Huffman Coding: Optimal Prefix-Free Codes

This lecture covers the concept of Huffman coding, focusing on the construction of optimal prefix-free codes. Starting with the proof of the upper bound, it compares Shannon-Fano codes with Huffman codes, demonstrating the optimality of the latter. The lecture explains the construction of Huffman codes from the leaves, emphasizing the prefix-free property and optimality in terms of average codeword length. It introduces the concept of a tree with probabilities and the path-length lemma to compute average path lengths efficiently. The proof of optimality for Huffman's construction is discussed, along with key facts guiding the code construction process. The lecture concludes with a detailed explanation of how Huffman's code construction guarantees the smallest average codeword-length for a given alphabet.