Heap and Trie in Data Structure

What is a Heap in Data Structures?

A heap is a data structure that consists of a complete binary tree. In a complete binary tree, all levels are filled except for the last, which may or may not be filled. A complete binary tree fills its elements from left to right.

Types of Heap Data Structure

There are two primary types of heap data structures:

1. Max Heap
2. Min Heap

Max Heap

A node is always larger than its child nodes, and the root node has the largest key among all nodes.

Min Heap

A node is always smaller than its child node(s), and the root node's key is the smallest of all nodes. 

Heap Operations in Data Structures

Following are the Heap Operations in Data Structures:

• Heapify: Heapify creates a heap from an array.
• Insertion: Add an element to an existing heap.
• Deletion: Deletion is the process of removing the top element in a heap or the highest priority element.
• Peek: Find the heap's most prior element.
• Extract-Max/Min: Extract-Max returns the node with the highest value after removing it from the Max Heap, whereas Extract-Min returns the node with the lowest value after removing it from the Min Heap.

Usages of Heap

• Dijkstra's Algorithms
• Heap Sort Algorithms

Applications of the Heap

• Priority Queues: Elements are dequeued according to their priority, and heaps are frequently used for efficient insertions and removals.
• Graph Algorithms: Heap permits efficient Dijkstra's and Prim's algorithms by storing vertices/edges according to distance/weight.
• Sorting: Heap sort removes the largest element iteratively, sorting using heap construction.
• Memory Management: In C/C++, heaps manage dynamic memory by keeping track of free and allocated blocks.
• Event-driven Simulation: The heap in event simulation keeps events for efficient processes.

What is a Trie?

A trie is a tree-based data structure that holds a collection of strings. The word trie derives from the verb reTRIEval, which means to find or obtain. If two strings share a prefix, they have the same ancestor in the trie. It stores a longer string and performs efficient searches on it. 

Need of Trie Data Structure

• Trie is used to store and retrieve data.
• Trie is more efficient than HashTable.
• Trie allows Prefix-based searching.

Trie Data Structure Properties

• Trie has an empty root node that relates to other nodes.
• Nodes represent strings, whereas edges represent characters.
• Nodes contain hashmaps or arrays of pointers.
• The index identifies a character, whereas the flag indicates the end of a string.
• Alphabets, numerals, and special characters may all be present.
• 26 pointers are required for each node (a-z).
• The path from root to node represents a word or string.

Working of Trie Data Structure

• Trie begins with an empty root node.
• Characters from a string are inserted one at a time, producing new nodes as needed.
• Each node represents a character, and edges connect them to other characters.
• Searching entails scanning nodes based on an input string and checking for existence.
• Trie, which enables for efficient prefix-based searching, is often used in autocomplete and dictionary applications.

Basic Operations on the Trie Data Structure

The following are the basic operations of the Trie data structure:

• Insertion
• Search
• Deletion

Insertion

Insertion is the process of adding a new string to the Trie by constructing nodes for each character that do not already exist and marking the last node to signal the string's end.

Search

Determines whether a given string exists in the Trie by traversing the nodes based on the string's characters and ensuring that the string finishes at a valid node.

Deletion

Removes a string from the Trie by traversing the nodes corresponding to the string's characters and eliminating them, optionally pruning unneeded nodes to ensure Trie's structural integrity.

Complexity Analysis of the Trie Data Structure

Applications of the Trie Data Structure

• Autocomplete: Autocomplete is powered by Trie, which suggests words as you type.
• Spell Checkers: Trie provides suggestions for misspelled words.
• Longest Prefix Matching: A trie-based technique improves network routing for IP addresses.

Advantages of the Trie Data Structure

• Trie has O(l) time complexity for input and search, where l represents word length.
• Faster than hash tables or binary search trees.
• Allows alphabetical filtering and printing.
• Each key requires a fixed amount of space, hence it takes up less space than BST.
• Efficient for prefix search and longest prefix match.
• Faster than hash tables for small keys.
• In contrast to the pseudorandom order of hash tables, it allows for ordered iteration.
• The time complexity of deletion is O(l), where l is the length of a word.

Disadvantages of the Trie Data Structure

• Due to the large number of node pointers, Trie consumes a lot of memory.
• Each node can have a large number of pointers, up to characters in the worst-case scenario.
• Efficient hash tables have an O(1) lookup time, which is quicker than Trie's O(l) lookup, where l is the string length.