prompter: Explain bubble sort

assistant: Bubble sort is a sorting algorithm to sort arrays of numbers or other comparable things. It works by swapping two elements that are next to each other so that the bigger one is on the right afterwards. This process is repeated for each position from left to right. After this the biggest number in the array is at the last position. Next, the algorithm starts at the very left again and swaps neighbours going to the right, but this time it can stop before the last position, as the biggest number is already there. It is called bubble sort as the biggest number always "bubbles" up.
Here is an example:
```
4 8 3 6 1
# don't swap 4, 8 as 8 > 4
4 8 3 6 1
# swap 8, 3
4 3 8 6 1
# swap 8, 6
4 3 6 8 1
# swap 8, 1; 8 is now correct
4 3 6 1 8
# swap 4, 3
3 4 6 1 8
# don't swap 4, 6 as 6 > 4
3 4 6 1 8
# swap 1, 6; 6 is now correct
3 4 1 6 8
# don't swap 3, 4 as 4 > 3
3 4 1 6 8
# swap 4, 1; 4 is now correct
3 1 4 6 8
# swap 3, 1; 3 is now correct
1 3 4 6 8
# array is sorted
```

prompter: Explain heap sort and heapify.

assistant: **Heap Sort**
Heap sort is a more efficient sorting algorithm than bubble sort that also sorts numbers or other similarly comparable things. It uses a heap, (can be max or min) which is a binary tree where a node is always greater or lesser than both of its children. It takes `n*log_2(n)` time to create the heap, because each item must be inserted, and can travel up at most `log_2(n)` levels of the tree.
Then, the root node is already max, so it can be taken out and placed in a sorted list instantly. Then, a leaf node is placed where the root node is, and the tree is re-sorted, which takes at most `log_2(n)` time. Then the root node is taken out and placed in the sorted list again. This process is repeated until the tree is empty, which will take a total of `n*log_2(n)` time.
Here's an example sorting 3 numbers:
```
##########################
# First, build the heap. #
##########################
7 2 9
# Add 7 to root: 
# (7)
2 9
# add two to next leaf node:
# 7
# └── 2
# is it a max heap? Are all child nodes less than their parent nodes? Yes.
9
# add nine to the next leaf node:
# 7
# ├── 2
# └── 9
# Are all child nodes less than their parent nodes? No.
# move nine up:
# 9
# ├── 2
# └── 7
# Are all child nodes less than their parent nodes? Yes.

# No more elements are left. Heap is created. 

###########################
# Now sort from the heap. #
###########################
# take the root node and place it into the list:
# -
# ├── 2
# └── 7
9
# replace the root node with the last leaf node:
# 7
# └── 2
# Are all child nodes less than their parent nodes? Yes.
# take the root node and place it into the list on the left:
# -
# └── 2
7 9
# replace the root node with the last leaf node:
# (2)
# Are all child nodes less than their parent nodes? Yes.
# take the root node and place it into the list on the left: 
# (_)
2 7 9
# the tree is empty, which means we're done sorting!
```

---

**Heapify**

Instead of adjusting upwards in heap creation like before, heapify looks at each node, starting with the rightmost leaf node, and goes left until it reaches something that isn't a heap. Once it reaches a node where the children aren't all smaller than the parent, it will switch out the value of the root with the largest child. This is repeated with the swapped child node until it reached the bottom of the tree. Then, this whole process is repeated until the root of the entire tree is reached. This process still takes `n*log_m(n)` time at most, if all items need to move to the bottom of the tree each time, but it's useful for generally transforming trees into heaps.
Here's an example with a binary tree with four elements:
```
# start with a non max heap binary tree:
2
├── 9
│   └── 11
└── 7
# 11 is a heap because it has no children, continue to 7
# 7 is a heap because it has no children, continue to 9
# 9 is not a heap, because 11 is greater. Swap.
2
├── 11
│   └── 9
└── 7
# 9 is now a heap. Continue to 2.
# 2 is not a heap. Both 11 and 7 are greater. Swap with 11 since it's highest:
11
├── 2
│   └── 9
└── 7
# 2 is still not a heap. Swap with 9:
11
├── 9
│   └── 2
└── 7
# 2 is a heap, and 11 is the root. We're done!
```

assistant: Heap sort is a sorting algorithm that works by using a binary heap data structure to sort elements. It is an in-place algorithm, meaning that it sorts the elements within the original array without creating a new array.

The algorithm works by first creating a binary heap from the input array using a process called "heapify". Once the binary heap is created, the algorithm repeatedly removes the root element from the heap and places it at the end of the array. This process continues until all elements have been removed from the heap and placed in the correct order in the array.

The "heapify" process involves rearranging the elements in the input array so that they form a binary heap. A binary heap is a tree-based data structure in which each parent node is greater than or equal to its children (in the case of a max heap), or less than or equal to its children (in the case of a min heap). To heapify an array, the algorithm starts by finding the last parent node in the tree and then repeatedly "sinks" each parent node down the tree until it is in the correct position.

The process of sinking a node involves comparing the node to its children and swapping it with the larger (or smaller, depending on whether it is a max or min heap) child if necessary. This process is repeated until the node is in the correct position in the heap. The algorithm then moves on to the next parent node in the tree and repeats the sinking process.

Once the heapify process is complete, the input array will be sorted in ascending order if it is a max heap, or descending order if it is a min heap. The heap sort algorithm then proceeds with the step of repeatedly removing the root element from the heap and placing it at the end of the array until all elements have been placed in the correct order.

assistant: The primary goal of heap sort is to rearrange the elements of an array in order of their magnitude. It is typically used in situations where sorting needs to be done on a large set of data. 

Heapify is a procedure in the heap sort algorithm that builds the heap data structure in-place. It takes an array, arr, of length n as input and uses the divide-and-conquer approach to create a heap structure out of the array. The process of Heapify involves maintaining the heap property of the heap structure as we insert elements into it. 
The main steps of heapify are first, to swap the parent node with its largest child node so that the parent node is always larger than its children, and secondly, to repeat the same process for the swapped node. By repeating this process for each node in the heap, the heap structure is constructed in a hierarchical fashion.