A discrete memoryless source has an alphabet of 5 symbols. The symbol probabilities are as following.
P(s0)=0.55,P(s1)=P(s2)=0.15,P(s3)=0.10,P(s4)=0.05.
What is the average codeword length of a Huffman code for this source ?
To find the average codeword length of a Huffman code for a discrete memoryless source, we first need to construct the Huffman code tree. The Huffman code is a prefix code that assigns variable-length codewords to each symbol based on their probabilities.
1. Start by listing all the symbols and their probabilities in decreasing order:
P(s0) = 0.55
P(s1) = P(s2) = 0.15
P(s3) = 0.10
P(s4) = 0.05
2. Merge the two symbols with the smallest probabilities and calculate their cumulative probabilities. Repeat this step until all symbols are merged into a single tree.
P(s1, s2) = P(s1) + P(s2) = 0.15 + 0.15 = 0.30
P(s1, s2, s4) = P(s1, s2) + P(s4) = 0.30 + 0.05 = 0.35
P(s0, s3, s1, s2, s4) = P(s0) + P(s3) + P(s1, s2, s4) = 0.55 + 0.10 + 0.35 = 1.00
3. Draw the Huffman code tree as shown below:
Root (1.00)
/ \
0.55 / \ 0.45
s0 Internal
/ \
0.35 / \ 0.10
s1, s2, s4 Internal s3
/ \
0.15 / \ 0.15
s1 s2
4. Assign codewords to each branch in the tree. Start from the root and assign a '0' to the left branch and '1' to the right branch. Repeat this step for each internal node until you reach the leaf nodes.
s0: 0
s1: 10
s2: 11
s3: 110
s4: 111
5. Calculate the average codeword length by multiplying each codeword length by its probability and summing the results.
Average codeword length = (0.55 * 1) + (0.15 * 2) + (0.15 * 2) + (0.10 * 3) + (0.05 * 3)
= 0.55 + 0.30 + 0.30 + 0.30 + 0.15
= 1.60
Therefore, the average codeword length of the Huffman code for this source is 1.60.