Lecture 15, Thu 05/21

Heaps

Heaps

• Recall
  • Binary tree: A tree structure where a node has two children
  • Binary Search tree: A tree structure where a node has two children AND
    • The left child < parent
    • The right child > parent
    • Insertion order affects the tree structure

A Heap is a complete binary tree

• Every level of the tree, except the deepest, must contain as many nodes as possible.
• The deepest level must have all nodes to the left as possible.

MaxHeap

• The value of a node is never less than the value of its children.
• Every level of the tree, except the deepest, must contain as many nodes as possible. The deepest level must have all nodes to the left as possible.

MinHeap

• Same as MaxHeap, EXCEPT
  • The value of a node is never greater than the value of its children.

Examples

• MaxHeap

• MinHeap

Heaps are an efficient way to implement a Priority Queue

• The only element we care about is the root (the min or max depending on MinHeap or MaxHeap).
• Insertion order should not affect the structure of the tree since operations need to preserve the complete balanced property.

Heaps represented as an array

• Since binary heaps have the complete property, representing this with an array is simple.
  • Easier to represent the heap where the 0th element in the array is meaningless.
  • The root of the binary heap is at index 1.
• A node’s index with respect to its parent and children can be generalized as:
  • A node’s parent index: index / 2 (integer division truncates decimal)
  • A node’s left child index: 2 * index
  • A node’s right child index: 2 * index + 1
• Example of an array representation of MaxHeap above

Insertion into a binary MaxHeap

• insert the element in the first available location.
  • Keeps the binary tree complete.
• While the element’s parent is less than the element
  • Swap the element with its parent
• Insertion is O(log n) (height of tree is log n)

• Note that MinHeap would be the same algorithm except we swap while the element’s parent is greater than the element.

Removing root element of a MaxHeap

• Since heaps are used to implement priority queues, removing the root element is a commonly used operation.
• Copy the root element into a variable
• Assign the last_element to the root position
• While last_element is less than one of its children
  • Swap the largest child with the last_element
• Return the original root element

Visualization tools

• Vis tool 1 and Vis tool 2

Simple implementation of a MaxHeap class

// main.cpp
#include <unistd.h>
#include <iostream>

using namespace std;

class HeapEmptyException {};

class MaxHeap {
	private:
		int* heapArray;
		int size;
		const static int CAPACITY = 100;
		void heapify(int index);
	public:
		MaxHeap();
		int removeMax() throw (HeapEmptyException);
		void insert(int e);
		int getSize() { return size; }
		void printHeap();
};

void MaxHeap::heapify(int index) {
    int leftChild = 2 * index;
    int rightChild = 2 * index + 1;
    int largestIndex = index;

    if (leftChild <= size && heapArray[leftChild] > heapArray[largestIndex]) {
        largestIndex = leftChild;
    }
    if (rightChild <= size && heapArray[rightChild] > heapArray[largestIndex]) {
        largestIndex = rightChild;
    }
    if (largestIndex != index) {
        int temp = heapArray[index];
        heapArray[index] = heapArray[largestIndex];
        heapArray[largestIndex] = temp;
        heapify(largestIndex);
    }
}

MaxHeap::MaxHeap() {
    size = 0;
    heapArray = new int[CAPACITY];
}

void MaxHeap::insert(int e) {
    size++;
    heapArray[size] = e;
    int temp;
    int index = size;
    while (index > 1 && heapArray[index / 2] < heapArray[index]) {
        // swap parent and current node
        temp = heapArray[index];
        heapArray[index] = heapArray[index / 2];
        heapArray[index / 2] = temp;
        index = index / 2;
    }
}
int MaxHeap::removeMax() throw (HeapEmptyException) {
    if (size <= 0)
        throw HeapEmptyException();

    if (size == 1) {
        size--;
        return heapArray[1];
    }

    int index = 1;
    int max = heapArray[index];
    heapArray[index] = heapArray[size];
    size--;

    heapify(index);
    return max;
}

void MaxHeap::printHeap() {
    for (int i = 1; i <= size; i++) {
        cout << "i = " << i << ": " << heapArray[i] << endl;
    }
}

int main() {
    MaxHeap heap;
    heap.insert(3);
    heap.insert(2);
    heap.insert(1);
    heap.insert(15);
    heap.insert(5);
    heap.insert(4);
    heap.insert(45);
    cout << "---" << endl;
    heap.printHeap();
    cout << "---" << endl;

    cout << heap.getSize() << endl;
    cout << "---" << endl;

    // heapSort
    int times = heap.getSize();
    for (int i = 0; i < times; i++) {
	cout << endl;
        cout << "Removed:" << heap.removeMax() << endl;
        heap.printHeap();
    }
}