Heap is a tree-based data structure in which all nodes in the tree are in the specific order. There are 2 types of heap, typically:

• Max Heap: all parent node's values are greater than or equal to children node's values, root node value is the largest.

• Min Heap: all parent node's values are less than or equal to children node's values, root node value is the smallest.

### Basic operations

• `insert` aka `push`, add a new node into the heap
• `remove` aka `pop`, retrieves and removes the min or the max node of the heap
• `examine` aka `peek`, retrieves, but does not remove, the min or the max node of the heap

### Internal operations

• `heapify`, maintains max or min-heap property (all parent node's values should be greater/less than or equal to the child node's values)

### Implementations

A common implementation of a heap is the binary heap which based on binary tree data structure

You can implement a binary heap with either a static array (capacity restricted) or a dynamic array

### Binary Max Heap implementation example with Static Array

#### Approach

• Represented by 1-based integer array A[N+1]
• With a node A[k] (1<=k<=N)
• Its parent is A[k/2]
• Left child is A[k*2] (k*2 <= N)
• Right child is A[k*2+1] (k*2+1 <= N)
• A[1] is root node, A[0] is `Integer.MAX_VALUE`
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### Binary Min Heap implementation example with Static Array

#### Approach

• Represented by 1-based integer array A[N+1]
• With a node A[k] (1<=k<=N)
• Its parent is A[k/2]
• Left child is A[k*2] (k*2 <= N)
• Right child is A[k*2+1] (k*2+1 <= N)
• A[1] is root node, A[0] is `Integer.MIN_VALUE`
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### Applications

• Heaps can be used to implement Priority Queues

• Find the shortest paths between vertices in a Graph Data Structure

The shortest path is a path between two vertices in a graph such that the total sum of the weights of the constituent edges is minimum

You can implement an algorithm to find the shortest path by using a Min-Heap and Dijkstra algorithm

### Heaps in Java

In Java, the binary heap data structure is used to implement priority queues including java.util.PriorityQueue and java.util.concurrent.PriorityBlockingQueue