Huffmans Rules : Introduction & Applications

Goals

Huffman's rules define a method for creating optimal, prefix-free variable-length codes, assigning shorter binary codes to frequent symbols and longer ones to rare symbols, achieved by repeatedly merging the two least frequent items into a new node in a binary tree until one root remains, ensuring no code is a prefix of another for unambiguous decoding. Key rules involve building a min-heap of frequencies, combining lowest-frequency nodes (0 for left, 1 for right), and the crucial Prefix Rule: no codeword is a prefix of another.
Core Principles

• Frequency-Based: More frequent symbols get shorter codes; less frequent get longer codes, minimizing total encoded length.
• Prefix-Free (No Ambiguity): No codeword is a prefix of another (e.g., if 'A' is '0', 'B' can't be '01'), preventing decoding errors.
• Greedy Approach: Repeatedly merge the two least probable symbols/nodes.

Algorithm Steps (Building the Tree)

1. Create Leaf Nodes: Start with a leaf node for each symbol, storing its frequency.
2. Use a Priority Queue (Min-Heap): Add all leaf nodes to a min-heap, ordered by frequency.
3. Combine Nodes: While more than one node is in the heap:
   • Extract the two nodes with the lowest frequencies (say, node1 and node2).
   • Create a new internal node with node1 as the left child (assigned '0') and node2 as the right child (assigned '1').
   • Set the new node's frequency to the sum of node1's and node2's frequencies.
   • Insert the new internal node back into the heap.
4. Final Tree: Repeat until only one node (the root) remains in the heap.

Getting the Codes

• Traverse the tree from the root to each leaf node.
• The path (sequence of '0's and '1's) forms the codeword for that symbol.

Canonical Huffman (Optional Standardization)

• Sort symbols first by code length (shortest to longest), then alphabetically/numerically for ties.
• Assign '0' to the first symbol (for its length).
• Assign subsequent codes by incrementing the previous code and left-shifting (adding zeros) as needed for length changes, ensuring codes remain numerically ordered.

Prerequisites

• Basic Computing Knowledge