Gamma Codes offer a powerful technique for compressing postings lists in inverted indexes, crucial for efficient information retrieval. This article delves into the mechanics of gamma encoding, exploring its efficiency, universality, and advantages over other compression methods.
Understanding Gamma Encoding
Gamma codes achieve variable-length encoding by representing a gap G (the difference between consecutive document IDs) as a pair: length and offset.
- Offset: The offset is the binary representation of
Gwith the leading 1 removed. For instance, the offset of 13 (binary 1101) is 101. - Length: The length signifies the number of bits in the offset, encoded using unary code. Unary code represents a number
nas a sequence ofn1s followed by a 0. Thus, the length of 101 (3 bits) is encoded as 1110.
The gamma code for 13 is the concatenation of the length and offset: 1110101.
Decoding involves reading the unary code until the terminating 0, determining the offset length. The offset is then read, the leading 1 is prepended, and the original gap is recovered. For example, decoding 1110101 involves reading 1110 (length 4), then reading the next 3 bits 101 (offset), adding the leading 1 back (1101), and converting from binary to decimal (13).
(Example of Gamma and Unary Codes)
Efficiency and Universality of Gamma Codes
The length of a gamma code for gap G is 2⌊log2G⌋ + 1 bits, close to the theoretical optimal encoding length. Remarkably, gamma codes are universal, performing efficiently even without prior knowledge of the gap distribution. This universality stems from the fact that gamma encoding is within a factor of 3 of the optimal encoding length for any probability distribution, as determined by entropy.
Furthermore, gamma codes are prefix-free, meaning no code is a prefix of another. This property allows unique decoding without delimiters, maximizing compression. They are also parameter-free, avoiding the complexities of model fitting and dynamic updates encountered with other methods.
Gamma Codes vs. Variable Byte Codes
While gamma codes often achieve higher compression ratios than variable byte codes (approximately 15% better for the Reuters-RCV1 corpus), they require more processing power to decode due to the bit-level operations involved. The choice between gamma and variable byte codes depends on the application’s specific requirements, balancing disk space conservation with query response time. For datasets with many small gaps (e.g., those containing stop words), gamma codes outperform variable byte codes. However, variable byte codes are more efficient for distributions dominated by larger gaps.
Gamma Codes in Practice
Applying Zipf’s law to the Reuters-RCV1 corpus, a theoretical estimation suggests gamma encoding compresses the inverted index to roughly one-fourth of its original size. In reality, the achieved compression is even greater, around one-tenth. This discrepancy arises from the limitations of Zipf’s law as a perfect model for real-world term distributions and the non-uniformity of gaps.
(Stratification of terms for estimating the size of a gamma encoded inverted index)
Conclusion
Gamma codes provide a robust and efficient solution for compressing postings lists in inverted indexes. Their universality, prefix-free nature, and parameter-free operation make them a compelling choice for various information retrieval applications. However, the decoding overhead should be considered when choosing between gamma and other compression techniques like variable byte codes. The optimal choice depends on the specific characteristics of the data and the prioritization of disk space versus query processing speed.
