(c) Find the Huffman encoding for the following alphabet and set of frequencies. \[ \{(a, 0.12),(b, 0.38),(c, 0.1),(e, 0.25),(f, 0.06),(d, 0.05),(g, 0.01),(h, 0.03)\} \]

To construct the Huffman encoding for the given alphabet and set of frequencies: \[ \{(a, 0.12),(b, 0.38),(c, 0.1),(e, 0.25),(f, 0.06),(d, 0.05),(g, 0.01),(h, 0.03)\} \] we'll follow the Huffman coding algorithm step by step. ### Step 1: List Symbols by Frequency First, sort the symbols in ascending order based on their frequencies: 1. **g**: 0.01 2. **h**: 0.03 3. **d**: 0.05 4. **f**: 0.06 5. **c**: 0.10 6. **a**: 0.12 7. **e**: 0.25 8. **b**: 0.38 ### Step 2: Build the Huffman Tree **Iteration 1:** - **Combine the two smallest frequencies:** **g (0.01)** and **h (0.03)** → **gh (0.04)** **Updated List:** - gh: 0.04 - d: 0.05 - f: 0.06 - c: 0.10 - a: 0.12 - e: 0.25 - b: 0.38 --- **Iteration 2:** - **Combine the next two smallest:** **gh (0.04)** and **d (0.05)** → **ghd (0.09)** **Updated List:** - ghd: 0.09 - f: 0.06 - c: 0.10 - a: 0.12 - e: 0.25 - b: 0.38 --- **Iteration 3:** - **Combine the next two smallest:** **f (0.06)** and **ghd (0.09)** → **fghd (0.15)** **Updated List:** - fghd: 0.15 - c: 0.10 - a: 0.12 - e: 0.25 - b: 0.38 --- **Iteration 4:** - **Combine the next two smallest:** **c (0.10)** and **a (0.12)** → **ca (0.22)** **Updated List:** - fghd: 0.15 - ca: 0.22 - e: 0.25 - b: 0.38 --- **Iteration 5:** - **Combine the next two smallest:** **fghd (0.15)** and **ca (0.22)** → **fghdca (0.37)** **Updated List:** - fghdca: 0.37 - e: 0.25 - b: 0.38 --- **Iteration 6:** - **Combine the next two smallest:** **e (0.25)** and **fghdca (0.37)** → **efghdca (0.62)** **Updated List:** - efghdca: 0.62 - b: 0.38 --- **Iteration 7:** - **Combine the remaining nodes:** **b (0.38)** and **efghdca (0.62)** → **Root (1.00)** ### Step 3: Assign Codes Starting from the root, assign binary codes by traversing the tree. Typically, assign **0** to the left branch and **1** to the right branch at each internal node. Here's the detailed assignment: 1. **Root (1.00):** - **Left Child (b):** `0` - **Right Child (efghdca):** `1` 2. **efghdca (0.62):** - **Left Child (e):** `10` - **Right Child (fghdca):** `11` 3. **fghdca (0.37):** - **Left Child (fghd):** `110` - **Right Child (ca):** `111` 4. **fghd (0.15):** - **Left Child (f):** `1100` - **Right Child (ghd):** `1101` 5. **ghd (0.09):** - **Left Child (gh):** `11010` - **Right Child (d):** `11011` 6. **gh (0.04):** - **Left Child (g):** `110100` - **Right Child (h):** `110101` 7. **ca (0.22):** - **Left Child (c):** `1110` - **Right Child (a):** `1111` ### Final Huffman Codes | Symbol | Frequency | Huffman Code | |--------|-----------|--------------| | **b** | 0.38 | `0` | | **e** | 0.25 | `10` | | **f** | 0.06 | `1100` | | **d** | 0.05 | `11011` | | **g** | 0.01 | `110100` | | **h** | 0.03 | `110101` | | **c** | 0.10 | `1110` | | **a** | 0.12 | `1111` | ### Verification To ensure the correctness of the Huffman codes: 1. **Prefix Property:** No code is a prefix of another code. 2. **Efficiency:** The most frequent symbols have the shortest codes. Indeed: - **b (0.38)** has the shortest code: `0` - **e (0.25)** has `10` - Less frequent symbols like **g (0.01)** have longer codes: `110100` This satisfies both the prefix property and ensures an efficient encoding based on the given frequencies.