### Binary Heap

A binary heap is a specific type of binary tree (more precisely, a complete binary tree) that satisfies the heap property:

- In a max heap, for any given node
*C*, if*P*is a parent node of*C*, then the key (the value) of*P*is greater than or equal to the key of*C*. - In a min heap, the key of
*P*is less than or equal to the key of*C*.

The primary purpose of a binary heap is to quickly access the highest or lowest element (depending on whether it’s a max heap or min heap).

### Complete Binary Tree

A complete binary tree is a binary tree in which every level, except possibly the last, is completely filled, and all nodes are as far left as possible. This property makes it an efficient structure for binary heaps because it ensures that the tree remains balanced, and operations like insertions, deletions, and accessing the top element can be done in logarithmic time, *O*(log*n*).

### Java’s PriorityQueue Uses a Binary Heap Implemented as a Complete Binary Tree

Java’s `PriorityQueue`

uses a binary heap because it efficiently supports the operations required by a priority queue, namely:

- Inserting an element while maintaining the heap property
*O*(log*n*). - Removing the root element (the minimum or maximum, depending on the heap type), which is the highest-priority element in a priority queue, in
*O*(log*n*) time. - Peeking at the root element to see the highest-priority item without removing it
*O*(1).

The binary heap is implemented internally as a complete binary tree because this structure supports efficient array-based storage, eliminating the need for pointers/references between nodes, which a typical tree structure requires. In Java, the complete binary tree structure of a heap is typically represented using an array:

- The root element is at the index 0.
- For any element at index
*i*, its children are at indices 2*i*+1 (left child) and 2*i*+2 (right child). - The parent of any element at index
*i*(where*i*>0) is at index (*i*−1)/2.

This array representation is space-efficient and makes the operations fast due to the locality of reference, which is beneficial for performance, especially with modern CPU caches.

. When we talk about binary heaps being implemented “as a complete binary tree,” we are referring to the logical structure and properties that the binary heap must maintain. However, this does not directly imply the physical storage mechanism (like a tree with node objects linked together). In Java’s `PriorityQueue`

, and in many binary heap implementations in general, the “complete binary tree” is indeed typically represented using an array for several reasons:

**Efficient Space Utilization**: An array-based representation of a complete binary tree is very space-efficient. There are no extra pointers required to link parent nodes to their children, which would be the case in a linked node-based tree structure. This makes the array representation more compact, especially since a binary heap is a complete binary tree, meaning it is perfectly balanced except possibly for the last level, which is filled from left to right.**Simplifies Index Calculations**: In a complete binary tree represented as an array, the children of the node at index`i`

can be easily found at indices`2*i + 1`

for the left child and`2*i + 2`

for the right child. Similarly, the parent of a node at index`i`

can be found at index`(i-1)/2`

. This property simplifies the algorithms for inserting and removing elements from the heap, as it allows for easy traversal up and down the tree.**Cache-friendly**: Access patterns to an array are more predictable and linear, especially when traversing the heap for operations like insertions and deletions. This makes the array-based representation more cache-friendly, as elements that are close together in the tree are also close together in memory, improving cache utilization and overall performance.**Simplicity**: Implementing a binary heap as an array simplifies the code. There is no need to manage pointers or allocate additional nodes dynamically. This reduces the complexity of the code and the potential for memory leaks or pointer errors.

So, when we say that Java’s `PriorityQueue`

uses a binary heap implemented “as a complete binary tree,” it means that the logical structure and properties of a complete binary tree are maintained, but the physical implementation uses an array for efficiency and simplicity. This is a common approach not just in Java but in most implementations of binary heaps across various programming languages and libraries.