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---------

Co-authored-by: Yudong Jin <krahets@163.com>
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Peng Chen
2025-03-09 06:28:27 +08:00
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en/docs/chapter_sorting/merge_sort.md
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<u>Merge sort</u> is a sorting algorithm based on the divide-and-conquer strategy, involving the "divide" and "merge" phases shown in the figure below.
1. **Divide phase**: Recursively split the array from the midpoint, transforming the sorting problem of a long array into that of shorter arrays.
2. **Merge phase**: Stop dividing when the length of the sub-array is 1, start merging, and continuously combine two shorter ordered arrays into one longer ordered array until the process is complete.
1. **Divide phase**: Recursively split the array from the midpoint, transforming the sorting problem of a long array into shorter arrays.
2. **Merge phase**: Stop dividing when the length of the sub-array is 1, and then begin merging. The two shorter sorted arrays are continuously merged into a longer sorted array until the process is complete.
![The divide and merge phases of merge sort](merge_sort.assets/merge_sort_overview.png)
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As shown in the figure below, the "divide phase" recursively splits the array from the midpoint into two sub-arrays from top to bottom.
1. Calculate the midpoint `mid`, recursively divide the left sub-array (interval `[left, mid]`) and the right sub-array (interval `[mid + 1, right]`).
2. Continue with step `1.` recursively until the sub-array interval length is 1 to stop.
2. Continue with step `1.` recursively until sub-array length becomes 1, then stops.
The "merge phase" combines the left and right sub-arrays into a single ordered array from bottom to top. Note that merging starts with sub-arrays of length 1, and each sub-array is ordered during the merge phase.
The "merge phase" combines the left and right sub-arrays into a sorted array from bottom to top. It is important to note that, merging starts with sub-arrays of length 1, and each sub-array is sorted during the merge phase.
=== "<1>"
![Merge sort process](merge_sort.assets/merge_sort_step1.png)
@@ -46,10 +46,10 @@ The "merge phase" combines the left and right sub-arrays into a single ordered a
=== "<10>"
![merge_sort_step10](merge_sort.assets/merge_sort_step10.png)
It is observed that the order of recursion in merge sort is consistent with the post-order traversal of a binary tree.
It can be observed that the order of recursion in merge sort is consistent with the post-order traversal of a binary tree.
- **Post-order traversal**: First recursively traverse the left subtree, then the right subtree, and finally handle the root node.
- **Merge sort**: First recursively handle the left sub-array, then the right sub-array, and finally perform the merge.
- **Post-order traversal**: First recursively traverse the left subtree, then the right subtree, and finally process the root node.
- **Merge sort**: First recursively process the left sub-array, then the right sub-array, and finally perform the merge.
The implementation of merge sort is shown in the following code. Note that the interval to be merged in `nums` is `[left, right]`, while the corresponding interval in `tmp` is `[0, right - left]`.
@@ -65,9 +65,9 @@ The implementation of merge sort is shown in the following code. Note that the i
## Linked List sorting
For linked lists, merge sort has significant advantages over other sorting algorithms, **optimizing the space complexity of the linked list sorting task to $O(1)$**.
For linked lists, merge sort has significant advantages over other sorting algorithms. **It can optimize the space complexity of the linked list sorting task to $O(1)$**.
- **Divide phase**: "Iteration" can be used instead of "recursion" to perform the linked list division work, thus saving the stack frame space used by recursion.
- **Merge phase**: In linked lists, node addition and deletion operations can be achieved by changing references (pointers), so no extra lists need to be created during the merge phase (combining two short ordered lists into one long ordered list).
- **Merge phase**: In linked lists, node insertion and deletion operations can be achieved by changing references (pointers), so no extra lists need to be created during the merge phase (combining two short ordered lists into one long ordered list).
Detailed implementation details are complex, and interested readers can consult related materials for learning.
The implementation details are relatively complex, and interested readers can consult related materials for learning.