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notebook/notes/algorithms/sorting/heapsort.md

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title TARGET DECK FILE TAGS tags
Heapsort Obsidian::STEM algorithm::sorting data_structure::heap
algorithm
heap
sorting

Overview

Property Value
Best Case O(n)
Worst Case O(n\lg{n})
Avg. Case O(n\lg{n})
Aux. Memory O(1)
Stable No
Adaptive Yes

!heapsort.gif

%%ANKI Basic Describe HEAPSORT in a single sentence. Back: Build a heap and then repeatedly extract the max to create a sorted array. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What is HEAPSORT's best case runtime? Back: \Omega(n) Reference: “Heapsort.” In Wikipedia, April 27, 2024. https://en.wikipedia.org/w/index.php?title=Heapsort&oldid=1220986714.

END%%

%%ANKI Basic What input produces HEAPSORT's best case runtime? Back: An array of equal keys. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

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

END%%

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

END%%

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

END%%

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

END%%

%%ANKI Basic Why does HEAPSORT have O(n\lg{n}) runtime? Back: Because BUILD_MAX_HEAP runs in O(n) time and MAX_HEAPIFY_DOWN runs in O(\lg{n}) time. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What sorting algorithm does the following demonstrate? !heapsort.gif Back: HEAPSORT Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

void heapsort(int n, int H[static n]) {
  build_max_heap(n, H);
  while (n > 1) {
    swap(A, 0, --n);
    max_heapify_down(n, A, 0);
  }
}

Refer to heaps for implementations of build_max_heap and max_heapify_down.

%%ANKI Basic Which element will HEAPSORT move to sorted?

[ heap | sorted ]

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

END%%

%%ANKI Basic How are elements of the following moved in an iteration of HEAPSORT?

[ heap | sorted ]

Back: The last element of heap is swapped with the first. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Cloze {HEAPSORT} is {SELECTION_SORT} using the right data structure. Reference: “Heapsort.” In Wikipedia, April 27, 2024. https://en.wikipedia.org/w/index.php?title=Heapsort&oldid=1220986714.

END%%

%%ANKI Basic What improvement does HEAPSORT introduce to SELECTION_SORT? Back: HEAPSORT avoids linear scanning by keeping unsorted elements in a heap. Reference: “Heapsort.” In Wikipedia, April 27, 2024. https://en.wikipedia.org/w/index.php?title=Heapsort&oldid=1220986714.

END%%

%%ANKI Basic What are the two high-level steps taken in HEAPSORT? Back: Heap construction and heap extraction. Reference: “Heapsort.” In Wikipedia, April 27, 2024. https://en.wikipedia.org/w/index.php?title=Heapsort&oldid=1220986714.

END%%

Loop Invariant

Consider loop-invariant P given by

A[0:i-1] is a max-heap containing the i smallest elements of A. A[i:n-1] contains the n - i largest elements of A sorted.

We prove P maintains the requisite properties:

  • Initialization
    • A[0:n-1] is a max-heap and A[n:n-1] is empty.
  • Maintenance
    • On each iteration, A[0] is swapped with A[i-1]. A[0] is originally the largest element of the max-heap and is smaller than the elements of A[i:n-1]. Thus A[i-1:n-1] is in sorted order. Decrementing i, decrementing the heap size, and invoking MAX_HEAPIFY_DOWN on A[0] fixes the max-heap property of A[0:i-1].
  • Termination
    • We terminate when i = 1. Since A[0:1] is a max-heap, it follows A[0] < A[1]. Furthermore, A[2:n-1] are the largest n - 2 elements of A in sorted order. Thus A is sorted.

%%ANKI Basic What loop invariant does HEAPSORT maintain on A[0:i-1]? Back: A[0:i-1] is a max-heap of the i smallest elements. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What loop invariant does HEAPSORT maintain on A[i:n-1]? Back: A[i:n-1] contains the n - i largest elements sorted. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What is initialization of HEAPSORT's extraction-based loop invariant? Back: The input array is a max-heap. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What is maintenance of HEAPSORT's extraction-based loop invariant? Back: Swap the root with the last position of the heap. Heapify the new root. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

Bibliography