156 lines
4.2 KiB
Markdown
156 lines
4.2 KiB
Markdown
|
---
|
||
|
title: Selection Sort
|
||
|
TARGET DECK: Obsidian::STEM
|
||
|
FILE TAGS: algorithm::sorting
|
||
|
tags:
|
||
|
- algorithm
|
||
|
- sorting
|
||
|
---
|
||
|
|
||
|
## Overview
|
||
|
|
||
|
Property | Value
|
||
|
---------- | --------
|
||
|
Best Case | $O(n^2)$
|
||
|
Worst Case | $O(n^2)$
|
||
|
Avg. Case | $O(n^2)$
|
||
|
Memory | $O(1)$
|
||
|
In Place | Yes
|
||
|
Stable | Yes
|
||
|
|
||
|
![[selection-sort.gif]]
|
||
|
|
||
|
%%ANKI
|
||
|
Basic
|
||
|
What is selection sort's best case runtime?
|
||
|
Back: $O(n^2)$
|
||
|
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
|
||
|
<!--ID: 1707398773323-->
|
||
|
END%%
|
||
|
|
||
|
%%ANKI
|
||
|
Basic
|
||
|
What is selection sort's worst case runtime?
|
||
|
Back: $O(n^2)$
|
||
|
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
|
||
|
<!--ID: 1707398773326-->
|
||
|
END%%
|
||
|
|
||
|
%%ANKI
|
||
|
Basic
|
||
|
What is selection sort's average case runtime?
|
||
|
Back: $O(n^2)$
|
||
|
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
|
||
|
<!--ID: 1707398773327-->
|
||
|
END%%
|
||
|
|
||
|
%%ANKI
|
||
|
Basic
|
||
|
Is selection sort in place?
|
||
|
Back: Yes
|
||
|
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
|
||
|
<!--ID: 1707398773328-->
|
||
|
END%%
|
||
|
|
||
|
%%ANKI
|
||
|
Basic
|
||
|
Is selection sort stable?
|
||
|
Back: Yes
|
||
|
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
|
||
|
<!--ID: 1707398773330-->
|
||
|
END%%
|
||
|
|
||
|
```c
|
||
|
void swap(int i, int j, int *A) {
|
||
|
int tmp = A[i];
|
||
|
A[i] = A[j];
|
||
|
A[j] = tmp;
|
||
|
}
|
||
|
|
||
|
void selection_sort(const int n, int A[static n]) {
|
||
|
for (int i = 0; i < n - 1; ++i) {
|
||
|
int mini = i;
|
||
|
for (int j = i + 1; j < n; ++j) {
|
||
|
if (A[j] < A[mini]) {
|
||
|
mini = j;
|
||
|
}
|
||
|
}
|
||
|
swap(i, mini, A);
|
||
|
}
|
||
|
}
|
||
|
```
|
||
|
|
||
|
%%ANKI
|
||
|
Basic
|
||
|
What sorting algorithm does the following demonstrate?
|
||
|
![[selection-sort.gif]]
|
||
|
Back: Selection sort.
|
||
|
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
|
||
|
<!--ID: 1707400943836-->
|
||
|
END%%
|
||
|
|
||
|
## Loop Invariant
|
||
|
|
||
|
Consider [[loop-invariant|loop invariant]] $P$ given by
|
||
|
|
||
|
> On each iteration, `A[0..i-1]` is a sorted array of the `i` least elements of `A`.
|
||
|
|
||
|
We prove $P$ maintains the requisite properties:
|
||
|
|
||
|
* Initialization
|
||
|
* When `i = 0`, `A[0..-1]` is an empty array. This trivially satisfies $P$.
|
||
|
* Maintenance
|
||
|
* Suppose $P$ holds for some `0 ≤ i < n - 1`. Then `A[0..i-1]` is a sorted array of the `i` least elements of `A`. Our inner loop then finds the smallest element in `A[i..n]` and swaps it with `A[i]`. Therefore `A[0..i]` is not a sorted array of the `i + 1` least elements of `A`. At the end of the iteration, `i` is incremented meaning `A[0..i-1]` still satisfies $P$.
|
||
|
* Termination
|
||
|
* On termination, `i = n - 1` and `A[0..n-2]` are the `n - 1` least elements of `A` in sorted order. But, by exhaustion, `A[n-1]` must be the largest element meaning `A[0..n-1]`, the entire array, is in sorted order.
|
||
|
|
||
|
%%ANKI
|
||
|
Basic
|
||
|
Given array `A[0..n-1]`, what is selection sort's loop invariant?
|
||
|
Back: `A[0..i-1]` is a sorted array of the `i` least elements of `A`.
|
||
|
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
|
||
|
<!--ID: 1707398773331-->
|
||
|
END%%
|
||
|
|
||
|
%%ANKI
|
||
|
Basic
|
||
|
What is initialization of selection sort's loop invariant?
|
||
|
Back: Sorting starts with an empty array which is trivially sorted.
|
||
|
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
|
||
|
<!--ID: 1707398773333-->
|
||
|
END%%
|
||
|
|
||
|
%%ANKI
|
||
|
Basic
|
||
|
What is maintenance of selection sort's loop invariant?
|
||
|
Back: Each iteration puts the next least element into the sorted subarray.
|
||
|
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
|
||
|
<!--ID: 1707398773334-->
|
||
|
END%%
|
||
|
|
||
|
%%ANKI
|
||
|
Basic
|
||
|
How does selection sort partition its input array?
|
||
|
Back:
|
||
|
```
|
||
|
[ sorted | unsorted ]
|
||
|
```
|
||
|
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
|
||
|
<!--ID: 1707399790952-->
|
||
|
END%%
|
||
|
|
||
|
%%ANKI
|
||
|
Basic
|
||
|
Which element will selection sort move to `sorted`?
|
||
|
```
|
||
|
[ sorted | unsorted ]
|
||
|
```
|
||
|
Back: The least element in `unsorted`.
|
||
|
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
|
||
|
<!--ID: 1707399790955-->
|
||
|
END%%
|
||
|
|
||
|
## References
|
||
|
|
||
|
* Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
|