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Lecture 16, Tue 12/03
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;
}
}