Huffman Coding

How Huffman coding works

Huffman coding operates by building a binary tree from the bottom up, based on symbol frequencies:

1. Calculate frequency/probability for each symbol
2. Create leaf nodes for each symbol
3. Iteratively combine the two lowest frequency nodes
4. Assign 0/1 to left/right branches
5. Extract codes by traversing paths from root to leaves

Mathematical foundations

The expected code length $L$ for a Huffman code is bounded by:

$H(X) \leq L < H(X) + 1$

Where $H(X)$ is the Shannon entropy of the source:

$H(X) = -\sum_{i=1}^n p_i \log_2(p_i)$

This proves Huffman coding achieves optimal compression among all prefix codes.

Applications in time-series data

In time-series databases, Huffman coding is often used to compress:

• Timestamp deltas
• Numerical value differences
• Categorical data fields
• Metadata and tags

The algorithm particularly excels when:

• Symbol frequencies are highly skewed
• The data contains repeating patterns
• Storage or transmission bandwidth is constrained

Compression efficiency

The compression ratio achieved depends on:

1. Distribution of symbol frequencies
2. Alphabet size
3. Input data size
4. Overhead of storing the code table

For time-series data, Huffman coding is often combined with other techniques like delta encoding and run-length encoding for optimal compression.

Implementation considerations

When implementing Huffman coding, key factors include:

• Code table representation and storage
• Tree building and traversal efficiency
• Memory usage during compression/decompression
• Handling dynamic vs static code tables
• Error detection and recovery

Modern implementations often use canonical Huffman codes to reduce the overhead of storing and transmitting code tables.

Performance characteristics

• Time Complexity

  • Tree construction: O(n log n)
  • Encoding: O(n)
  • Decoding: O(n)

• Space Complexity

  • Tree storage: O(k) where k is alphabet size
  • Encoded data: Variable based on compression ratio

The algorithm provides excellent compression with relatively low computational overhead, making it suitable for real-time applications.

Related topics

• Arithmetic coding for alternative entropy coding
• Shannon entropy for theoretical compression limits
• Information gain in compression evaluation
• Minimum description length for model selection