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* a max heap's **root node must** have all its children either **greater than or equal** to its children. a min heap is the opposite. duplicate values are allowed.
* since you always remove the root, insertion and deletion takes `O(log(n))`. the maximum/minimum value in the heap can be obtained with `O(1)` time complexity.
* heaps can be represented with linked lists or queues (arrays).
* in the case of arrays, the index of the parent `node = [(n-1)/2]`.
* heaps can be represented with linked lists or queues (arrays) or binary trees.
* in the case of a tree to array heap:
* the parent node index is given by `n / 2`
* the left children index is `2 * n`
* the right children index is `2 * n + 1`
* a node is a leaf when its index `> n / 2`
<br>
----
### priority queues
<br>
* a priority queue is an abstract data type with the following properties:
1. every item has a priority (usually an integer).
2. an item with a high priority is dequeued before an item with low priority.
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### min heaps
<br>
* a **min heap** is a complete binary tree where each node is smaller than its children (the root is the min element).
* `insert`:
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- then bubble down, swapping it with one of its children until the min-heap is properly restored
- there is no need for order between right and left, so this operation would only take `O(log n)` runtime.
* the code below is an example:
* the code below is an example of an ad-hoc heap class in python.
<br>
```python
class MinHeap:
@ -128,6 +141,8 @@ class MinHeap:
### max heaps
<br>
* a **max heap** is a complete binary tree where each node is larger than its children (the root is the max element).
* `insert`:
@ -138,7 +153,9 @@ class MinHeap:
* remove/return the top and then replace the tree's top with its bottom rightmost element.
* swap up until the max element is on the top.
* the code below is an example:
* the code below is an example of a max heap class built in python:
<br>
```python
class MaxHeap: