Question

What is the maximum number of bits that Huffman's greedy algorithm might use to encode a single symbol? (As usual, n = |Σ| denotes the alphabet size.)

a) n

b) n - 1

c) 2n - 1

d) 2n

Answer 1

In Huffman coding, the maximum number of bits to encode a single symbol from an alphabet of size n is n - 1, which occurs when the Huffman tree is extremely skewed.

Step-by-step explanation:

The maximum number of bits that Huffman's greedy algorithm might use to encode a single symbol, given an alphabet size n, is n - 1. This is because Huffman's algorithm constructs a binary tree where each symbol is represented by a leaf node. The worst-case scenario occurs when the tree is skewed, meaning one branch is much longer than the others. In such a skewed binary tree, the maximum depth is n - 1, where n is the number of symbols in the alphabet (the number of leaf nodes).

The greedy Huffman algorithm ensures no prefix rule is violated, and hence each symbol's encoding is unique. When the tree is completely balanced, the maximum depth would be log2(n), which represents the ideal scenario with the shortest average bit length per symbol. However, when considering the maximum bit length for any single symbol, n - 1 is the correct answer.