## heaps
* a heap is a binary tree with these properties:
* it must have all of **its nodes in a specific order**, and
* its shape must be **complete** (all the levels of the tree must be completely filled except maybe for the last one and the last level must have the left-most nodes filled, always).
* 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]`.
----
### priority queues
* 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.
3. two items with an equal priority are dequeued based on their order in the queue.
* priority queues can be implemented with a stack, queue, or linked list data structures. however, heaps are the structure that guarantees both insertion and deletion to have time complexity `O(log N)` (while maintaining get_max/get_min at `O(1)`).
---
### min heaps
* a **min heap** is a complete binary tree where each node is smaller than its children (the root is the min element).
* `insert`:
- insert the element at the bottom, by finding the most rightmost node and checking its children: if left is empty, insert there, otherwise, insert on right.
- then compare this node to each parent, exchanging them until the tree's properties are corret.
* `extract_min`:
- first, remove/return the top and then replace the tree's top with its latest element (the bottom most rightmost).
- 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:
```python
class MinHeap:
def __init__(self, size):
self.heapsize = size
self.minheap = [0] * (size + 1)
self.realsize = 0
def add(self, element):
if self.realsize + 1 > self.heapsize:
print("Too many elements!")
return False
self.realsize += 1
self.minheap[self.realsize] = element
index = self.realsize
parent = index // 2
while self.minheap[index] < self.minheap[parent] and index > 1:
self.minheap[parent], self.minheap[index] = self.minheap[index], self.minheap[parent]
index = parent
parent = index // 2
def peek(self):
return self.minheap[1]
def pop(self):
if self.realsize < 1:
print("Heap is empty.")
return False
else:
remove_element = self.minheap[1]
self.minheap[1] = self.minheap[self.realsize]
self.realsize -= 1
index = 1
while index <= self.realsize // 2:
left_children = index * 2
right_children = (index * 2) + 1
if self.minheap[index] > self.minheap[left_children] or \
self.minheap[index] > self.minheap[right_children]:
if self.minheap[left_children] < self.minheap[right_children]:
self.minheap[left_children], self.minheap[index] = self.minheap[index], self.minheap[left_children]
index = left_children
else:
self.minheap[right_children], self.minheap[index] = self.minheap[index], self.minheap[right_children]
index = right_children
else:
break
return remove_element
def size(self):
return self.realsize
def __str__(self):
return str(self.minheap[1 : self.realsize + 1])
```
---
### max heaps
* a **max heap** is a complete binary tree where each node is larger than its children (the root is the max element).
* `insert`:
* insert the element at the bottom, at the leftmost node.
* then compare the node to each parent, exchanging them until the tree's properties are correct.
* `extreact_max`:
* 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:
```python
class MaxHeap:
def __init__(self, heapsize):
self.heapsize = heapsize
self.maxheap = [0] * (heapsize + 1)
self.realsize = 0
def add(self, element):
self.realsize += 1
if self.realsize > self.heapsize:
print("Too many elements!")
self.realsize -= 1
return False
self.maxheap[self.realsize] = element
index = self.realsize
parent = index // 2
while self.maxheap[index] > self.maxheap[parent] and index > 1:
self.maxheap[parent], self.maxheap[index] = self.maxheap[index], self.maxheap[parent]
index = parent
parent = index // 2
def peek(self):
return self.maxheap[1]
def pop(self):
if self.realsize < 1:
print("Heap is empty.")
return False
else:
remove_element = self.maxheap[1]
self.maxheap[1] = self.maxheap[self.realsize]
self.realsize -= 1
index = 1
while (index <= self.realsize // 2):
left_children = index * 2
right_children = (index * 2) + 1
if (self.maxheap[index] < self.maxheap[left_children] or self.maxheap[index] < self.maxheap[right_children]):
if self.maxheap[left_children] > self.maxheap[right_children]:
self.maxheap[left_children], self.maxheap[index] = self.maxheap[index], self.maxheap[left_children]
index = left_children
else:
self.maxheap[right_children], self.maxheap[index] = self.maxheap[index], self.maxheap[right_children]
index = right_children
else:
break
return remove_element
def size(self):
return self.realsize
def __str__(self):
return str(self.maxheap[1 : self.realsize + 1])
```