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

title	TARGET DECK	FILE TAGS	tags
Bubble 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	Yes
Adaptive	Yes

!

%%ANKI Basic Describe bubble sort in a single sentence. Back: Repeatedly swap the smaller of adjacent records downward. Reference: Thomas H. Cormen et al., Introduction to Algorithms, 3rd ed (Cambridge, Mass: MIT Press, 2009).

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%%ANKI Basic What is bubble sort's best case runtime? Back: \Omega(n) Reference: Thomas H. Cormen et al., Introduction to Algorithms, 3rd ed (Cambridge, Mass: MIT Press, 2009).

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%%ANKI Basic How is it bubble sort achieves best case linear runtime? Back: By terminating when no swaps occurred on a given iteration. Reference: Thomas H. Cormen et al., Introduction to Algorithms, 3rd ed (Cambridge, Mass: MIT Press, 2009).

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%%ANKI Basic What input value does bubble sort perform best on? Back: An already sorted array. Reference: Thomas H. Cormen et al., Introduction to Algorithms, 3rd ed (Cambridge, Mass: MIT Press, 2009).

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%%ANKI Basic What is bubble 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).

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%%ANKI Basic What input value does bubble sort perform worst on? Back: An array in reverse-sorted order. Reference: Thomas H. Cormen et al., Introduction to Algorithms, 3rd ed (Cambridge, Mass: MIT Press, 2009).

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%%ANKI Basic What is bubble 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).

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%%ANKI Basic Is bubble sort in place? Back: Yes Reference: Thomas H. Cormen et al., Introduction to Algorithms, 3rd ed (Cambridge, Mass: MIT Press, 2009).

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%%ANKI Basic Is bubble sort stable? Back: Yes Reference: Thomas H. Cormen et al., Introduction to Algorithms, 3rd ed (Cambridge, Mass: MIT Press, 2009).

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%%ANKI Basic Is bubble sort adaptive? Back: Yes Reference: Thomas H. Cormen et al., Introduction to Algorithms, 3rd ed (Cambridge, Mass: MIT Press, 2009).

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void swap(int i, int j, int *A) {
  int tmp = A[i];
  A[i] = A[j];
  A[j] = tmp;
}

void bubble_sort(const int n, int A[static n]) {
  bool swapped = true;
  for (int i = 0; swapped && i < n - 1; ++i) {
    swapped = false;
    for (int j = n - 1; j > i; --j) {
      if (A[j] < A[j - 1]) {
	    swap(j, j - 1, A);
	    swapped = true;
      }
    }
  }
}

%%ANKI Basic What sorting algorithm does the following demonstrate? ! Back: Bubble sort. Reference: Thomas H. Cormen et al., Introduction to Algorithms, 3rd ed (Cambridge, Mass: MIT Press, 2009).

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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 now starts at the end of the array and swaps each adjacent pair, putting the smaller of the two closer to position i. Repeating this process across all pairs from n - 1 to i + 1 ensures A[i] is the smallest element of A[i..n-1]. 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
  • Termination happens when i = n - 1. Then P implies A[0..n-2] is a sorted array of the n - 1 least elements of A. But then A[n-1] must be the greatest element of A meaning A[0..n-1], the entire array, is in sorted order.

%%ANKI Basic Given array A[0..n-1], what is bubble 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).

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%%ANKI Basic What is initialization of bubble 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).

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%%ANKI Basic What is maintenance of bubble 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).

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

[ sorted | unsorted ]

Reference: Thomas H. Cormen et al., Introduction to Algorithms, 3rd ed (Cambridge, Mass: MIT Press, 2009).

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%%ANKI Basic Which element will bubble 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).

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%%ANKI Cloze Selection sort makes fewer {swaps} than bubble sort in the average case. Reference: Thomas H. Cormen et al., Introduction to Algorithms, 3rd ed (Cambridge, Mass: MIT Press, 2009).

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References

• Thomas H. Cormen et al., Introduction to Algorithms, 3rd ed (Cambridge, Mass: MIT Press, 2009).