Question

Use Huffman's algorithm to construct an optimal binary prefix code for the letters in the following table. Letter \(: A B I M S \times Z\) Frequency : 1271810952

Short answer

The binary prefix (Huffman) code for the letters will be generated by Huffman's Algorithm with the result differing based on the construction of the Huffman Tree. Generally, the character \(A\) with the highest frequency will have the shortest binary prefix code.

Step-by-step solution

Setup

Create a priority queue of nodes. Each node has as elements one of the characters \(A, B, I, M, S, \times, Z\) and its frequency \(127, 18, 10, 9, 5, 2\). The elements should be ordered by frequency, from the least frequent to the most frequent. The node structure could be \(character, frequency, leftChild, rightChild\).

Building the Huffman Tree

Remove the two nodes with the smallest frequencies from the queue. Create a new node with these two nodes as children and with frequency equal to the sum of the two nodes' frequencies. Insert this new node into the queue. Repeat this step until the queue has only one node left, this will be the root node of the Huffman Tree.

Deriving the Huffman Code

Starting from the root of the Huffman Tree, move to the left child assigning a value of '0' and to the right child assigning a value of '1'. Concatenate these values upwards to form binary codewords based on the paths from the root to each character.

Key concepts

Optimal Binary Prefix Code

When it comes to data compression, finding the most efficient way to represent information can lead to significant savings in space and time. An optimal binary prefix code plays a crucial role in this, and that's where Huffman's algorithm comes into the spotlight.

Huffman's algorithm generates an optimal binary prefix code by assigning shorter codes to more frequently occurring characters and longer codes to less frequent characters. What makes them 'prefix' codes is that no code is a prefix of another, meaning that the encoded data can be unambiguously decoded. This optimal arrangement is crucial for efficient data compression, ensuring that on average the smallest number of bits are used.

The example in the exercise presents various letters with corresponding frequencies. The goal is to use Huffman's algorithm to assign the shortest possible binary codes to the most frequently occurring characters (A, B, I, M, S, ×, and Z in this case) while ensuring these codes are uniquely decodable. The method is centered on building a tree structure, the Huffman tree, which provides a visual and functional representation of the prefix codes.

Priority Queue

Central to executing Huffman's algorithm effectively is the use of a priority queue. This isn't just any queue; it's a specialized data structure that orders elements based on their priority — in this instance, the frequency of the characters.

Imagine standing in line where your position isn't determined by when you arrived but by the priority of your task—this is how a priority queue functions. For Huffman's algorithm, it starts by creating a node for each character and sorting them in the priority queue by their frequency, with the least frequent at the front of the queue.

Here's why this is important: as you build the Huffman tree, you repeatedly extract the two nodes with the lowest frequency (highest priority in this context) from the queue and combine them. This process makes sure that the new nodes are correctly placed in the tree according to their composite frequencies and reinserted into the queue in the proper position. It's a neat and systematic way to ensure the tree is built in an optimal fashion.

Huffman Tree

The Huffman tree itself is the crown jewel of Huffman's algorithm. Think of it as a family tree, but instead of relatives, you have characters and their frequencies. This tree is binary, meaning that each node has zero, one, or two children, and is built by the specific rules of Huffman's algorithm.

The process begins small, like planting a seed, by removing the two nodes from the priority queue — these nodes are the least frequent characters. They're combined into a new node with a frequency equal to their sum, and this new node is then put back into the priority queue. You repeat this process—pluck, combine, replant—until a sole node remains. This node, now the root of the Huffman tree, holds within its branches the ideal binary prefix codes for the characters based on their paths from the root to each leaf.As Huffman trees grow, they reveal a pathway to each character. A left turn along the branch corresponds to a binary '0', a right turn to a '1'. Following these paths, you can derive the binary codes used to compress data. In the exercise, for example, characters like 'A' with high frequency would be positioned closer to the root to minimize their code lengths, making the overall compressed data smaller.