notebook/notes/algorithms/sorting/merge-sort.md

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---
title: Merge Sort
TARGET DECK: Obsidian::STEM
FILE TAGS: algorithm::sorting
tags:
- algorithm
- sorting
---
## Overview
Property | Value
----------- | --------
Best Case | $\Omega(n\lg{n})$
Worst Case | $O(n\lg{n})$
Avg. Case | $O(n\lg{n})$
Aux. Memory | -
Stable | -
Adaptive | -
![[merge-sort.gif]]
%%ANKI
Basic
What does the term "merge" in merge sort refer to?
Back: The primary operation used to combine array halves.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
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END%%
%%ANKI
Basic
What is merge sort's best case runtime?
Back: $\Omega(n\lg{n})$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
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END%%
%%ANKI
Basic
What is merge sort's worst case runtime?
Back: $O(n\lg{n})$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
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END%%
%%ANKI
Basic
What is merge sort's average case runtime?
Back: $O(n\lg{n})$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
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END%%
%%ANKI
Basic
What is the "divide" step of merge sort?
Back: Divide the input array into two subarrays of half size.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
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END%%
%%ANKI
Basic
What is the "conquer" step of merge sort?
Back: Call merge sort on the "divide"-step's two subarrays.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
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END%%
%%ANKI
Basic
What is the "combine" step of merge sort?
2024-03-01 14:12:43 +00:00
Back: Merge the sorted subarrays produced by the "conquer" step.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
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END%%
%%ANKI
Basic
When does merge sort "bottom out"?
Back: When the sequence to be sorted has length 1 or less.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
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END%%
```c
static void merge(int i, int mid, int j, int *A) {
int si = mid - i + 1;
int sj = j - (mid + 1) + 1;
int *L = malloc(sizeof(int) * (si + 1));
int *R = malloc(sizeof(int) * (sj + 1));
L[si] = INT_MAX;
R[sj] = INT_MAX;
for (int k = 0; k < si; ++k) {
L[k] = A[i + k];
}
for (int k = 0; k < sj; ++k) {
R[k] = A[mid + 1 + k];
}
int topL = 0, topR = 0;
for (int k = 0; k < j - i + 1; ++k) {
if (L[topL] < R[topR]) {
A[i + k] = L[topL++];
} else {
A[i + k] = R[topR++];
}
}
free(L);
free(R);
}
void merge_sort(int i, int j, int *A) {
if (j <= i) {
return;
}
int mid = (i + j) / 2;
merge_sort(i, mid, A);
merge_sort(mid + 1, j, A);
merge(i, mid, j, A);
}
```
%%ANKI
Basic
Where in merge sort's implementation are sentinels useful?
Back: As the last elements of the two arrays to combine.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
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END%%
%%ANKI
Basic
What sentinel values are typically used in merge sort's "merge" operation?
Back: $\infty$ or the record type's equivalent.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
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END%%
%%ANKI
Basic
What sorting algorithm does the following demonstrate?
![[merge-sort.gif]]
Back: Merge sort.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
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END%%
## References
* Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).