🧩Intro to Algorithms Unit 5 – Heap and Heap Sort

What's a Heap?

• A heap is a specialized tree-based data structure that satisfies the heap property
• Heap property states that for a max heap, the key of each node is always greater than or equal to the keys of its children
• For a min heap, the key of each node is always less than or equal to the keys of its children
• Heaps are commonly represented using an array, where the parent-child relationship is defined by the index
  • For a node at index $i$, its left child is at index $2i+1$ and its right child is at index $2i+2$
  • The parent of a node at index $i$ is located at index $\lfloor(i-1)/2\rfloor$
• Heaps are used to implement priority queues and for efficient sorting algorithms like heap sort
• The two main types of heaps are max heap and min heap, which differ in their ordering of elements
• Heaps are complete binary trees, meaning all levels except possibly the last are fully filled and nodes are as far left as possible

Building a Heap

• To build a heap from an array, we start by inserting elements into the tree in the order they appear
• After insertion, we perform a "heapify" operation to restore the heap property if it is violated
• Heapify operation compares a node with its parent and swaps them if the heap property is not satisfied
  • For a max heap, if a node is greater than its parent, they are swapped
  • For a min heap, if a node is smaller than its parent, they are swapped
• Heapify operation is performed recursively up the tree until the root is reached or the heap property is satisfied
• Building a heap has a time complexity of $O(n)$, where $n$ is the number of elements in the array
• The process of building a heap is also known as "heapification"
• Heapify can be performed in a bottom-up manner, starting from the last non-leaf node and moving towards the root
• After the heap is built, the maximum element (for a max heap) or the minimum element (for a min heap) is always located at the root

Heap Operations

• The two main operations performed on a heap are insertion and extraction
• Insertion adds a new element to the heap while maintaining the heap property
  • New element is added at the end of the heap (last position in the array)
  • Heapify operation is performed to move the element up the tree until the heap property is satisfied
  • Insertion has a time complexity of $O(\log n)$ in the worst case, where $n$ is the number of elements in the heap
• Extraction removes the root element (maximum for max heap, minimum for min heap) from the heap
  • Root element is replaced by the last element in the heap
  • Heapify operation is performed to move the new root element down the tree until the heap property is satisfied
  • Extraction has a time complexity of $O(\log n)$ in the worst case
• Other operations like finding the maximum/minimum element, deleting an arbitrary element, and updating an element's value are also possible
• Heaps support efficient retrieval of the maximum (for max heap) or minimum (for min heap) element in $O(1)$ time
• The space complexity of a heap is $O(n)$, where $n$ is the number of elements in the heap

Heap Sort Algorithm

• Heap sort is a comparison-based sorting algorithm that uses a heap data structure
• The algorithm consists of two main phases: building a max heap and repeatedly extracting the maximum element
• In the first phase, the input array is transformed into a max heap using the heapify operation
  • This step ensures that the largest element is at the root of the heap
• In the second phase, the root element (maximum) is swapped with the last element of the heap
  • The heap size is reduced by one, excluding the last element which is now in its final sorted position
  • Heapify operation is performed on the reduced heap to restore the max heap property
• The process of swapping the root with the last element and heapifying is repeated until the heap is empty
• At the end of the algorithm, the array is sorted in ascending order
• Heap sort has a time complexity of $O(n \log n)$ in all cases (best, average, and worst)
• The space complexity of heap sort is $O(1)$ as it operates in-place, requiring only a constant amount of additional memory

Time Complexity

• The time complexity of building a heap from an array is $O(n)$, where $n$ is the number of elements
  • This is because each element is inserted and heapified, which takes $O(\log n)$ time in the worst case
  • However, the amortized time complexity of building a heap is $O(n)$ due to the properties of a complete binary tree
• Insertion and extraction operations on a heap have a time complexity of $O(\log n)$ in the worst case
  • This is because the height of a heap is logarithmic in the number of elements, and heapify operation traverses the height of the tree
• Heap sort algorithm has a time complexity of $O(n \log n)$ in all cases (best, average, and worst)
  • Building the initial max heap takes $O(n)$ time
  • Each extraction and heapify operation takes $O(\log n)$ time, and there are $n$ such operations
• The space complexity of a heap and heap sort is $O(n)$ and $O(1)$ respectively
• Heap operations like finding the maximum/minimum element have a time complexity of $O(1)$
• Deleting an arbitrary element or updating an element's value in a heap takes $O(\log n)$ time in the worst case

Practical Applications

• Heaps are commonly used to implement priority queues, where elements have associated priorities
  • Priority queues are used in scheduling algorithms, such as task scheduling in operating systems
  • They are also used in graph algorithms like Dijkstra's shortest path algorithm and Prim's minimum spanning tree algorithm
• Heap sort is an efficient sorting algorithm, particularly when the input size is large
  • It is an in-place sorting algorithm, meaning it does not require additional memory for sorting
  • Heap sort is often used when memory usage is a concern and stability is not required
• Heaps are used in the selection of the top k elements from a large dataset
  • By constructing a min heap of size k and iterating through the dataset, the top k elements can be efficiently determined
• Heaps are utilized in the implementation of the Heap's algorithm for finding the k-th smallest/largest element in an array
• Heaps are employed in the construction of Huffman trees for data compression
• Heaps find applications in event simulation, where events with the earliest timestamp are processed first
• Heaps are used in the A* search algorithm for pathfinding and graph traversal problems

Pros and Cons

• Pros of using heaps:
  • Heaps provide efficient insertion and extraction of the maximum/minimum element in logarithmic time
  • Heaps are useful for implementing priority queues, where elements have associated priorities
  • Heap sort is an efficient, in-place sorting algorithm with a guaranteed time complexity of $O(n \log n)$
  • Heaps are simple to implement and understand compared to other self-balancing tree structures
  • Heaps have a space complexity of $O(n)$, making them space-efficient
• Cons of using heaps:
  • Heaps do not support efficient search operations for arbitrary elements
    • Finding an element in a heap requires traversing the entire heap, which takes $O(n)$ time
  • Heaps do not maintain the relative order of elements with equal priorities
    • If stability is required, additional measures need to be taken
  • Heaps are not suitable for dynamically changing priorities of elements
    • Updating the priority of an element requires deleting and reinserting the element, which takes $O(\log n)$ time
  • Heaps are not efficient for finding the k-th smallest/largest element directly
    • Additional operations or data structures are needed to solve such problems efficiently
  • Heaps are not ideal for problems that require frequent merging of two heaps
    • Merging two heaps requires rebuilding the resulting heap, which takes $O(n)$ time

Key Takeaways

• Heaps are tree-based data structures that satisfy the heap property, where the key of each node is greater than or equal to (max heap) or less than or equal to (min heap) the keys of its children
• Heaps are commonly represented using an array, with the parent-child relationship defined by the index
• Building a heap from an array has a time complexity of $O(n)$, while insertion and extraction operations take $O(\log n)$ time
• Heap sort is an efficient, in-place sorting algorithm with a time complexity of $O(n \log n)$ in all cases
• Heaps are used to implement priority queues, which have applications in scheduling algorithms, graph algorithms, and event simulation
• Heaps provide efficient access to the maximum/minimum element but do not support efficient search or dynamic priority updates
• Understanding the properties, operations, and applications of heaps is crucial for designing efficient algorithms and solving problems in computer science