Understanding the First Round of Merge Sort on an Array: A Step-by-Step Guide

Introduction to Merge Sort

In computer science, merge sort is a highly efficient, general-purpose, and comparison-based sorting algorithm. Merge sort utilizes the divide and conquer technique to sort data. This technique works by recursively breaking down the problem into smaller subproblems until the subproblems can be easily solved, and then merging those solutions to form the final sorted array. Merge sort divides the input array into two halves, sorts each half independently, and then merges the two sorted halves to produce the final sorted array.

In this article, we will focus on the first round of the merge sort algorithm applied to the array: [3 1 8 7 6 4 5 2]. We will break down the process step-by-step, from the division phase to the merging phase, to help you fully understand how merge sort operates.

The Initial State

The given array is: [3 1 8 7 6 4 5 2]. In the first phase of the merge sort, the array will be split into two equal halves. If the array length is even, each half will have an equal number of elements. If the length is odd, one half will have one more element than the other.

For the array [3 1 8 7 6 4 5 2], the two halves would be:

First half: [3 1 8 7] (4 elements) Second half: [6 4 5 2] (4 elements)

The First Round of Conquer: Sorting the Halves

In the first phase of the merge sort, each half of the array is sorted independently. This process is known as the 'conquer' phase.

Let's sort the first half, [3 1 8 7], and the second half, [6 4 5 2], step-by-step:

Sorting the First Half: [3 1 8 7]

1. Compare the first and second elements: [3 1] -> [1 3] 2. Compare the third and fourth elements: [1 3 8 7] -> [1 3 7 8] 3. Compare the second and third elements: [1 3 7 8]

After sorting, the first half of the array is [1 3 7 8].

Sorting the Second Half: [6 4 5 2]

1. Compare the first and second elements: [6 4] -> [4 6] 2. Compare the third and fourth elements: [4 6 5 2] -> [4 6 5 2]3. Compare the second and third elements: [4 5 6 2] -> [4 5 2 6]4. Compare the first and second elements: [4 2 5 6] -> [2 4 5 6]

After sorting, the second half of the array is [2 4 5 6].

The state of our array after the sorting (conquer) phase is now:

First half: [1 3 7 8]

Second half: [2 4 5 6]

The First Round of Combine: Merging the Two Sorted Halves

Once the halves are sorted, they are merged back together to form a sorted array. This process is known as the 'combine' phase. At each step, two smallest elements are compared and placed in the correct position in the merged array until all elements are placed.

Merging the Two Sorted Halves

Let's start merging the two halves: [1 3 7 8] and [2 4 5 6] into a single sorted array:

1. Compare and place the smallest element. The smallest element is 1, so it goes first: [1]. 2. Compare and place the next smallest element. The next smallest element is 2: [1, 2]. 3. Compare and place the next smallest element. The next smallest element is 3: [1, 2, 3]. 4. Continue this process until all elements are placed in the correct order.

The final merged array is: [1 2 3 4 5 6 7 8].

Conclusion: Understanding Merge Sort

Understanding how merge sort works in the first round is crucial for mastering this powerful sorting algorithm. By dividing the problem into smaller subproblems, sorting each subproblem, and then merging them, merge sort guarantees a sorted array in O(n log n) time complexity.

If you have any questions or need further clarification, feel free to ask in the comments below. Happy coding!