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# Min Heap

Min Heap is a type of data structure used in computer science to store elements. It is a binary tree-like structure where each node can have at most two children. Min Heap is also known as a Min Priority Queue since its elements are stored in a manner such that the smallest element is always at the root (top) level. It is used in a wide range of algorithms including sorting, priority queue implementation, graph traversal, and more. In this article, we will take a look at the structure of a Min Heap, how to create and add elements to a Min Heap, the Time Complexity and Space Complexity of Min Heap operations, and how to code a Min Heap in Python.

## What is a Min Heap?

A min heap is a type of data structure used to store and organize data. It is a special type of binary tree that satisfies the heap property, which requires that the root node of the tree have the smallest value of any node in the tree. A min heap is often used to implement Priority Queues, which are used to store elements with associated priorities. The priority for each element is determined by the value of the element. In this article, we will discuss what a min heap is, how it works, its time and space complexities, and provide some examples of Python code. We will also discuss the use of min heaps in Priority Queues and provide several coding exercises to test your understanding.

## What is a Min Heap?

A min heap is a data structure that uses a tree-like representation to store elements. It is a type of binary tree, meaning that each node has two children, a left and a right. Each node contains a value, which is used to determine the order of the elements in the tree. The root node of the tree contains the smallest value of any node in the tree. This is known as the heap property. The min heap is a type of complete binary tree, meaning that all levels of the tree are filled from left to right.

The min heap has several important applications in data structures and algorithms. When used to store elements, it can be used to quickly find the smallest element in the tree. When used to implement Priority Queues, it can be used to store elements with associated priorities and retrieve the highest priority element.

## How Does the Min Heap Algorithm Work?

The min heap algorithm works by maintaining the heap property, which requires that the root node of the tree have the smallest value of any node in the tree. To maintain this property, the algorithm must perform two types of operations: insertion and removal.

When an element is inserted into the min heap, it is first added to the bottom level of the tree. The algorithm then checks to see if the element is smaller than its parent node. If it is, the element is swapped with its parent node. This process is repeated until the element is no longer smaller than its parent node. This ensures that the heap property is maintained.

When an element is removed from the min heap, the root node is removed and the last element in the tree is moved to the root node. The algorithm then checks to see if the element is larger than its children nodes. If it is, the element is swapped with the smallest of its children nodes. This process is repeated until the element is no longer larger than its children nodes. This ensures that the heap property is maintained.

## Time and Space Complexity of the Min Heap

The time complexity of the min heap algorithm is O(log n) for insertion and removal, and O(1) for searching. The space complexity of the min heap algorithm is O(n), where n is the number of elements in the heap.

## Python Code Examples

Below are some examples of Python code that demonstrate how the min heap algorithm works.

The following code shows how to insert an element into the min heap:

def insert_min_heap(heap, element):
heap.append(element)
# compare element with its parent node and swap if necessary
current_node = len(heap) - 1
parent_node = (current_node - 1) // 2
while heap[parent_node] > element:
heap[current_node], heap[parent_node] = heap[parent_node], heap[current_node]
current_node = parent_node
parent_node = (current_node - 1) // 2

The following code shows how to remove an element from the min heap:

def remove_min_heap(heap):
# remove the root node and move the last element to the root node
heap, heap[-1] = heap[-1], heap
removed_element = heap.pop(-1)
# compare root node with its children and swap if necessary
current_node = 0
left_child = (2 * current_node) + 1
right_child = (2 * current_node) + 2
while (left_child < len(heap) and heap[current_node] > heap[left_child]) or (right_child < len(heap) and heap[current_node] > heap[right_child]):
if right_child < len(heap) and heap[right_child] < heap[left_child]:
heap[current_node], heap[right_child] = heap[right_child], heap[current_node]
current_node = right_child
else:
heap[current_node], heap[left_child] = heap[left_child], heap[current_node]
current_node = left_child
left_child = (2 * current_node) + 1
right_child = (2 * current_node) + 2
return removed_element

## Using Min Heaps in Priority Queues

Min heaps can be used to implement Priority Queues, which are used to store elements with associated priorities. The priority for each element is determined by the value of the element. The elements with the highest priority are placed at the top of the queue, while the elements with the lowest priority are placed at the bottom of the queue.

When using a min heap to implement a Priority Queue, the priority of each element is determined by the value of the element. The elements with the lowest values are placed at the top of the queue, while the elements with the highest values are placed at the bottom of the queue. The min heap algorithm can be used to quickly find the highest priority element in the queue.

## Conclusion

In this article, we discussed what a min heap is, how it works, its time and space complexities, and provided some examples of Python code. We also discussed the use of min heaps in Priority Queues.

The min heap is an important data structure used in a variety of algorithms and applications. It is a type of binary tree that satisfies the heap property, which requires that the root node of the tree have the smallest value of any node in the tree. The min heap algorithm has a time complexity of O(log n) for insertion and removal, and a space complexity of O(n).

## Exercises

#### Create a function that takes a min heap and returns the highest priority element in the heap.

def highest_priority_element(heap):
return heap

#### Create a function that takes a min heap and an element and adds the element to the heap while maintaining the heap property.

def insert_min_heap(heap, element):
heap.append(element)
# compare element with its parent node and swap if necessary
current_node = len(heap) - 1
parent_node = (current_node - 1) // 2
while heap[parent_node] > element:
heap[current_node], heap[parent_node] = heap[parent_node], heap[current_node]
current_node = parent_node
parent_node = (current_node - 1) // 2

#### Create a function that takes a min heap and removes the highest priority element from the heap while maintaining the heap property.

def remove_min_heap(heap):
# remove the root node and move the last element to the root node
heap, heap[-1] = heap[-1], heap
removed_element = heap.pop(-1)
# compare root node with its children and swap if necessary
current_node = 0
left_child = (2 * current_node) + 1
right_child = (2 * current_node) + 2
while (left_child < len(heap) and heap[current_node] > heap[left_child]) or (right_child < len(heap) and heap[current_node] > heap[right_child]):
if right_child < len(heap) and heap[right_child] < heap[left_child]:
heap[current_node], heap[right_child] = heap[right_child], heap[current_node]
current_node = right_child
else:
heap[current_node], heap[left_child] = heap[left_child], heap[current_node]
current_node = left_child
left_child = (2 * current_node) + 1
right_child = (2 * current_node) + 2
return removed_element

#### Create a function that takes a list of elements and creates a min heap.

def create_min_heap(elements):
heap = []
for element in elements:
insert_min_heap(heap, element)
return heap

#### Create a function that takes a min heap and a priority and returns the element in the heap with the highest priority that is less than or equal to the given priority.

def highest_priority_element_less_than(heap, priority):
if heap > priority:
return None
else:
element = heap
current_node = 0
left_child = (2 * current_node) + 1
right_child = (2 * current_node) + 2
while left_child < len(heap) and heap[left_child] <= priority:
if heap[right_child] <= priority and heap[right_child] > heap[left_child]:
element = heap[right_child]
current_node = right_child
else:
element = heap[left_child]
current_node = left_child
left_child = (2 * current_node) + 1
right_child = (2 * current_node) + 2
return element