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

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

Overview

Property	Value
Best Case	\Omega(n)
Worst Case	O(n^2)
Avg. Case	O(n^2)
Aux. Memory	O(1)
Stable	Yes
Adaptive	Yes

!

%%ANKI Basic Describe INSERTION_SORT in a single sentence. Back: Repeatedly put the next record into a sorted array from right to left. 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 INSERTION_SORT's best case runtime? Back: \Omega(n) Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

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%%ANKI Basic What input value does INSERTION_SORT perform best on? Back: An already 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 INSERTION_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 input value does INSERTION_SORT perform worst on? Back: An array in reverse-sorted order. 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 INSERTION_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 INSERTION_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 INSERTION_SORT stable? 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 INSERTION_SORT adaptive? Back: Yes. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

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void insertion_sort(const int n, int A[static n]) {
  for (int i = 1; i < n; ++i) {
	int key = A[i];
	int j = i - 1;
	for (; j >= 0 && A[j] > key; --j) {
	  A[j + 1] = A[j];
	}
	A[j + 1] = key;
  }
}

%%ANKI Basic What sorting algorithm does the following demonstrate? ! Back: INSERTION_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] consists of the original A[0..i-1] elements but in sorted order.

We prove P maintains the requisite properties:

• Initialization
  • When i = 1, A[0..0] contains a single element. This trivially satisfies P.
• Maintenance
  • Suppose P holds for some 1 ≤ i < n. Then A[0..i-1] consists of the original A[0..i-1] elements but in sorted order. On iteration i + 1, the nested for loop puts A[0..i] in sorted order. At the end of the iteration, i is incremented meaning A[0..i-1] still satisfies P.
• Termination
  • The loop ends because i < n is no longer true. Then i = n. Since P holds, this means A[0..n-1], the entire array, is in sorted order.

%%ANKI Basic Given array A[0..n-1], what is INSERTION_SORT's loop invariant? Back: A[0..i-1] consists of the original A[0..i-1] elements but in sorted order.

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%%ANKI Basic What is initialization of INSERTION_SORT's loop invariant? Back: Sorting starts with an singleton 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 INSERTION_SORT's loop invariant? Back: Each iteration puts the current element into sorted order. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

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%%ANKI Cloze INSERTION_SORT makes fewer {comparisons} than SELECTION_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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Analogy

Suppose you have a shuffled deck of playing cards face-down on a table. Start by grabbing a card from the deck with your left hand. For the remainder of the cards, use your right hand to transition the topmost card to the end of your left hand. If the newly placed card isn't in sorted order, move it one position closer to the start. Repeat until it's in sorted order.

If you repeat this process for every card in the deck, your left hand will eventually contain the entire deck in sorted order.

%%ANKI Basic What analogy does Cormen et al. use to explain INSERTION_SORT? Back: Sorting a shuffled deck of playing cards. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

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%%ANKI Basic What invariant does the left hand maintain in Cormen et al.'s INSERTION_SORT analogy? Back: It contains all drawn cards in sorted order. 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 INSERTION_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 How many comparisons does INSERTION_SORT typically perform with x?

[ sorted | x : unsorted ]

Back: One plus however many elements in sorted are greater than x. 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 INSERTION_SORT move to sorted?

[ sorted | unsorted ]

Back: The first element of unsorted. 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).