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Data Structures & Algorithms in Swift

Fourth Edition · iOS 15 · Swift 5.5 · Xcode 13

Before You Begin

Section 0: 6 chapters
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Section I: Introduction

Section 1: 3 chapters
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Section II: Elementary Data Structures

Section 2: 6 chapters
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Section III: Trees

Section 3: 16 chapters
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Section IV: Sorting Algorithms

Section 4: 10 chapters
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Section V: Graphs

Section 5: 11 chapters
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22. Heaps
Written by Vincent Ngo

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Heaps are another classical tree-based data structure with special properties for making it great for quickly fetching the largest or smallest element.

In this chapter, you will focus on creating and manipulating heaps. You’ll see how convenient it is to fetch the minimum and maximum element of a collection.

What is a heap?

A heap is a complete binary tree, also known as a binary heap, that can be constructed using an array.

Note: Don’t confuse these heaps with memory heaps. The term heap is sometimes confusingly used in computer science to refer to a pool of memory. Memory heaps are a different concept and not what you are studying here.

Heaps come in two flavors:

  1. Max heap, in which elements with a higher value have a higher priority.
  2. Min heap, in which elements with a lower value have a higher priority.

The heap property

A heap has an essential characteristic that must always be satisfied. This characteristic is known as the heap invariant or heap property.

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Heap applications

Some practical applications of a heap include:

Common heap operations

Open the empty starter playground for this chapter. Start by defining the following basic Heap type:

struct Heap<Element: Equatable> {

  var elements: [Element] = []
  let sort: (Element, Element) -> Bool

  init(sort: @escaping (Element, Element) -> Bool) {
    self.sort = sort
  }
}

How do you represent a heap?

Trees hold nodes that store references to their children. In the case of a binary tree, these are references to a left and right child. Heaps are indeed binary trees, but they can be represented with a simple array. This representation might seem like an unusual way to build a tree. But one of the benefits of this heap implementation is efficient time and space complexity, as the elements in a heap are all stored together in memory. You will see later on that swapping elements will play a big part in heap operations. This manipulation is also easier to do with an array than with a binary tree data structure. Take a look at how you can represent a heap using an array. Take the following binary heap:

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52 6 9 2 1 2 9 2 0 aqsan 0 2 4 2 7 2 4 cayuv 6 lucep 1 cicos 2 paxiw 1

65 9 1 9 0 0 4 2 4 uygab 9 6 2 8 9 4 1 o = 4 Cidj: 7i + 8 = 8 o = 8 Heqmc: 1a + 1 = 1 Luhb: 4u + 6 = 1 a = 1 Ziqhh: 1o + 4 = 8 a = 8 

var isEmpty: Bool {
  elements.isEmpty
}

var count: Int {
  elements.count
}

func peek() -> Element? {
  elements.first
}

func leftChildIndex(ofParentAt index: Int) -> Int {
  (2 * index) + 1
}

func rightChildIndex(ofParentAt index: Int) -> Int {
  (2 * index) + 2
}

func parentIndex(ofChildAt index: Int) -> Int {
  (index - 1) / 2
}

Removing from a heap

A basic remove operation removes the root node from the heap.

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5 4 3 2 9 4 6 1 3 6 0 1 0 1

9 5 3 9 8 2 8 7 9 0 3 4 2 5

2 1 7 8 5 2 5

Implementation of remove

Add the following method to Heap:

mutating func remove() -> Element? {
  guard !isEmpty else { // 1
    return nil
  }
  elements.swapAt(0, count - 1) // 2
  defer {
    siftDown(from: 0) // 4
  }
  return elements.removeLast() // 3
}

mutating func siftDown(from index: Int) {
  var parent = index // 1
  while true { // 2
    let left = leftChildIndex(ofParentAt: parent) // 3
    let right = rightChildIndex(ofParentAt: parent)
    var candidate = parent // 4
    if left < count && sort(elements[left], elements[candidate]) {
      candidate = left // 5
    }
    if right < count && sort(elements[right], elements[candidate]) {
      candidate = right // 6
    }
    if candidate == parent {
      return // 7
    }
    elements.swapAt(parent, candidate) // 8
    parent = candidate
  }
}

Inserting into a heap

Let’s say you insert a value of 7 to the heap below:

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4 6 1 2 1 9 4 9

1 5 1 4 6 4 4 2 7 5 9 4 7 4 6 1

6 1 4 9 0 4 4 3 7 6 1 5 6 1 9 0

1 3 3 9 6 5 8 0 8 5 5 1 6 6 6 0

Implementation of insert

Add the following method to Heap:

mutating func insert(_ element: Element) {
  elements.append(element)
  siftUp(from: elements.count - 1)
}

mutating func siftUp(from index: Int) {
  var child = index
  var parent = parentIndex(ofChildAt: child)
  while child > 0 && sort(elements[child], elements[parent]) {
    elements.swapAt(child, parent)
    child = parent
    parent = parentIndex(ofChildAt: child)
  }
}

Removing from an arbitrary index

Add the following to Heap:

mutating func remove(at index: Int) -> Element? {
  guard index < elements.count else {
    return nil // 1
  }
  if index == elements.count - 1 {
    return elements.removeLast() // 2
  } else {
    elements.swapAt(index, elements.count - 1) // 3
    defer {
      siftDown(from: index) // 5
      siftUp(from: index)
    }
    return elements.removeLast() // 4
  }
}
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Searching for an element in a heap

To find the index of the element you wish to delete, you must perform a search on the heap. Unfortunately, heaps are not designed for fast searches. With a binary search tree, you can perform a search in O(log n) time, but since heaps are built using an array, and the node ordering in an array is different, you can’t even perform a binary search.

func index(of element: Element, startingAt i: Int) -> Int? {
  if i >= count {
    return nil // 1
  }
  if sort(element, elements[i]) {
    return nil // 2
  }
  if element == elements[i] {
    return i // 3
  }
  if let j = index(of: element, startingAt: leftChildIndex(ofParentAt: i)) {
    return j // 4
  }
  if let j = index(of: element, startingAt: rightChildIndex(ofParentAt: i)) {
    return j // 5
  }
  return nil // 6
}

Building a heap

You now have all the necessary tools to represent a heap. To wrap up this chapter, you’ll build a heap from an existing array of elements and test it out. Update the initializer of Heap as follows:

init(sort: @escaping (Element, Element) -> Bool,
     elements: [Element] = []) {
  self.sort = sort
  self.elements = elements

  if !elements.isEmpty {
    for i in stride(from: elements.count / 2 - 1, through: 0, by: -1) {
      siftDown(from: i)
    }
  }
}
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Testing

Time to try it out. Add the following to your playground:

var heap = Heap(sort: >, elements: [1,12,3,4,1,6,8,7])

while !heap.isEmpty {
  print(heap.remove()!)
}

12
8
7
6
4
3
1
1

Key points

  • Here is a summary of the algorithmic complexity of the heap operations you implemented in this chapter:
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21. Binary Search Challenges
23. Heap Challenges
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