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# 8.1   Heap

A <u>heap</u> is a complete binary tree that satisfies specific conditions and can be mainly categorized into two types, as shown in Figure 8-1.

- <u>min heap</u>: The value of any node $\leq$ the values of its child nodes.
- <u>max heap</u>: The value of any node $\geq$ the values of its child nodes.

![Min heap and max heap](heap.assets/min_heap_and_max_heap.png){ class="animation-figure" }

<p align="center"> Figure 8-1 &nbsp; Min heap and max heap </p>

As a special case of a complete binary tree, heaps have the following characteristics.

- The bottom layer nodes are filled from left to right, and nodes in other layers are fully filled.
- We call the root node of the binary tree the "heap top" and the bottom-rightmost node the "heap bottom."
- For max heaps (min heaps), the value of the heap top element (root node) is the largest (smallest).

## 8.1.1 &nbsp; Common Heap Operations

It should be noted that many programming languages provide a <u>priority queue</u>, an abstract data structure defined as a queue whose elements are ordered by priority.

In fact, **heaps are typically used to implement priority queues, with max heaps corresponding to priority queues where elements are dequeued in descending order**. From a usage perspective, we can regard "priority queue" and "heap" as equivalent data structures. Therefore, this book does not make a special distinction between the two and uniformly refers to them as "heap."

Common heap operations are shown in Table 8-1, and method names need to be determined based on the programming language.

<p align="center"> Table 8-1 &nbsp; Efficiency of Heap Operations </p>

<div class="center-table" markdown>

| Method name | Description                                                       | Time complexity |
| ----------- | ----------------------------------------------------------------- | --------------- |
| `push()`    | Insert an element into the heap                                   | $O(\log n)$     |
| `pop()`     | Remove the heap top element                                       | $O(\log n)$     |
| `peek()`    | Access the heap top element (max/min value for max/min heap)     | $O(1)$          |
| `size()`    | Get the number of elements in the heap                            | $O(1)$          |
| `isEmpty()` | Check if the heap is empty                                        | $O(1)$          |

</div>

In practical applications, we can directly use the heap class (or priority queue class) provided by programming languages.

Similar to "ascending order" and "descending order" in sorting algorithms, we can implement conversion between "min heap" and "max heap" by setting a `flag` or modifying the `Comparator`. The code is as follows:

=== "Python"

    ```python title="heap.py"
    # Initialize a min heap
    min_heap, flag = [], 1
    # Initialize a max heap
    max_heap, flag = [], -1

    # Python's heapq module implements a min heap by default
    # Consider negating elements before pushing them to the heap, which inverts the size relationship and thus implements a max heap
    # In this example, flag = 1 corresponds to a min heap, flag = -1 corresponds to a max heap

    # Push elements into the heap
    heapq.heappush(max_heap, flag * 1)
    heapq.heappush(max_heap, flag * 3)
    heapq.heappush(max_heap, flag * 2)
    heapq.heappush(max_heap, flag * 5)
    heapq.heappush(max_heap, flag * 4)

    # Get the heap top element
    peek: int = flag * max_heap[0] # 5

    # Remove the heap top element
    # The removed elements will form a descending sequence
    val = flag * heapq.heappop(max_heap) # 5
    val = flag * heapq.heappop(max_heap) # 4
    val = flag * heapq.heappop(max_heap) # 3
    val = flag * heapq.heappop(max_heap) # 2
    val = flag * heapq.heappop(max_heap) # 1

    # Get the heap size
    size: int = len(max_heap)

    # Check if the heap is empty
    is_empty: bool = not max_heap

    # Build a heap from an input list
    min_heap: list[int] = [1, 3, 2, 5, 4]
    heapq.heapify(min_heap)
    ```

=== "C++"

    ```cpp title="heap.cpp"
    /* Initialize a heap */
    // Initialize a min heap
    priority_queue<int, vector<int>, greater<int>> minHeap;
    // Initialize a max heap
    priority_queue<int, vector<int>, less<int>> maxHeap;

    /* Push elements into the heap */
    maxHeap.push(1);
    maxHeap.push(3);
    maxHeap.push(2);
    maxHeap.push(5);
    maxHeap.push(4);

    /* Get the heap top element */
    int peek = maxHeap.top(); // 5

    /* Remove the heap top element */
    // The removed elements will form a descending sequence
    maxHeap.pop(); // 5
    maxHeap.pop(); // 4
    maxHeap.pop(); // 3
    maxHeap.pop(); // 2
    maxHeap.pop(); // 1

    /* Get the heap size */
    int size = maxHeap.size();

    /* Check if the heap is empty */
    bool isEmpty = maxHeap.empty();

    /* Build a heap from an input list */
    vector<int> input{1, 3, 2, 5, 4};
    priority_queue<int, vector<int>, greater<int>> minHeap(input.begin(), input.end());
    ```

=== "Java"

    ```java title="heap.java"
    /* Initialize a heap */
    // Initialize a min heap
    Queue<Integer> minHeap = new PriorityQueue<>();
    // Initialize a max heap (use lambda expression to modify Comparator)
    Queue<Integer> maxHeap = new PriorityQueue<>((a, b) -> b - a);

    /* Push elements into the heap */
    maxHeap.offer(1);
    maxHeap.offer(3);
    maxHeap.offer(2);
    maxHeap.offer(5);
    maxHeap.offer(4);

    /* Get the heap top element */
    int peek = maxHeap.peek(); // 5

    /* Remove the heap top element */
    // The removed elements will form a descending sequence
    peek = maxHeap.poll(); // 5
    peek = maxHeap.poll(); // 4
    peek = maxHeap.poll(); // 3
    peek = maxHeap.poll(); // 2
    peek = maxHeap.poll(); // 1

    /* Get the heap size */
    int size = maxHeap.size();

    /* Check if the heap is empty */
    boolean isEmpty = maxHeap.isEmpty();

    /* Build a heap from an input list */
    minHeap = new PriorityQueue<>(Arrays.asList(1, 3, 2, 5, 4));
    ```

=== "C#"

    ```csharp title="heap.cs"
    /* Initialize a heap */
    // Initialize a min heap
    PriorityQueue<int, int> minHeap = new();
    // Initialize a max heap (use lambda expression to modify Comparer)
    PriorityQueue<int, int> maxHeap = new(Comparer<int>.Create((x, y) => y.CompareTo(x)));

    /* Push elements into the heap */
    maxHeap.Enqueue(1, 1);
    maxHeap.Enqueue(3, 3);
    maxHeap.Enqueue(2, 2);
    maxHeap.Enqueue(5, 5);
    maxHeap.Enqueue(4, 4);

    /* Get the heap top element */
    int peek = maxHeap.Peek();//5

    /* Remove the heap top element */
    // The removed elements will form a descending sequence
    peek = maxHeap.Dequeue();  // 5
    peek = maxHeap.Dequeue();  // 4
    peek = maxHeap.Dequeue();  // 3
    peek = maxHeap.Dequeue();  // 2
    peek = maxHeap.Dequeue();  // 1

    /* Get the heap size */
    int size = maxHeap.Count;

    /* Check if the heap is empty */
    bool isEmpty = maxHeap.Count == 0;

    /* Build a heap from an input list */
    minHeap = new PriorityQueue<int, int>([(1, 1), (3, 3), (2, 2), (5, 5), (4, 4)]);
    ```

=== "Go"

    ```go title="heap.go"
    // In Go, we can construct a max heap of integers by implementing heap.Interface
    // Implementing heap.Interface also requires implementing sort.Interface
    type intHeap []any

    // Push implements the heap.Interface method for pushing an element into the heap
    func (h *intHeap) Push(x any) {
        // Push and Pop use pointer receiver as parameters
        // because they not only adjust the slice contents but also modify the slice length
        *h = append(*h, x.(int))
    }

    // Pop implements the heap.Interface method for popping the heap top element
    func (h *intHeap) Pop() any {
        // The element to be removed is stored at the end
        last := (*h)[len(*h)-1]
        *h = (*h)[:len(*h)-1]
        return last
    }

    // Len is a sort.Interface method
    func (h *intHeap) Len() int {
        return len(*h)
    }

    // Less is a sort.Interface method
    func (h *intHeap) Less(i, j int) bool {
        // To implement a min heap, change this to a less-than sign
        return (*h)[i].(int) > (*h)[j].(int)
    }

    // Swap is a sort.Interface method
    func (h *intHeap) Swap(i, j int) {
        (*h)[i], (*h)[j] = (*h)[j], (*h)[i]
    }

    // Top gets the heap top element
    func (h *intHeap) Top() any {
        return (*h)[0]
    }

    /* Driver Code */
    func TestHeap(t *testing.T) {
        /* Initialize a heap */
        // Initialize a max heap
        maxHeap := &intHeap{}
        heap.Init(maxHeap)
        /* Push elements into the heap */
        // Call heap.Interface methods to add elements
        heap.Push(maxHeap, 1)
        heap.Push(maxHeap, 3)
        heap.Push(maxHeap, 2)
        heap.Push(maxHeap, 4)
        heap.Push(maxHeap, 5)

        /* Get the heap top element */
        top := maxHeap.Top()
        fmt.Printf("Heap top element is %d\n", top)

        /* Remove the heap top element */
        // Call heap.Interface methods to remove elements
        heap.Pop(maxHeap) // 5
        heap.Pop(maxHeap) // 4
        heap.Pop(maxHeap) // 3
        heap.Pop(maxHeap) // 2
        heap.Pop(maxHeap) // 1

        /* Get the heap size */
        size := len(*maxHeap)
        fmt.Printf("Number of heap elements is %d\n", size)

        /* Check if the heap is empty */
        isEmpty := len(*maxHeap) == 0
        fmt.Printf("Is the heap empty? %t\n", isEmpty)
    }
    ```

=== "Swift"

    ```swift title="heap.swift"
    /* Initialize a heap */
    // Swift's Heap type supports both max heaps and min heaps, and requires importing swift-collections
    var heap = Heap<Int>()

    /* Push elements into the heap */
    heap.insert(1)
    heap.insert(3)
    heap.insert(2)
    heap.insert(5)
    heap.insert(4)

    /* Get the heap top element */
    var peek = heap.max()!

    /* Remove the heap top element */
    peek = heap.removeMax() // 5
    peek = heap.removeMax() // 4
    peek = heap.removeMax() // 3
    peek = heap.removeMax() // 2
    peek = heap.removeMax() // 1

    /* Get the heap size */
    let size = heap.count

    /* Check if the heap is empty */
    let isEmpty = heap.isEmpty

    /* Build a heap from an input list */
    let heap2 = Heap([1, 3, 2, 5, 4])
    ```

=== "JS"

    ```javascript title="heap.js"
    // JavaScript does not provide a built-in Heap class
    ```

=== "TS"

    ```typescript title="heap.ts"
    // TypeScript does not provide a built-in Heap class
    ```

=== "Dart"

    ```dart title="heap.dart"
    // Dart does not provide a built-in Heap class
    ```

=== "Rust"

    ```rust title="heap.rs"
    use std::collections::BinaryHeap;
    use std::cmp::Reverse;

    /* Initialize a heap */
    // Initialize a min heap
    let mut min_heap = BinaryHeap::<Reverse<i32>>::new();
    // Initialize a max heap
    let mut max_heap = BinaryHeap::new();

    /* Push elements into the heap */
    max_heap.push(1);
    max_heap.push(3);
    max_heap.push(2);
    max_heap.push(5);
    max_heap.push(4);

    /* Get the heap top element */
    let peek = max_heap.peek().unwrap();  // 5

    /* Remove the heap top element */
    // The removed elements will form a descending sequence
    let peek = max_heap.pop().unwrap();   // 5
    let peek = max_heap.pop().unwrap();   // 4
    let peek = max_heap.pop().unwrap();   // 3
    let peek = max_heap.pop().unwrap();   // 2
    let peek = max_heap.pop().unwrap();   // 1

    /* Get the heap size */
    let size = max_heap.len();

    /* Check if the heap is empty */
    let is_empty = max_heap.is_empty();

    /* Build a heap from an input list */
    let min_heap = BinaryHeap::from(vec![Reverse(1), Reverse(3), Reverse(2), Reverse(5), Reverse(4)]);
    ```

=== "C"

    ```c title="heap.c"
    // C does not provide a built-in Heap class
    ```

=== "Kotlin"

    ```kotlin title="heap.kt"
    /* Initialize a heap */
    // Initialize a min heap
    var minHeap = PriorityQueue<Int>()
    // Initialize a max heap (use lambda expression to modify Comparator)
    val maxHeap = PriorityQueue { a: Int, b: Int -> b - a }

    /* Push elements into the heap */
    maxHeap.offer(1)
    maxHeap.offer(3)
    maxHeap.offer(2)
    maxHeap.offer(5)
    maxHeap.offer(4)

    /* Get the heap top element */
    var peek = maxHeap.peek() // 5

    /* Remove the heap top element */
    // The removed elements will form a descending sequence
    peek = maxHeap.poll() // 5
    peek = maxHeap.poll() // 4
    peek = maxHeap.poll() // 3
    peek = maxHeap.poll() // 2
    peek = maxHeap.poll() // 1

    /* Get the heap size */
    val size = maxHeap.size

    /* Check if the heap is empty */
    val isEmpty = maxHeap.isEmpty()

    /* Build a heap from an input list */
    minHeap = PriorityQueue(mutableListOf(1, 3, 2, 5, 4))
    ```

=== "Ruby"

    ```ruby title="heap.rb"
    # Ruby does not provide a built-in Heap class
    ```

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## 8.1.2 &nbsp; Implementation of the Heap

The following implementation is for a max heap. To convert it to a min heap, simply reverse all comparison logic related to ordering (for example, replace $\geq$ with $\leq$). Interested readers are encouraged to implement this on their own.

### 1. &nbsp; Heap Storage and Representation

As mentioned in the "Binary Tree" chapter, complete binary trees are well-suited for array representation. Since heaps are a type of complete binary tree, **we will use arrays to store heaps**.

When representing a binary tree with an array, elements represent node values, and indexes represent node positions in the binary tree. **Parent-child relationships are represented through index-mapping formulas**.

As shown in Figure 8-2, given an index $i$, the index of its left child is $2i + 1$, the index of its right child is $2i + 2$, and the index of its parent is $(i - 1) / 2$ (floor division). When an index is out of bounds, it indicates a null node or that the node does not exist.

![Representation and storage of heaps](heap.assets/representation_of_heap.png){ class="animation-figure" }

<p align="center"> Figure 8-2 &nbsp; Representation and storage of heaps </p>

We can encapsulate the index mapping formula into functions for convenient subsequent use:

=== "Python"

    ```python title="my_heap.py"
    def left(self, i: int) -> int:
        """Get index of left child node"""
        return 2 * i + 1

    def right(self, i: int) -> int:
        """Get index of right child node"""
        return 2 * i + 2

    def parent(self, i: int) -> int:
        """Get index of parent node"""
        return (i - 1) // 2  # Floor division
    ```

=== "C++"

    ```cpp title="my_heap.cpp"
    /* Get index of left child node */
    int left(int i) {
        return 2 * i + 1;
    }

    /* Get index of right child node */
    int right(int i) {
        return 2 * i + 2;
    }

    /* Get index of parent node */
    int parent(int i) {
        return (i - 1) / 2; // Floor division
    }
    ```

=== "Java"

    ```java title="my_heap.java"
    /* Get index of left child node */
    int left(int i) {
        return 2 * i + 1;
    }

    /* Get index of right child node */
    int right(int i) {
        return 2 * i + 2;
    }

    /* Get index of parent node */
    int parent(int i) {
        return (i - 1) / 2; // Floor division
    }
    ```

=== "C#"

    ```csharp title="my_heap.cs"
    /* Get index of left child node */
    int Left(int i) {
        return 2 * i + 1;
    }

    /* Get index of right child node */
    int Right(int i) {
        return 2 * i + 2;
    }

    /* Get index of parent node */
    int Parent(int i) {
        return (i - 1) / 2; // Floor division
    }
    ```

=== "Go"

    ```go title="my_heap.go"
    /* Get index of left child node */
    func (h *maxHeap) left(i int) int {
        return 2*i + 1
    }

    /* Get index of right child node */
    func (h *maxHeap) right(i int) int {
        return 2*i + 2
    }

    /* Get index of parent node */
    func (h *maxHeap) parent(i int) int {
        // Floor division
        return (i - 1) / 2
    }
    ```

=== "Swift"

    ```swift title="my_heap.swift"
    /* Get index of left child node */
    func left(i: Int) -> Int {
        2 * i + 1
    }

    /* Get index of right child node */
    func right(i: Int) -> Int {
        2 * i + 2
    }

    /* Get index of parent node */
    func parent(i: Int) -> Int {
        (i - 1) / 2 // Floor division
    }
    ```

=== "JS"

    ```javascript title="my_heap.js"
    /* Get index of left child node */
    #left(i) {
        return 2 * i + 1;
    }

    /* Get index of right child node */
    #right(i) {
        return 2 * i + 2;
    }

    /* Get index of parent node */
    #parent(i) {
        return Math.floor((i - 1) / 2); // Floor division
    }
    ```

=== "TS"

    ```typescript title="my_heap.ts"
    /* Get index of left child node */
    left(i: number): number {
        return 2 * i + 1;
    }

    /* Get index of right child node */
    right(i: number): number {
        return 2 * i + 2;
    }

    /* Get index of parent node */
    parent(i: number): number {
        return Math.floor((i - 1) / 2); // Floor division
    }
    ```

=== "Dart"

    ```dart title="my_heap.dart"
    /* Get index of left child node */
    int _left(int i) {
      return 2 * i + 1;
    }

    /* Get index of right child node */
    int _right(int i) {
      return 2 * i + 2;
    }

    /* Get index of parent node */
    int _parent(int i) {
      return (i - 1) ~/ 2; // Floor division
    }
    ```

=== "Rust"

    ```rust title="my_heap.rs"
    /* Get index of left child node */
    fn left(i: usize) -> usize {
        2 * i + 1
    }

    /* Get index of right child node */
    fn right(i: usize) -> usize {
        2 * i + 2
    }

    /* Get index of parent node */
    fn parent(i: usize) -> usize {
        (i - 1) / 2 // Floor division
    }
    ```

=== "C"

    ```c title="my_heap.c"
    /* Get index of left child node */
    int left(MaxHeap *maxHeap, int i) {
        return 2 * i + 1;
    }

    /* Get index of right child node */
    int right(MaxHeap *maxHeap, int i) {
        return 2 * i + 2;
    }

    /* Get index of parent node */
    int parent(MaxHeap *maxHeap, int i) {
        return (i - 1) / 2; // Round down
    }
    ```

=== "Kotlin"

    ```kotlin title="my_heap.kt"
    /* Get index of left child node */
    fun left(i: Int): Int {
        return 2 * i + 1
    }

    /* Get index of right child node */
    fun right(i: Int): Int {
        return 2 * i + 2
    }

    /* Get index of parent node */
    fun parent(i: Int): Int {
        return (i - 1) / 2 // Floor division
    }
    ```

=== "Ruby"

    ```ruby title="my_heap.rb"
    ### Get left child index ###
    def left(i)
      2 * i + 1
    end

    ### Get right child index ###
    def right(i)
      2 * i + 2
    end

    ### Get parent node index ###
    def parent(i)
      (i - 1) / 2     # Floor division
    end
    ```

### 2. &nbsp; Accessing the Heap Top Element

The heap top element is the root node of the binary tree, which is also the first element of the list:

=== "Python"

    ```python title="my_heap.py"
    def peek(self) -> int:
        """Access top element"""
        return self.max_heap[0]
    ```

=== "C++"

    ```cpp title="my_heap.cpp"
    /* Access top element */
    int peek() {
        return maxHeap[0];
    }
    ```

=== "Java"

    ```java title="my_heap.java"
    /* Access top element */
    int peek() {
        return maxHeap.get(0);
    }
    ```

=== "C#"

    ```csharp title="my_heap.cs"
    /* Access top element */
    int Peek() {
        return maxHeap[0];
    }
    ```

=== "Go"

    ```go title="my_heap.go"
    /* Access top element */
    func (h *maxHeap) peek() any {
        return h.data[0]
    }
    ```

=== "Swift"

    ```swift title="my_heap.swift"
    /* Access top element */
    func peek() -> Int {
        maxHeap[0]
    }
    ```

=== "JS"

    ```javascript title="my_heap.js"
    /* Access top element */
    peek() {
        return this.#maxHeap[0];
    }
    ```

=== "TS"

    ```typescript title="my_heap.ts"
    /* Access top element */
    peek(): number {
        return this.maxHeap[0];
    }
    ```

=== "Dart"

    ```dart title="my_heap.dart"
    /* Access top element */
    int peek() {
      return _maxHeap[0];
    }
    ```

=== "Rust"

    ```rust title="my_heap.rs"
    /* Access top element */
    fn peek(&self) -> Option<i32> {
        self.max_heap.first().copied()
    }
    ```

=== "C"

    ```c title="my_heap.c"
    /* Access top element */
    int peek(MaxHeap *maxHeap) {
        return maxHeap->data[0];
    }
    ```

=== "Kotlin"

    ```kotlin title="my_heap.kt"
    /* Access top element */
    fun peek(): Int {
        return maxHeap[0]
    }
    ```

=== "Ruby"

    ```ruby title="my_heap.rb"
    ### Access heap top element ###
    def peek
      @max_heap[0]
    end
    ```

### 3. &nbsp; Inserting an Element Into the Heap

Given an element `val`, we first add it to the bottom of the heap. After insertion, because `val` may be larger than other elements in the heap, the heap property may be violated. **Therefore, we need to restore the heap property along the path from the inserted node to the root**. This operation is called <u>heapify</u>.

Starting from the inserted node, **perform heapify from bottom to top**. As shown in Figure 8-3, we compare the inserted node with its parent, and if the inserted node is larger, we swap them. We continue this process from bottom to top until we move past the root or reach a node that no longer needs to be swapped.

=== "<1>"
    ![Steps of inserting an element into the heap](heap.assets/heap_push_step1.png){ class="animation-figure" }

=== "<2>"
    ![heap_push_step2](heap.assets/heap_push_step2.png){ class="animation-figure" }

=== "<3>"
    ![heap_push_step3](heap.assets/heap_push_step3.png){ class="animation-figure" }

=== "<4>"
    ![heap_push_step4](heap.assets/heap_push_step4.png){ class="animation-figure" }

=== "<5>"
    ![heap_push_step5](heap.assets/heap_push_step5.png){ class="animation-figure" }

=== "<6>"
    ![heap_push_step6](heap.assets/heap_push_step6.png){ class="animation-figure" }

=== "<7>"
    ![heap_push_step7](heap.assets/heap_push_step7.png){ class="animation-figure" }

=== "<8>"
    ![heap_push_step8](heap.assets/heap_push_step8.png){ class="animation-figure" }

=== "<9>"
    ![heap_push_step9](heap.assets/heap_push_step9.png){ class="animation-figure" }

<p align="center"> Figure 8-3 &nbsp; Steps of inserting an element into the heap </p>

Given a total of $n$ nodes, the tree height is $O(\log n)$. Thus, the number of loop iterations in the heapify operation is at most $O(\log n)$, **making the time complexity of the element insertion operation $O(\log n)$**. The code is as follows:

=== "Python"

    ```python title="my_heap.py"
    def push(self, val: int):
        """Element enters heap"""
        # Add node
        self.max_heap.append(val)
        # Heapify from bottom to top
        self.sift_up(self.size() - 1)

    def sift_up(self, i: int):
        """Starting from node i, heapify from bottom to top"""
        while True:
            # Get parent node of node i
            p = self.parent(i)
            # When "crossing root node" or "node needs no repair", end heapify
            if p < 0 or self.max_heap[i] <= self.max_heap[p]:
                break
            # Swap two nodes
            self.swap(i, p)
            # Loop upward heapify
            i = p
    ```

=== "C++"

    ```cpp title="my_heap.cpp"
    /* Element enters heap */
    void push(int val) {
        // Add node
        maxHeap.push_back(val);
        // Heapify from bottom to top
        siftUp(size() - 1);
    }

    /* Starting from node i, heapify from bottom to top */
    void siftUp(int i) {
        while (true) {
            // Get parent node of node i
            int p = parent(i);
            // When "crossing root node" or "node needs no repair", end heapify
            if (p < 0 || maxHeap[i] <= maxHeap[p])
                break;
            // Swap two nodes
            swap(maxHeap[i], maxHeap[p]);
            // Loop upward heapify
            i = p;
        }
    }
    ```

=== "Java"

    ```java title="my_heap.java"
    /* Element enters heap */
    void push(int val) {
        // Add node
        maxHeap.add(val);
        // Heapify from bottom to top
        siftUp(size() - 1);
    }

    /* Starting from node i, heapify from bottom to top */
    void siftUp(int i) {
        while (true) {
            // Get parent node of node i
            int p = parent(i);
            // When "crossing root node" or "node needs no repair", end heapify
            if (p < 0 || maxHeap.get(i) <= maxHeap.get(p))
                break;
            // Swap two nodes
            swap(i, p);
            // Loop upward heapify
            i = p;
        }
    }
    ```

=== "C#"

    ```csharp title="my_heap.cs"
    /* Element enters heap */
    void Push(int val) {
        // Add node
        maxHeap.Add(val);
        // Heapify from bottom to top
        SiftUp(Size() - 1);
    }

    /* Starting from node i, heapify from bottom to top */
    void SiftUp(int i) {
        while (true) {
            // Get parent node of node i
            int p = Parent(i);
            // If 'past root node' or 'node needs no repair', end heapify
            if (p < 0 || maxHeap[i] <= maxHeap[p])
                break;
            // Swap two nodes
            Swap(i, p);
            // Loop upward heapify
            i = p;
        }
    }
    ```

=== "Go"

    ```go title="my_heap.go"
    /* Element enters heap */
    func (h *maxHeap) push(val any) {
        // Add node
        h.data = append(h.data, val)
        // Heapify from bottom to top
        h.siftUp(len(h.data) - 1)
    }

    /* Starting from node i, heapify from bottom to top */
    func (h *maxHeap) siftUp(i int) {
        for true {
            // Get parent node of node i
            p := h.parent(i)
            // When "crossing root node" or "node needs no repair", end heapify
            if p < 0 || h.data[i].(int) <= h.data[p].(int) {
                break
            }
            // Swap two nodes
            h.swap(i, p)
            // Loop upward heapify
            i = p
        }
    }
    ```

=== "Swift"

    ```swift title="my_heap.swift"
    /* Element enters heap */
    func push(val: Int) {
        // Add node
        maxHeap.append(val)
        // Heapify from bottom to top
        siftUp(i: size() - 1)
    }

    /* Starting from node i, heapify from bottom to top */
    func siftUp(i: Int) {
        var i = i
        while true {
            // Get parent node of node i
            let p = parent(i: i)
            // When "crossing root node" or "node needs no repair", end heapify
            if p < 0 || maxHeap[i] <= maxHeap[p] {
                break
            }
            // Swap two nodes
            swap(i: i, j: p)
            // Loop upward heapify
            i = p
        }
    }
    ```

=== "JS"

    ```javascript title="my_heap.js"
    /* Element enters heap */
    push(val) {
        // Add node
        this.#maxHeap.push(val);
        // Heapify from bottom to top
        this.#siftUp(this.size() - 1);
    }

    /* Starting from node i, heapify from bottom to top */
    #siftUp(i) {
        while (true) {
            // Get parent node of node i
            const p = this.#parent(i);
            // When "crossing root node" or "node needs no repair", end heapify
            if (p < 0 || this.#maxHeap[i] <= this.#maxHeap[p]) break;
            // Swap two nodes
            this.#swap(i, p);
            // Loop upward heapify
            i = p;
        }
    }
    ```

=== "TS"

    ```typescript title="my_heap.ts"
    /* Element enters heap */
    push(val: number): void {
        // Add node
        this.maxHeap.push(val);
        // Heapify from bottom to top
        this.siftUp(this.size() - 1);
    }

    /* Starting from node i, heapify from bottom to top */
    siftUp(i: number): void {
        while (true) {
            // Get parent node of node i
            const p = this.parent(i);
            // When "crossing root node" or "node needs no repair", end heapify
            if (p < 0 || this.maxHeap[i] <= this.maxHeap[p]) break;
            // Swap two nodes
            this.swap(i, p);
            // Loop upward heapify
            i = p;
        }
    }
    ```

=== "Dart"

    ```dart title="my_heap.dart"
    /* Element enters heap */
    void push(int val) {
      // Add node
      _maxHeap.add(val);
      // Heapify from bottom to top
      siftUp(size() - 1);
    }

    /* Starting from node i, heapify from bottom to top */
    void siftUp(int i) {
      while (true) {
        // Get parent node of node i
        int p = _parent(i);
        // When "crossing root node" or "node needs no repair", end heapify
        if (p < 0 || _maxHeap[i] <= _maxHeap[p]) {
          break;
        }
        // Swap two nodes
        _swap(i, p);
        // Loop upward heapify
        i = p;
      }
    }
    ```

=== "Rust"

    ```rust title="my_heap.rs"
    /* Element enters heap */
    fn push(&mut self, val: i32) {
        // Add node
        self.max_heap.push(val);
        // Heapify from bottom to top
        self.sift_up(self.size() - 1);
    }

    /* Starting from node i, heapify from bottom to top */
    fn sift_up(&mut self, mut i: usize) {
        loop {
            // Node i is already the heap root, end heapification
            if i == 0 {
                break;
            }
            // Get parent node of node i
            let p = Self::parent(i);
            // When "node needs no repair", end heapification
            if self.max_heap[i] <= self.max_heap[p] {
                break;
            }
            // Swap two nodes
            self.swap(i, p);
            // Loop upward heapify
            i = p;
        }
    }
    ```

=== "C"

    ```c title="my_heap.c"
    /* Element enters heap */
    void push(MaxHeap *maxHeap, int val) {
        // By default, should not add this many nodes
        if (maxHeap->size == MAX_SIZE) {
            printf("heap is full!");
            return;
        }
        // Add node
        maxHeap->data[maxHeap->size] = val;
        maxHeap->size++;

        // Heapify from bottom to top
        siftUp(maxHeap, maxHeap->size - 1);
    }

    /* Starting from node i, heapify from bottom to top */
    void siftUp(MaxHeap *maxHeap, int i) {
        while (true) {
            // Get parent node of node i
            int p = parent(maxHeap, i);
            // When "crossing root node" or "node needs no repair", end heapify
            if (p < 0 || maxHeap->data[i] <= maxHeap->data[p]) {
                break;
            }
            // Swap two nodes
            swap(maxHeap, i, p);
            // Loop upward heapify
            i = p;
        }
    }
    ```

=== "Kotlin"

    ```kotlin title="my_heap.kt"
    /* Element enters heap */
    fun push(_val: Int) {
        // Add node
        maxHeap.add(_val)
        // Heapify from bottom to top
        siftUp(size() - 1)
    }

    /* Starting from node i, heapify from bottom to top */
    fun siftUp(it: Int) {
        // Kotlin function parameters are immutable, so create temporary variable
        var i = it
        while (true) {
            // Get parent node of node i
            val p = parent(i)
            // When "crossing root node" or "node needs no repair", end heapify
            if (p < 0 || maxHeap[i] <= maxHeap[p]) break
            // Swap two nodes
            swap(i, p)
            // Loop upward heapify
            i = p
        }
    }
    ```

=== "Ruby"

    ```ruby title="my_heap.rb"
    ### Push element to heap ###
    def push(val)
      # Add node
      @max_heap << val
      # Heapify from bottom to top
      sift_up(size - 1)
    end

    ### Heapify from node i, bottom to top ###
    def sift_up(i)
      loop do
        # Get parent node of node i
        p = parent(i)
        # When "crossing root node" or "node needs no repair", end heapify
        break if p < 0 || @max_heap[i] <= @max_heap[p]
        # Swap two nodes
        swap(i, p)
        # Loop upward heapify
        i = p
      end
    end
    ```

### 4. &nbsp; Removing the Heap Top Element

The heap top element is the root node of the binary tree, which is the first element of the list. If we directly remove the first element from the list, all node indexes in the binary tree would change, making subsequent repair with heapify difficult. To minimize changes in element indexes, we use the following steps.

1. Swap the heap top element with the heap bottom element (swap the root node with the rightmost leaf node).
2. After swapping, remove the heap bottom from the list (note that since we've swapped, we're actually removing the original heap top element).
3. Starting from the root node, **perform heapify from top to bottom**.

As shown in Figure 8-4, **the direction of "top-to-bottom heapify" is opposite to "bottom-to-top heapify"**. We compare the root node's value with its two children and swap it with the largest child. Then loop this operation until we pass a leaf node or encounter a node that doesn't need swapping.

=== "<1>"
    ![Steps of removing the heap top element](heap.assets/heap_pop_step1.png){ class="animation-figure" }

=== "<2>"
    ![heap_pop_step2](heap.assets/heap_pop_step2.png){ class="animation-figure" }

=== "<3>"
    ![heap_pop_step3](heap.assets/heap_pop_step3.png){ class="animation-figure" }

=== "<4>"
    ![heap_pop_step4](heap.assets/heap_pop_step4.png){ class="animation-figure" }

=== "<5>"
    ![heap_pop_step5](heap.assets/heap_pop_step5.png){ class="animation-figure" }

=== "<6>"
    ![heap_pop_step6](heap.assets/heap_pop_step6.png){ class="animation-figure" }

=== "<7>"
    ![heap_pop_step7](heap.assets/heap_pop_step7.png){ class="animation-figure" }

=== "<8>"
    ![heap_pop_step8](heap.assets/heap_pop_step8.png){ class="animation-figure" }

=== "<9>"
    ![heap_pop_step9](heap.assets/heap_pop_step9.png){ class="animation-figure" }

=== "<10>"
    ![heap_pop_step10](heap.assets/heap_pop_step10.png){ class="animation-figure" }

<p align="center"> Figure 8-4 &nbsp; Steps of removing the heap top element </p>

Similar to the element insertion operation, the time complexity of the heap top element removal operation is also $O(\log n)$. The code is as follows:

=== "Python"

    ```python title="my_heap.py"
    def pop(self) -> int:
        """Element exits heap"""
        # Handle empty case
        if self.is_empty():
            raise IndexError("Heap is empty")
        # Swap root node with rightmost leaf node (swap first element with last element)
        self.swap(0, self.size() - 1)
        # Delete node
        val = self.max_heap.pop()
        # Heapify from top to bottom
        self.sift_down(0)
        # Return top element
        return val

    def sift_down(self, i: int):
        """Starting from node i, heapify from top to bottom"""
        while True:
            # Find node with largest value among i, l, r, denoted as ma
            l, r, ma = self.left(i), self.right(i), i
            if l < self.size() and self.max_heap[l] > self.max_heap[ma]:
                ma = l
            if r < self.size() and self.max_heap[r] > self.max_heap[ma]:
                ma = r
            # If node i is largest or indices l, r are out of bounds, no need to continue heapify, break
            if ma == i:
                break
            # Swap two nodes
            self.swap(i, ma)
            # Loop downward heapify
            i = ma
    ```

=== "C++"

    ```cpp title="my_heap.cpp"
    /* Element exits heap */
    void pop() {
        // Handle empty case
        if (isEmpty()) {
            throw out_of_range("Heap is empty");
        }
        // Delete node
        swap(maxHeap[0], maxHeap[size() - 1]);
        // Remove node
        maxHeap.pop_back();
        // Return top element
        siftDown(0);
    }

    /* Starting from node i, heapify from top to bottom */
    void siftDown(int i) {
        while (true) {
            // If node i is largest or indices l, r are out of bounds, no need to continue heapify, break
            int l = left(i), r = right(i), ma = i;
            if (l < size() && maxHeap[l] > maxHeap[ma])
                ma = l;
            if (r < size() && maxHeap[r] > maxHeap[ma])
                ma = r;
            // Swap two nodes
            if (ma == i)
                break;
            swap(maxHeap[i], maxHeap[ma]);
            // Loop downwards heapification
            i = ma;
        }
    }
    ```

=== "Java"

    ```java title="my_heap.java"
    /* Element exits heap */
    int pop() {
        // Handle empty case
        if (isEmpty())
            throw new IndexOutOfBoundsException();
        // Delete node
        swap(0, size() - 1);
        // Remove node
        int val = maxHeap.remove(size() - 1);
        // Return top element
        siftDown(0);
        // Return heap top element
        return val;
    }

    /* Starting from node i, heapify from top to bottom */
    void siftDown(int i) {
        while (true) {
            // If node i is largest or indices l, r are out of bounds, no need to continue heapify, break
            int l = left(i), r = right(i), ma = i;
            if (l < size() && maxHeap.get(l) > maxHeap.get(ma))
                ma = l;
            if (r < size() && maxHeap.get(r) > maxHeap.get(ma))
                ma = r;
            // Swap two nodes
            if (ma == i)
                break;
            // Swap two nodes
            swap(i, ma);
            // Loop downwards heapification
            i = ma;
        }
    }
    ```

=== "C#"

    ```csharp title="my_heap.cs"
    /* Element exits heap */
    int Pop() {
        // Handle empty case
        if (IsEmpty())
            throw new IndexOutOfRangeException();
        // Delete node
        Swap(0, Size() - 1);
        // Remove node
        int val = maxHeap.Last();
        maxHeap.RemoveAt(Size() - 1);
        // Return top element
        SiftDown(0);
        // Return heap top element
        return val;
    }

    /* Starting from node i, heapify from top to bottom */
    void SiftDown(int i) {
        while (true) {
            // If node i is largest or indices l, r are out of bounds, no need to continue heapify, break
            int l = Left(i), r = Right(i), ma = i;
            if (l < Size() && maxHeap[l] > maxHeap[ma])
                ma = l;
            if (r < Size() && maxHeap[r] > maxHeap[ma])
                ma = r;
            // If 'node i is largest' or 'past leaf node', end heapify
            if (ma == i) break;
            // Swap two nodes
            Swap(i, ma);
            // Loop downwards heapification
            i = ma;
        }
    }
    ```

=== "Go"

    ```go title="my_heap.go"
    /* Element exits heap */
    func (h *maxHeap) pop() any {
        // Handle empty case
        if h.isEmpty() {
            fmt.Println("error")
            return nil
        }
        // Delete node
        h.swap(0, h.size()-1)
        // Remove node
        val := h.data[len(h.data)-1]
        h.data = h.data[:len(h.data)-1]
        // Return top element
        h.siftDown(0)

        // Return heap top element
        return val
    }

    /* Starting from node i, heapify from top to bottom */
    func (h *maxHeap) siftDown(i int) {
        for true {
            // Find node with maximum value among nodes i, l, r, denoted as max
            l, r, max := h.left(i), h.right(i), i
            if l < h.size() && h.data[l].(int) > h.data[max].(int) {
                max = l
            }
            if r < h.size() && h.data[r].(int) > h.data[max].(int) {
                max = r
            }
            // Swap two nodes
            if max == i {
                break
            }
            // Swap two nodes
            h.swap(i, max)
            // Loop downwards heapification
            i = max
        }
    }
    ```

=== "Swift"

    ```swift title="my_heap.swift"
    /* Element exits heap */
    func pop() -> Int {
        // Handle empty case
        if isEmpty() {
            fatalError("Heap is empty")
        }
        // Delete node
        swap(i: 0, j: size() - 1)
        // Remove node
        let val = maxHeap.remove(at: size() - 1)
        // Return top element
        siftDown(i: 0)
        // Return heap top element
        return val
    }

    /* Starting from node i, heapify from top to bottom */
    func siftDown(i: Int) {
        var i = i
        while true {
            // If node i is largest or indices l, r are out of bounds, no need to continue heapify, break
            let l = left(i: i)
            let r = right(i: i)
            var ma = i
            if l < size(), maxHeap[l] > maxHeap[ma] {
                ma = l
            }
            if r < size(), maxHeap[r] > maxHeap[ma] {
                ma = r
            }
            // Swap two nodes
            if ma == i {
                break
            }
            // Swap two nodes
            swap(i: i, j: ma)
            // Loop downwards heapification
            i = ma
        }
    }
    ```

=== "JS"

    ```javascript title="my_heap.js"
    /* Element exits heap */
    pop() {
        // Handle empty case
        if (this.isEmpty()) throw new Error('Heap is empty');
        // Delete node
        this.#swap(0, this.size() - 1);
        // Remove node
        const val = this.#maxHeap.pop();
        // Return top element
        this.#siftDown(0);
        // Return heap top element
        return val;
    }

    /* Starting from node i, heapify from top to bottom */
    #siftDown(i) {
        while (true) {
            // If node i is largest or indices l, r are out of bounds, no need to continue heapify, break
            const l = this.#left(i),
                r = this.#right(i);
            let ma = i;
            if (l < this.size() && this.#maxHeap[l] > this.#maxHeap[ma]) ma = l;
            if (r < this.size() && this.#maxHeap[r] > this.#maxHeap[ma]) ma = r;
            // Swap two nodes
            if (ma === i) break;
            // Swap two nodes
            this.#swap(i, ma);
            // Loop downwards heapification
            i = ma;
        }
    }
    ```

=== "TS"

    ```typescript title="my_heap.ts"
    /* Element exits heap */
    pop(): number {
        // Handle empty case
        if (this.isEmpty()) throw new RangeError('Heap is empty.');
        // Delete node
        this.swap(0, this.size() - 1);
        // Remove node
        const val = this.maxHeap.pop();
        // Return top element
        this.siftDown(0);
        // Return heap top element
        return val;
    }

    /* Starting from node i, heapify from top to bottom */
    siftDown(i: number): void {
        while (true) {
            // If node i is largest or indices l, r are out of bounds, no need to continue heapify, break
            const l = this.left(i),
                r = this.right(i);
            let ma = i;
            if (l < this.size() && this.maxHeap[l] > this.maxHeap[ma]) ma = l;
            if (r < this.size() && this.maxHeap[r] > this.maxHeap[ma]) ma = r;
            // Swap two nodes
            if (ma === i) break;
            // Swap two nodes
            this.swap(i, ma);
            // Loop downwards heapification
            i = ma;
        }
    }
    ```

=== "Dart"

    ```dart title="my_heap.dart"
    /* Element exits heap */
    int pop() {
      // Handle empty case
      if (isEmpty()) throw Exception('Heap is empty');
      // Delete node
      _swap(0, size() - 1);
      // Remove node
      int val = _maxHeap.removeLast();
      // Return top element
      siftDown(0);
      // Return heap top element
      return val;
    }

    /* Starting from node i, heapify from top to bottom */
    void siftDown(int i) {
      while (true) {
        // If node i is largest or indices l, r are out of bounds, no need to continue heapify, break
        int l = _left(i);
        int r = _right(i);
        int ma = i;
        if (l < size() && _maxHeap[l] > _maxHeap[ma]) ma = l;
        if (r < size() && _maxHeap[r] > _maxHeap[ma]) ma = r;
        // Swap two nodes
        if (ma == i) break;
        // Swap two nodes
        _swap(i, ma);
        // Loop downwards heapification
        i = ma;
      }
    }
    ```

=== "Rust"

    ```rust title="my_heap.rs"
    /* Element exits heap */
    fn pop(&mut self) -> i32 {
        // Handle empty case
        if self.is_empty() {
            panic!("index out of bounds");
        }
        // Delete node
        self.swap(0, self.size() - 1);
        // Remove node
        let val = self.max_heap.pop().unwrap();
        // Return top element
        self.sift_down(0);
        // Return heap top element
        val
    }

    /* Starting from node i, heapify from top to bottom */
    fn sift_down(&mut self, mut i: usize) {
        loop {
            // If node i is largest or indices l, r are out of bounds, no need to continue heapify, break
            let (l, r, mut ma) = (Self::left(i), Self::right(i), i);
            if l < self.size() && self.max_heap[l] > self.max_heap[ma] {
                ma = l;
            }
            if r < self.size() && self.max_heap[r] > self.max_heap[ma] {
                ma = r;
            }
            // Swap two nodes
            if ma == i {
                break;
            }
            // Swap two nodes
            self.swap(i, ma);
            // Loop downwards heapification
            i = ma;
        }
    }
    ```

=== "C"

    ```c title="my_heap.c"
    /* Element exits heap */
    int pop(MaxHeap *maxHeap) {
        // Handle empty case
        if (isEmpty(maxHeap)) {
            printf("heap is empty!");
            return INT_MAX;
        }
        // Delete node
        swap(maxHeap, 0, size(maxHeap) - 1);
        // Remove node
        int val = maxHeap->data[maxHeap->size - 1];
        maxHeap->size--;
        // Return top element
        siftDown(maxHeap, 0);

        // Return heap top element
        return val;
    }

    /* Starting from node i, heapify from top to bottom */
    void siftDown(MaxHeap *maxHeap, int i) {
        while (true) {
            // Find node with maximum value among nodes i, l, r, denoted as max
            int l = left(maxHeap, i);
            int r = right(maxHeap, i);
            int max = i;
            if (l < size(maxHeap) && maxHeap->data[l] > maxHeap->data[max]) {
                max = l;
            }
            if (r < size(maxHeap) && maxHeap->data[r] > maxHeap->data[max]) {
                max = r;
            }
            // Swap two nodes
            if (max == i) {
                break;
            }
            // Swap two nodes
            swap(maxHeap, i, max);
            // Loop downwards heapification
            i = max;
        }
    }
    ```

=== "Kotlin"

    ```kotlin title="my_heap.kt"
    /* Element exits heap */
    fun pop(): Int {
        // Handle empty case
        if (isEmpty()) throw IndexOutOfBoundsException()
        // Delete node
        swap(0, size() - 1)
        // Remove node
        val _val = maxHeap.removeAt(size() - 1)
        // Return top element
        siftDown(0)
        // Return heap top element
        return _val
    }

    /* Starting from node i, heapify from top to bottom */
    fun siftDown(it: Int) {
        // Kotlin function parameters are immutable, so create temporary variable
        var i = it
        while (true) {
            // If node i is largest or indices l, r are out of bounds, no need to continue heapify, break
            val l = left(i)
            val r = right(i)
            var ma = i
            if (l < size() && maxHeap[l] > maxHeap[ma]) ma = l
            if (r < size() && maxHeap[r] > maxHeap[ma]) ma = r
            // Swap two nodes
            if (ma == i) break
            // Swap two nodes
            swap(i, ma)
            // Loop downwards heapification
            i = ma
        }
    }
    ```

=== "Ruby"

    ```ruby title="my_heap.rb"
    ### Pop element from heap ###
    def pop
      # Handle empty case
      raise IndexError, "Heap is empty" if is_empty?
      # Delete node
      swap(0, size - 1)
      # Remove node
      val = @max_heap.pop
      # Return top element
      sift_down(0)
      # Return heap top element
      val
    end

    ### Heapify from node i, top to bottom ###
    def sift_down(i)
      loop do
        # If node i is largest or indices l, r are out of bounds, no need to continue heapify, break
        l, r, ma = left(i), right(i), i
        ma = l if l < size && @max_heap[l] > @max_heap[ma]
        ma = r if r < size && @max_heap[r] > @max_heap[ma]

        # Swap two nodes
        break if ma == i

        # Swap two nodes
        swap(i, ma)
        # Loop downwards heapification
        i = ma
      end
    end
    ```

## 8.1.3 &nbsp; Common Applications of Heaps

- **Priority queue**: Heaps are typically the preferred data structure for implementing priority queues. The time complexity of both enqueue and dequeue operations is $O(\log n)$, and heap construction has a time complexity of $O(n)$, making these operations highly efficient.
- **Heap sort**: Given a set of data, we can build a heap with them and then continuously perform element removal operations to obtain sorted data. However, we usually use a more elegant approach to implement heap sort, as detailed in the "Heap Sort" chapter.
- **Getting the largest $k$ elements**: This is a classic algorithm problem and also a typical application, such as selecting the top 10 trending news items for Weibo Hot Search or the top 10 best-selling products.