Trie Data Structure

When it comes to storing and retrieving information efficiently, what data structure comes to mind? Is it the commonly used arrays, lists, or hash tables? While these structures serve their purposes, there is one lesser-known data structure that excels in certain scenarios – the Trie.

The Trie, short for retrieval trie, is a specialized tree-like data structure specifically designed for efficient information retrieval and string manipulation. Its unique characteristics make it an invaluable tool in various domains, ranging from search engines to routing algorithms.

But what exactly is a Trie, and what sets it apart from other data structures? How does it work, and why is it so effective for string-related tasks? In this article, we will explore the Trie data structure in depth, unraveling its inner workings, operations, optimizations, and real-world applications.

Table of Contents

Key Takeaways:

  • The Trie data structure is optimized for efficient information retrieval and string manipulation.
  • It differs from other data structures, such as arrays and lists, in its unique design and characteristics.
  • Trie operations include insertion, searching, and deletion, making it a powerful tool for various applications.
  • Tries outperform Hash Tables in scenarios that involve prefix-based operations, such as autocomplete and spell checking.
  • Optimizations and variations, such as compression techniques and Radix Tries, enhance the Trie’s performance and scalability.

What is a Trie?

A Trie, also known as a prefix tree, is a specialized data structure used for efficient information retrieval and string manipulation. Unlike other data structures such as arrays, linked lists, and hash tables, Tries are specifically designed to handle sets of strings and provide efficient search operations.

The structure of a Trie allows for fast searching, insertion, and deletion of strings, making it ideal for applications involving dictionary systems, autocomplete functionalities, spell checking algorithms, and IP routing.

A Trie consists of a collection of nodes, where each node represents a character of a string. The nodes are arranged in a hierarchical manner, with the root node at the top and the child nodes branching out based on the characters they represent.

Each node in the Trie contains a set of child nodes, representing the possible characters that can follow in the string. This structure allows for efficient retrieval of strings by traversing down the Trie based on the input characters. By utilizing a Trie, the time complexity of string operations such as searching and insertion can be significantly reduced.

Here is an example of a Trie structure:

Node Child Nodes
Root A, B, C, D, E
A P, T
C A, A
D O, O
E N, L
P P, L, E
A C, T
O G, G

In the example above, the Trie represents a set of strings: “A”, “AB”, “ACA”, “ACAT”, “ADO”, and “ENL”. Each node in the Trie represents a character, and the child nodes indicate the possible characters that can follow.

By traversing the Trie based on the input characters, string operations can be performed efficiently, making Tries a powerful tool for various text processing tasks.

Trie Operations

In a Trie data structure, various operations can be performed to manipulate the stored data efficiently. The most commonly used operations are Trie insert, Trie search, and Trie delete.

Trie Insert

Trie insert operation allows adding a new word or string to the Trie. It starts at the root node and traverses through the Trie following the path that matches the characters of the word being inserted.

Here is a step-by-step breakdown of the Trie insert operation:

  1. Start at the root node.
  2. Check if the current character of the word exists as a child of the current node.
  3. If the character exists, move to the next node and continue to the next character.
  4. If the character doesn’t exist, create a new node for that character and link it to the current node.
  5. Repeat steps 2-4 until all characters of the word are processed.
  6. Mark the last node of the word as a terminal node to indicate the end of the word.

Trie Search

Trie search operation is used to find whether a given word or string exists in the Trie. It follows the same path as the Trie insert operation but stops if any character of the word is not found.

Here is a step-by-step breakdown of the Trie search operation:

  1. Start at the root node.
  2. Check if the current character of the word exists as a child of the current node.
  3. If the character exists, move to the next node and continue to the next character.
  4. If the character doesn’t exist, the word doesn’t exist in the Trie.
  5. If all characters are processed and the last node is marked as a terminal node, the word exists in the Trie.

Trie Delete

Trie delete operation allows removing a word or string from the Trie. It follows a similar path as the Trie search operation but additionally removes any unnecessary nodes to optimize the Trie structure.

Here is a step-by-step breakdown of the Trie delete operation:

  1. Start at the root node.
  2. Check if the current character of the word exists as a child of the current node.
  3. If the character exists, move to the next node and continue to the next character.
  4. If the character doesn’t exist, the word doesn’t exist in the Trie.
  5. If all characters are processed and the last node is marked as a terminal node, remove the terminal marker and check for child nodes.
  6. If the last node has no child nodes, remove the node and traverse back to the previous node.
  7. If the previous node has no other child nodes, remove it and continue to traverse back until reaching the root node or a node with other child nodes.

By utilizing these Trie operations, developers can efficiently store, search, and remove words or strings in the Trie data structure. These operations provide a powerful foundation for various applications such as autocomplete systems, spell checking algorithms, and IP routing.

Trie vs. Hash Table

When it comes to data structures for efficient information retrieval and string manipulation, two popular options that often come up are Tries and Hash Tables. While both have their strengths and weaknesses, a comparison between the two showcases the advantages of using a Trie over a Hash Table for certain applications.

Let’s delve into the details:

Trie Advantages

“Tries provide fast retrieval of stored information, especially for prefix-based searches.”

Tries excel in scenarios where we need to quickly search for words or strings based on their prefixes. This makes them ideal for autocomplete systems, spell checking algorithms, and other applications that involve searching based on partial matches. With a Trie, the search operation can be completed in a time complexity of O(m), where m is the length of the search key.

Additionally, Tries provide efficient memory utilization as they store only the necessary information in their nodes. This allows for space optimization and reduces the overall memory footprint of the data structure.

Hash Table Disadvantages

“Hash Tables have high memory consumption and suffer from collision-related issues.”

One of the main disadvantages of using a Hash Table is the potential for high memory consumption. Hash Tables need to allocate a fixed amount of memory for storing a predetermined number of slots, regardless of the amount of data actually stored. This can result in wasteful memory usage, especially when dealing with sparse data or when expecting the data to grow dynamically.

Moreover, Hash Tables are susceptible to collision-related issues. Collisions occur when multiple keys hash to the same location, forcing the Hash Table to handle these collisions through techniques like chaining or open addressing. This extra processing can have an impact on the overall performance of the data structure, especially when the number of collisions increases.

Trie vs. Hash Table Comparison

Trie Hash Table
Efficient for prefix-based searches Not optimized for prefix-based searches
Low memory consumption High memory consumption
Fast retrieval for partial matches Relies on complete key matches
Ideal for autocomplete, spell checking, and IP routing General-purpose data structure

As shown in the comparison table, Tries offer distinct advantages over Hash Tables when it comes to efficient prefix-based searches and memory utilization. However, it’s important to consider the specific requirements of your application before choosing between the two data structures. While Tries excel in certain scenarios, Hash Tables have their own strengths and may be better suited for other use cases.

Trie Implementation

Implementing a Trie involves defining a Trie node structure and understanding the complexity of Trie operations. The Trie data structure is commonly used for efficient string manipulation and information retrieval tasks.

Trie Node

A Trie node is the building block of the Trie data structure. It contains the following components:

  1. A character value to represent the current node
  2. A boolean flag to indicate whether the current node is the end of a word
  3. A collection of child nodes, represented as pointers or references

The Trie node structure allows for the efficient storage and retrieval of strings, with each node representing a character in the string.

Complexity of Trie Operations

The complexity of Trie operations depends on the length of the longest word in the Trie, denoted as n.

Operation Time Complexity
Trie Insert O(n)
Trie Search O(n)
Trie Delete O(n)

Table: Complexity of Trie Operations

Trie operations such as insertion, search, and deletion have a time complexity of O(n), where n represents the length of the input string. This makes Trie a suitable choice for scenarios where fast string manipulation and retrieval are required.

Applications of Tries

The Trie data structure is widely used in various applications, showcasing its versatility and efficiency in solving different computational problems. Let’s explore some of the key applications of Tries:

1. Autocomplete Systems

In the age of digital communication, autocomplete systems have become an essential component of user interfaces. Tries provide a fast and efficient way to implement autocomplete functionality, suggesting word completions based on user input. By storing all possible words or phrases in a Trie, autocomplete systems can quickly retrieve and present relevant suggestions, enhancing user experience and productivity.

2. Spell Checking

Spell checking algorithms heavily rely on Tries to validate the correctness of words in a text. By maintaining a Trie with a dictionary of valid words, spell checking systems can efficiently identify and correct misspelled words, offering suggestions for alternative spellings. Tries enable high-performance spell checking by providing fast access to words and efficiently detecting spelling errors.

3. IP Routing

In computer networks, IP routing algorithms play a crucial role in determining the path through which data packets should be transmitted. Tries are used to store and search routing tables efficiently, enabling routers to make rapid routing decisions. By using a Trie data structure, IP routing algorithms can scale well and handle large routing tables, ensuring efficient packet forwarding across complex network topologies.

Tries are also employed in other domains, such as:

  • Data Compression: Tries can be used to compress data by storing repetitive sequences as shortened representations.
  • Text Editors: Tries can assist in implementing features like word highlighting and pattern matching within text editors.
  • Genomics: Tries are utilized in DNA sequencing and analysis algorithms for efficient pattern searching.

These are just a few examples of the diverse applications of Tries. Their ability to quickly retrieve and manipulate information makes them indispensable in various computational tasks.

Optimizations for Tries

In the quest for efficient information retrieval and string manipulation, optimizations play a crucial role in improving the performance of Trie data structures. By implementing compression techniques and utilizing Ternary Search Tries, developers can unlock enhanced efficiency and reduced memory consumption.

Compression Techniques

One of the key optimizations for Tries is the implementation of compression techniques. These techniques aim to reduce the memory footprint of the Trie while maintaining its functionality. Two commonly used approaches in Trie compression are:

  1. Prefix Compression: This technique involves compressing common prefixes in Trie nodes into a single prefix node. By storing shared prefixes only once, the overall memory usage can be significantly reduced.
  2. Patricia Compression: Also known as radix compression, this method reduces the number of nodes in the Trie by merging nodes that have a single child. By eliminating redundant nodes, space efficiency is greatly improved.

Ternary Search Tries

Ternary Search Tries are another optimization technique that can enhance the performance of Tries. Instead of using a separate pointer or child node for every character in the alphabet, Ternary Search Tries employ a binary search tree-like structure with three child pointers: less than, equal to, and greater than. This structure allows for quicker traversal and search operations, resulting in improved efficiency.

By implementing compression techniques and using Ternary Search Tries, developers can optimize Tries to achieve faster lookup times, reduced memory consumption, and overall improved performance.

Variations of Tries

In the realm of Trie structures, there exist fascinating variations that offer unique features and cater to specific use cases. Two prominent variations worth exploring are Radix Trie and Patricia Trie.

Radix Trie

Radix Trie, also known as Radix Patricia Trie, is a modified version of the Trie data structure that optimizes space utilization. It achieves this by merging nodes with a single child, reducing the storage overhead compared to a traditional Trie.

By compressing common prefixes into a single node, Radix Trie effectively eliminates redundant information, leading to improved memory efficiency. This makes Radix Trie particularly suitable for scenarios where memory consumption is a concern.

Patricia Trie

Patricia Trie, or Practical Algorithm to Retrieve Information Coded in Alphanumeric, is another variation of the Trie data structure. It was designed to address some limitations of the regular Trie, such as excessive space usage for storing individual characters.

Patricia Trie achieves space optimization by storing entire strings, rather than individual characters, in each node. This approach significantly reduces the memory overhead and results in a compact structure that is often more efficient than traditional Tries for storing large sets of strings.

Moreover, Patricia Trie’s search and insert operations have a time complexity of O(m), where m is the length of the search string, making it an appealing choice for applications that involve frequent string lookups or insertions.

Use Cases

Both Radix Trie and Patricia Trie find applications in various domains, including:

  • Autocomplete systems: Tries, particularly Radix Tries, are commonly employed to quickly suggest completions as users type in search queries or input fields.
  • Spell checking: Patricia Tries are often utilized to efficiently verify the correctness of words or suggest alternatives by leveraging the compactness of their structure.
  • IP routing: Radix Tries’ memory efficiency makes them ideal for managing large routing tables in network infrastructure devices.

These variations of Tries offer valuable alternatives to the conventional Trie data structure, addressing specific needs in terms of space optimization and performance. Their unique features make them indispensable components in various applications, enhancing efficiency and enabling seamless information retrieval.

Challenges and Limitations

In the journey of utilizing Trie data structures, there are certain challenges and limitations that developers need to be aware of. This section highlights some of the key obstacles in Trie implementation, focusing on memory consumption and scalability concerns.

Memory Consumption

One of the major challenges in using Tries is their memory consumption. The Trie data structure requires a significant amount of memory to store the entire set of keys. In scenarios where the keys are large or numerous, this can lead to high memory usage, potentially impacting the overall performance of the system.

For example, consider a Trie used in a spell checking application that needs to store a large dictionary of words. Each word in the dictionary will be represented by a separate node in the Trie, contributing to memory overhead. As the size of the dictionary grows, so does the memory consumption of the Trie.

To mitigate this issue, developers can apply various memory optimization techniques. One common approach is to use compressed Trie structures, such as Patricia Tries or Burst Tries, which store prefixes compactly, reducing memory requirements. Another technique is to implement prefix compression, where common prefixes are shared among multiple keys, further minimizing memory consumption.


Scalability is another critical aspect to consider when working with Tries. As the number of keys increases, the time complexity of operations like insertion, deletion, and search in the Trie can potentially impact the overall system performance.

Due to the hierarchical nature of Tries, the time complexity of these operations is mainly dependent on the length of the key being processed. In worst-case scenarios, where the key length is high, the performance of Trie operations may degrade, resulting in slower response times.

Developers can address scalability challenges by employing various optimization techniques. One such technique is the use of Ternary Search Tries, which strike a balance between space efficiency and search performance. Ternary Search Tries reduce the branching factor of the Trie, leading to improved scalability without sacrificing much memory consumption.

“When working with Tries, it is essential to carefully balance memory consumption and scalability. Employing memory optimization techniques and leveraging specialized Trie variations can help overcome these challenges, providing efficient and scalable solutions for a wide range of applications.”

Challenges Impact
Memory Consumption High memory usage, potential impact on system performance
Scalability Potential degradation in performance as the number of keys increases

Trie in Real-World Systems

In real-world systems, Tries play a crucial role in optimizing search functionality and URL routing algorithms. Leading companies like Google harness the power of Trie data structure to enhance their search engine performance and provide users with accurate and swift search results.

Google Search

Google’s search engine relies heavily on Tries to efficiently retrieve relevant search results from its vast index of web pages. By constructing a Trie of keywords and their corresponding web page references, Google can quickly match user queries against the Trie, allowing for speedy and accurate search results.

The use of Tries allows Google to handle complex search queries and provide valuable features like autocomplete suggestions. As users type their search queries, the Trie is used to predict the rest of the query, offering real-time suggestions that help users find the information they need more quickly and conveniently.

URL Routing

URL routing algorithms in web applications take advantage of Tries to efficiently map and route incoming URLs to the appropriate web pages or resources. By building a Trie of URL patterns and associating them with the corresponding handlers, web servers can process incoming requests with high performance and minimal overhead.

When a user visits a specific URL, the server can traverse the Trie to match the incoming URL against the defined patterns, quickly determining the appropriate handler for processing the request. This enables web applications to handle a large number of requests simultaneously while maintaining optimal performance.

Overall, Tries have proven to be invaluable in real-world systems, enabling seamless search functionality and efficient URL routing. Their ability to handle complex tasks with speed and accuracy makes them a vital component in modern computing.

Use Case Benefits of Tries
Google Search
  • Fast search results
  • Autocomplete suggestions
  • Efficient query handling
URL Routing
  • High-performance request processing
  • Minimal overhead
  • Scalability for handling large volumes of requests

Performance Analysis

In this section, a comprehensive performance analysis of the Trie data structure will be conducted, focusing on its time complexity and space complexity in various scenarios. Understanding the performance characteristics of Tries is essential for determining the efficiency and suitability of this data structure for different applications.

Time Complexity

The time complexity of Trie operations, including insertion, search, and deletion, plays a crucial role in assessing its efficiency. By analyzing the time complexity, we can gain insights into how the Trie data structure performs as the size of the dataset and the length of the strings increase.

Below is a table that summarizes the time complexity of common Trie operations:

Operation Time Complexity
Trie Insertion O(L)
Trie Search O(L)
Trie Deletion O(L)

In the table, “L” represents the length of the string being inserted, searched, or deleted. As seen, the time complexity of Trie operations is directly proportional to the length of the strings. This characteristic makes Tries particularly efficient for tasks that involve string manipulation and retrieval, such as autocomplete systems and spell checking algorithms.

Space Complexity

The space complexity of Tries refers to the amount of memory required to store the Trie data structure and the associated data. Analyzing the space complexity helps assess the memory footprint and scalability of Tries.

The space complexity of Tries is influenced by factors such as the number of unique strings, their lengths, and the number of characters in the alphabet. In general, the space complexity of a Trie can be considered as O(N*M), where N is the number of unique strings and M is the maximum length of the strings.

It is important to note that Tries consume more memory compared to other data structures, such as Hash Tables, due to the need to store each character individually. However, advancements in compression techniques, such as prefix compression, can significantly reduce the memory consumption of Tries without compromising their efficiency.

By evaluating the time complexity and space complexity of Tries, we can make informed decisions about their suitability for specific applications. The performance analysis provides valuable insights into the efficiency, scalability, and trade-offs associated with using Tries in various scenarios.

Tips for Using Tries Effectively

When working with Trie data structures, there are several tips and techniques that can help optimize their performance and maximize their efficiency. By implementing prefix compression and memory optimization strategies, developers can significantly improve the speed and memory usage of their Trie implementations.

1. Prefix Compression

Prefix compression is a technique that reduces the memory footprint of a Trie by compressing common prefixes in the keys. Instead of storing each character individually, the shared prefixes are represented as a single node with a count or a flag indicating the number of keys that share that prefix.

“Prefix compression in Tries allows for significant memory savings, especially when dealing with large datasets. By efficiently storing and compressing common prefixes, the Trie structure becomes more compact and consumes less memory.”

2. Memory Optimization

Tries can consume a considerable amount of memory, especially when dealing with large datasets or long keys. To optimize memory usage, developers can employ various techniques:

  • Implement compressed Trie structures that use less memory for each node, such as Ternary Search Tries.
  • Use bit-level compression techniques to minimize memory usage, such as bitwise tries.
  • Consider memory-efficient node representations, such as using arrays instead of pointers.

3. Efficient Trie Operations

To further enhance the performance of Trie operations, consider the following best practices:

  • Use tail end compression to reduce the number of nodes and improve search efficiency.
  • Implement Trie traversal algorithms in an iterative manner instead of recursive, to save memory and avoid stack overflow errors.
  • Consider optimizing specific operations based on the use case. For example, if searching is more frequent than insertion, choose a Trie variant optimized for search operations.

4. Testing and Benchmarking

When using Tries in performance-critical applications, it’s crucial to conduct thorough testing and benchmarking to identify potential bottlenecks and optimize accordingly. Proper load testing with realistic datasets can help identify areas for improvement and ensure the Trie implementation meets the desired performance requirements.

5. Documentation and Code Maintenance

Lastly, maintaining detailed documentation and appropriately organizing the codebase is essential for long-term success. Clear documentation helps other developers understand the Trie implementation, its optimizations, and best practices. Additionally, consistent code maintenance ensures that any optimizations or improvements made to the Trie structure are properly incorporated.

By following these tips and techniques, developers can leverage the power of Tries while optimizing memory usage and enhancing overall performance. The table below summarizes the key tips for using Tries effectively.

Tips for Using Tries Effectively
Prefix Compression
Memory Optimization
Efficient Trie Operations
Testing and Benchmarking
Documentation and Code Maintenance

Future Trends and Developments

In the ever-evolving landscape of computing, the Trie data structure continues to garner significant attention from researchers and developers alike. Ongoing research and exploration have paved the way for exciting advancements and future trends in the field of Tries. These developments hold the potential to revolutionize information retrieval and string manipulation, further enhancing the efficiency and effectiveness of computing systems.

Advancements in Trie Research

Researchers have been actively investigating new approaches and techniques to improve the performance of Tries. These advancements aim to address some of the challenges associated with Tries, such as memory consumption and scalability. By devising novel strategies and optimization techniques, Trie researchers are pushing the boundaries of what can be achieved with this powerful data structure.

“Advancements in Trie research are unlocking new possibilities and expanding the realm of applications where Tries can be effectively utilized.”

Expanding Applications of Tries

As technology continues to advance, so does the range of applications that can benefit from Tries. While Tries have already proven their value in areas like autocomplete systems, spell checking, and IP routing algorithms, there is a growing interest in exploring new domains where Tries can be leveraged. From natural language processing to network analysis, Tries have the potential to play a crucial role in various real-world systems.

Efficiency and Scalability Improvements

Efforts are underway to enhance the efficiency and scalability of Tries, making them more viable for large-scale applications. Researchers are exploring new compression techniques and variations of Tries, such as Radix Tries and Patricia Tries, to reduce memory requirements and improve performance. These advancements not only open doors to new possibilities but also make Tries more accessible and practical for a broader range of use cases.

Potential Challenges and Future Direction

While exciting developments are underway, it is essential to recognize the potential challenges that lie ahead. Memory consumption and scalability remain significant concerns, especially when dealing with massive datasets and high-speed networks. Researchers are actively addressing these challenges, and future advancements will likely focus on optimizing Trie structures for resource-constrained environments and handling ever-increasing data volumes.

The Road Ahead

The future of Tries holds immense potential for transforming various areas of computing. From advancements in research to expanded applications and improved efficiency, Tries are poised to play a vital role in shaping the next generation of computing systems. As developers and researchers continue to push the boundaries, the impact of Tries is likely to be felt far and wide, paving the way for a more efficient and intelligent digital landscape.


In conclusion, the Trie Data Structure is a powerful tool in modern computing, offering efficient information retrieval and string manipulation capabilities. Throughout this article, we have explored the key aspects of Tries, including their definition, structure, operations, and implementation.

We have also examined the advantages of using Tries over Hash Tables for certain applications, as well as their various practical uses, such as autocomplete systems, spell checking, and IP routing algorithms. Additionally, we have discussed optimizations and variations of Tries, highlighting their potential for improved performance and flexibility.

However, it is important to acknowledge the challenges and limitations of using Tries, such as memory consumption and scalability concerns. Despite these challenges, Tries continue to be widely employed in real-world systems, including popular search engines like Google and URL routing algorithms.

In conclusion, the Trie Data Structure is a fundamental concept in modern computing, with ongoing research and development aiming to enhance its capabilities and address its limitations. As technology advances, it is expected that Tries will play an even more crucial role in efficiently managing and manipulating large volumes of data.


What is a Trie?

A Trie is a data structure that is used for efficient information retrieval and string manipulation. It is also known as a prefix tree or digital tree.

What operations can be performed on a Trie?

Various operations can be performed on a Trie, including insertion, searching, and deletion. These operations make Tries highly useful for tasks such as autocomplete, spell checking, and IP routing.

How does a Trie compare to a Hash Table?

Tries have several advantages over Hash Tables. Tries provide efficient prefix-based searching, while Hash Tables offer constant-time searching. Tries are particularly useful when dealing with tasks that involve word-based or prefix-based searching.

How is a Trie implemented?

A Trie is implemented using nodes, where each node represents a character in a string. The Trie complexity is determined by the number of nodes and the length of the strings being stored.

What are some practical applications of Tries?

Tries have various practical applications, including autocomplete systems, spell checking, and IP routing algorithms. They are widely used in systems like Google search and URL routing.

Are there any optimizations for improving Trie performance?

Yes, there are several optimizations that can be applied to Tries. These include compression techniques that reduce memory consumption and the use of specialized Trie variants, such as Ternary Search Tries.

What are some variations of Tries?

There are different variations of Tries, including Radix Tries and Patricia Tries. These variations have unique features and different use cases.

What are the challenges and limitations of using Tries?

Using Tries can present challenges, such as high memory consumption and scalability concerns. Careful design and optimization techniques are necessary to mitigate these limitations.

How are Tries used in real-world systems?

Tries are used extensively in real-world systems like Google’s search engine and URL routing algorithms. They play a crucial role in efficient information retrieval and routing decisions.

How is the performance of Tries analyzed?

Trie performance is analyzed by evaluating its time complexity, which measures the efficiency of operations, and space complexity, which assesses the memory requirements of the structure.

What are some tips for using Tries effectively?

To use Tries effectively, consider implementing prefix compression techniques to reduce memory usage. Optimization strategies such as memory allocation and deallocation can also enhance Trie performance.

What are the future trends and developments in the field of Tries?

Ongoing research and advancements are being made in the field of Tries. These developments aim to improve Trie performance, enhance scalability, and explore new applications for the data structure.

What is the significance of the Trie Data Structure in modern computing?

The Trie Data Structure is highly significant in modern computing due to its efficiency in information retrieval, string manipulation, and its practical applications in various systems and algorithms.

Avatar Of Deepak Vishwakarma
Deepak Vishwakarma


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