Lecture 14, Wed 03/06

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

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 << heap.removeMax() << endl;
	}
}