Question

15.3-6 Generalize Huffman's algorithm to ternary codewords (i.e., codewords using the symbols 0,1 , and 2 ), and prove that it yields optimal ternary codes.

Answer 1

Huffman's algorithm can be generalized for ternary codewords using symbols 0, 1, and 2. Symbols with higher frequencies are assigned shorter codewords, resulting in optimal ternary codes. The algorithm involves creating a frequency table, combining symbols into a single tree, and assigning codewords based on the tree structure. It is proven that Huffman's algorithm yields optimal codes due to the property of no codeword being a prefix of another codeword.

Step-by-step explanation:

Huffman's algorithm is a method for constructing optimal prefix codes. In the case of ternary codewords, where the symbols used are 0, 1, and 2, the algorithm can be generalized by assigning shorter codewords to more frequently occurring symbols. Here is a step-by-step explanation of the generalized Huffman's algorithm:

1. Create a frequency table of the symbols in the input data.
2. Sort the symbols in decreasing order of frequency.
3. Take the two symbols with the lowest frequencies and combine them into a parent node, assigning the sum of their frequencies as the frequency of the parent node.
4. Repeat step 3 until all symbols are combined into a single tree.
5. Assign 0, 1, and 2 as codewords to the left, middle, and right branches of the tree, respectively.
6. Traverse the tree to assign unique ternary codewords to each symbol, following the path from the root to each symbol.

To prove that Huffman's algorithm yields optimal ternary codes, we can use the property of Huffman codes that no codeword is a prefix of another codeword. Since the codewords are assigned based on frequency, more frequently occurring symbols will have shorter codewords, resulting in a more efficient encoding and decoding process.