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Learn how to implement a max-heap in Python.

Max-Heaps: Python

Now that we understand how to determine the relationship of elements with the internal list, we're ready to finish `.heapify_up()`. 

We need to make sure that every child is smaller in value than their parent. Say we add an element to the following heap:

```py
print(heap.heap_list)
# Output: [None, 99, 22, 61, 10, 21, 13, 23]
heap.add(90)

# ( new_element )
# { parent_element }
# [None, 99, 22, 61, {10}, 21, 13, 23, (90)]
```

Oh no! We've violated the heap property: since `90`'s parent is `10`, **the parent element is smaller than the child**.

Don't fret; we can fix this. We'll exchange the parent and the child elements.

```py
# [None, 99, 22, 61, {10}, 21, 13, 23, (90)]
# SWAP 90 and 10
# [None, 99, 22, 61, (90), 21, 13, 23, {10}]
```

`90`'s parent is now `22`, so **the parent element is still smaller than the child**. Keep on swappin'!

```py
# [None, 99, 22, 61, (90), 21, 13, 23, 10]
# SWAP 22 and 90
# [None, 99, (90), 61, {22}, 21, 13, 23, 10]
```

Okay, you can let out that sigh of relief. We've restored the heap properties!

```py
# [None, {99}, (90), 61, 22, 21, 13, 23, 10]
# The parent, 99, is greater than the child, 90!
```

Let's recap our strategy for `.heapify_up()`:

```pseudo
start at the last element of the list
while there's a parent element available:
  if the parent element is smaller:
    swap the elements
  set the target element index to be the parent's index
```

Adding an Element: Heapify Up II

We're going to implement a max-heap in Python. Max-heaps efficiently keep track of the **maximum value** in a dataset, even as we add and remove elements. 

Max-heaps are nearly identical to a min-heap; however, min-heaps keep track of the minimum value in a data set. We'll be focusing on max-heaps in this lesson, but much of what we cover in this lesson can be applied to a min-heap as well.

Heaps enable solutions for complex problems such as finding the shortest path (**Dijkstra's Algorithm**) or efficiently sorting a dataset (**heapsort**). 

They're an essential tool for confidently navigating some of the difficult questions posed in a technical interview.

By understanding the operations of a heap, you will have made a valuable addition to your problem-solving toolkit. 


Introduction to Max-Heaps

Great work so far! Our `MaxHeap` adds elements to `heap_list`, keeps count of the number elements inside the heap, and has the beginnings of `.heapify_up()`.

Before we dive into the logic for `.heapify_up()`, let's review how heaps track elements. We use a list for storing internal elements, but we're modeling it on a binary tree, where every "parent" element has up to two "child" elements.

"child" and "parent" elements are determined by their relative indices within the internal list. By doing some arithmetic on an element's index, we can determine the indices for parent and child elements (if they exist).

* Parent: `index // 2`
* Left Child: `index * 2`
* Right Child: `(index * 2) + 1`

```py
print(heap.heap_list)
# Outputs: [None, 99, 22, 61, 10, 21, 13, 23]
# Indices: [0, 1, 2, 3, 4, 5, 6, 7]

heap.parent_idx(4)
# (4 // 2) == 2
# Element at index 4 is 10 
# Element at index 2 is 22
# The parent element of 10 is 22

heap.left_child(3)
# (3 * 2) == 6
# Element at index 3 is 61
# Element at index 6 is 13
# The left child element of 61 is 13
```

These calculations are important for the efficiency of the heap, but they're not necessary to memorize, so we've added three helper methods: `.parent_idx()`, `.left_child_idx()`, and `.right_child_idx()`. 

These helpers take an index as the argument and return the corresponding parent or child index.

Translating Parent and Child Elements Into Indices

Our `MaxHeap` class will store two pieces of information in the form of instance attributes: 
* A Python list of the elements within the heap.
* A count of the elements within the heap.

To make our lives easier, we'll always keep one _sentinel_ element at the beginning inside the list: `None`. 

```py
heap = MaxHeap()
print(heap.heap_list)
# [None]
print(heap.count)
# 0
```

A sentinel value will terminate a loop when read as input. This sentinel element of `None` will save us the trouble of checking whether the list is empty and simplify the methods we define in later lessons.

Defining Max-Heap

Nice work! You've implemented a `MaxHeap` class in Python. Heaps are useful because they're **efficient** in maintaining their heap properties.

Let's recap what we learned:
* A max-heap tracks the maximum element as the element at index `1` within an internal Python list. 
* Max-heaps must maintain the heap property that **the parent values must be greater than their children**.
* When adding elements, we use `.heapify_up()` to compare the new element with its parent; if it violates the heap property, then we must swap the two values.

In the next lesson, we'll learn how to extract the root value of a heap as well as how to sort data using heapsort!

Review

Our `MaxHeap` class isn't very useful if all it ever contains is `None`. Let's build the functionality to add elements to our max-heap.

Our `MaxHeap` must abide by two principles:
* The element at index `1` is the **maximum** value in the entire list.
* Every "child" element in the list must be smaller than their "parent".

The first element we add to the list will automatically be the maximum value since there are no other elements in our heap. We'll tackle the trickier aspects of maintaining these principles in the coming exercises

For now, lets define two class instance methods called `.add()` and `.heapify_up()`:
- `.add()` will handle adding an element to the heap via the `.heap_list` property.
- `.heapify_up()` will do the work of maintaining the heap properties as we add additional elements.