Question

Design an optimum binary prefix code for the alphabet {a, b, c,

d, e, f, g, h} with corresponding probabilities of occurrence at
{0.05, 0.1, 0.05, 0.15, 0.25, 0.1, 0.2, 0.1}

Answer 1

To design an optimum binary prefix code, we need to use the Huffman coding algorithm. Here are the steps to follow:

1. Arrange the alphabet and their corresponding probabilities in decreasing order of probabilities: - Alphabet: {e, g, h, d, f, b, c, a} - Probabilities: {0.25, 0.2, 0.1, 0.15, 0.1, 0.1, 0.05, 0.05}

2. Create a binary tree where each leaf node represents an alphabet and its probability. Start by combining the two smallest probabilities to form a parent node. Assign 0 to the left child and 1 to the right child.
3. Repeat step 2 until all the nodes are combined into a single tree. At each step, merge the two smallest probabilities to form a new parent node.
4. Assign binary codes to each alphabet by traversing the tree. Assign 0 when moving left and 1 when moving right.

Following the above steps, we get the optimum binary prefix code:

e: 00 g: 01 h: 10 d: 110 f: 1110 b: 11110 c: 11111 a: 111110 In this code, the more frequent alphabets have shorter codes, reducing the average code length and maximizing the efficiency of the code.