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

title	TARGET DECK	FILE TAGS	tags
Selection Sort	Obsidian::STEM	algorithm::sorting	algorithm sorting

Overview

Property	Value
Best Case	\Omega(n^2)
Worst Case	O(n^2)
Avg. Case	O(n^2)
Aux. Memory	O(1)
Stable	No
Adaptive	No

!

%%ANKI Basic Describe SELECTION_SORT in a single sentence. Back: Repeatedly put the smallest unsorted record at the end of a sorted array. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

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%%ANKI Basic What is SELECTION_SORT's best case runtime? Back: \Omega(n^2) Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

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%%ANKI Basic What is SELECTION_SORT's worst case runtime? Back: O(n^2) Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

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%%ANKI Basic What is SELECTION_SORT's average case runtime? Back: O(n^2) Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

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%%ANKI Basic Is SELECTION_SORT in place? Back: Yes. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

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%%ANKI Basic Is SELECTION_SORT stable? Back: No. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

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%%ANKI Basic Is SELECTION_SORT adaptive? Back: No. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

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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? ! Back: SELECTION_SORT Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

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Loop Invariant

Consider loop-invariant P given by

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 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, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

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%%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, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

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%%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, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

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%%ANKI Basic How does SELECTION_SORT partition its input array? Back:

[ sorted | unsorted ]

Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

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%%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, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

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%%ANKI Cloze SELECTION_SORT makes fewer {swaps} than INSERTION_SORT in the average case. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

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Bibliography

• Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).