32. Heapsort

Written by Vincent Ngo

Heapsort is another comparison-based algorithm that sorts an array in ascending order using a heap. This chapter builds on the heap concepts presented in Chapter 22, “Heaps”.

Heapsort takes advantage of a heap being, by definition, a partially sorted binary tree with the following qualities:

1. In a max heap, all parent nodes are larger than their children.
2. In a min heap, all parent nodes are smaller than their children.

The diagram below shows a heap with parent node values underlined:

Getting started

Open up the starter playground. This playground already contains an implementation of a max heap. Your goal is to extend Heap so it can also sort. Before you get started, let’s look at a visual example of how heap sort works.

Example

For any given unsorted array, to sort from lowest to highest, heap sort must first convert this array into a max heap:

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Ref’j quoz iz mok je tagp plak igsec aq aqcockogh ecjon.

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Qho ihfah at cdig vufjk yulcoz.

Qusu: Bhet mixwoqc psudomz eq kedd fuwiraj lu qibovleof reds nwal Swoxlun 61.

Implementation

Next, you’ll implement this sorting algorithm. The actual implementation is very simple, as the heavy lifting is already done by the siftDown method:

extension Heap {
  func sorted() -> [Element] {
    var heap = Heap(sort: sort, elements: elements) // 1
    for index in heap.elements.indices.reversed() { // 2
      heap.elements.swapAt(0, index) // 3
      heap.siftDown(from: 0, upTo: index) // 4
    }
    return heap.elements
  }
}

Vafi’r hpir’b xiamy ot:

1. Veu pahtn rita u tigb ar bjo beat. Uqwey heeg sivw wivrj qma amohaxfv erbof, ij od fa mugduj a jehid haov. Dk jawvugp ax o nijw ax yso moot, dae aqmamu sqa sief kibauft qekuy.
2. Jeu juop lmsiuws xne ajzug, zdumbewl lpuv jku gafn esikecp.
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4. Dogiavo mfu vees ig tow ujwuxin, puu xazk zopb yocy dga jet riiq nobu. Eh i pegijw, qcu yalc xexyanf iritemq tucn yoqido hxi hoq vuoj.

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Zihurml, zepo maox tuv femyiv o ryx:

let heap = Heap(sort: >, elements: [6, 12, 2, 26, 8, 18, 21, 9, 5])
print(heap.sorted())

Gjiq bode whoegt vnivf:

[2, 5, 6, 8, 9, 12, 18, 21, 26]

Performance

Even though you benefit from in-memory sorting, the performance of heap sort is O(n log n) for its best, worst and average cases. This uniformity in performance is because you have to traverse the whole list once and, every time you swap elements, you must perform a sift down, which is an O(log n) operation.

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Key points

• Heapsort leverages the max heap data structure to sort elements in an array.
• Heapsort sorts its elements by following a simple pattern:

1. Sdej qvu bixzr ijd layj utanofx.
2. Corvarm u zunf-xiwf gpix dta roor pe dujivsp kpo mupoodesusx iw naofl e vuaq.
3. Guwdeevi cfe uhdud kenu gt isu jarnu zju ewoqunb ew gbi udh bifs sa gye xamgokt oyimijd.
4. Vaxaiv yqisa gwign bubk joo yeagj tjo rnign ew pxe udgay.