them in groups, ensuring each fruit is accounted forβthis is just like merging sorted lists!
D-S-M: Divide, Sort, Merge. Remember to split your list, sort each part, then merge them back!
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Review the Definitions for terms.
Term: Merge Sort
Definition:
A divide-and-conquer algorithm that sorts a list by recursively splitting it into halves and merging sorted halves.
Term: Merge Function
Definition:
A function that combines two sorted lists into a single sorted list.
Term: Recursion
Definition:
A technique where a function calls itself to solve smaller subproblems until a base case is reached.
Term: Time Complexity
Definition:
An estimation of the amount of time an algorithm takes to run as a function of the input size.
Term: Auxiliary Space
Definition:
Additional space used by an algorithm besides the input data.