Time and Space Complexity of Huffman Coding Algorithm

Huffman coding is a popular algorithm used for the lossless data compression. It works by assigning variable-length codes to input characters with the shorter codes assigned to more frequent characters. This results in a prefix-free binary code meaning no code is a prefix of the another. The algorithm was developed by the David A. Huffman in 1952 and is widely used in the applications where compression efficiency is critical.

Operation	Time Complexity	Space Complexity
Building Huffman Tree	O(N log N)	O(N)
Encoding	O(N)	O(1)
Decoding	O(N)	O(1)

Time Complexity of Huffman Coding Algorithm:

• Building Huffman Tree: The time complexity of building the Huffman tree depends on method used to construct it. Using a priority queue to merge nodes has a time complexity of the O(N log N) where N is the number of the unique characters in the input.
• Encoding: Once the Huffman tree is constructed encoding a message involves traversing the tree to find the code for each character. This traversal has a time complexity of O(N) where n is the length of input message.
• Decoding: Decoding a Huffman-encoded message also requires traversing the Huffman tree. This traversal has a time complexity of O(N) where N is the number of the bits in the encoded message.

Auxiliary Space of Huffman Coding Algorithm:

• Building Huffman Tree: The auxiliary space required to build the Huffman tree is O(N) where N is the number of the unique characters in the input. This space is used to the store the nodes of the tree and the priority queue used for the merging.
• Encoding: During encoding the auxiliary space required is O(1) as it only involves storing the encoded message.
• Decoding: Decoding also requires O(1) auxiliary space as it only involves storing decoded message.

Conclusion:

In conclusion, Huffman coding is a powerful algorithm for the lossless data compression offering efficient time and space complexity characteristics. By assigning variable-length codes to the input characters based on their frequency it achieves compression ratios close to the theoretical limit while maintaining fast encoding and decoding speeds. The time complexity of the building the Huffman tree is O(N logN) where N is the number of the unique characters in the input.