Fibonacci hashing offers a faster alternative to the typical modulo operator (%) for distributing items into hash tables, especially when the table size is a power of two. It leverages the golden ratio's properties by multiplying the hash key by a large constant derived from the golden ratio and then bit-shifting the result, effectively achieving a modulo operation without the expensive division. This method produces a more even distribution compared to modulo with prime table sizes, particularly when dealing with keys exhibiting sequential patterns, thus reducing collisions and improving performance. While theoretically superior, its benefits may be negligible in modern systems due to compiler optimizations and branch prediction for modulo with powers of two.
The blog post "Fibonacci Hashing: The Optimization That the World Forgot (or a Better Alternative to Integer Modulo)" by Christopher Wellons explores a highly efficient hashing technique based on the golden ratio, arguing that it's often superior to the commonly used modulo operator for distributing hash values across a hash table. Wellons begins by explaining the shortcomings of the modulo operator, particularly when the table size is not a prime number. If the table size has common factors with the hash values, the modulo operation can lead to clustering and reduced performance. This is because the modulo will effectively only distribute the keys among a subset of the available slots, proportional to the greatest common divisor of the table size and the hash.
He then introduces the concept of Fibonacci hashing, which utilizes a specific multiplication and bitwise shift operation as a replacement for modulo. This technique relies on the properties of the golden ratio, an irrational number closely approximated by the ratio of consecutive Fibonacci numbers. The golden ratio's inherent connection to relatively prime numbers allows for more even distribution of hash values even when the table size is not prime, and especially when it’s a power of two. This is achieved by multiplying the hash value by a large integer representation of the golden ratio's fractional part (specifically 264 * φf where φf is the fractional part of the golden ratio) and then taking the high bits of the result, equivalent to a right bitwise shift. This operation effectively mimics the behavior of modulo a prime number, spreading the hashed values more uniformly across the hash table.
Wellons delves into the mathematical underpinnings of why this method works, explaining how the multiplication with the golden ratio's fractional part and the subsequent bitwise shift are analogous to rotating a circle by an irrational angle, ensuring points are never aligned and thus promoting even distribution. He contrasts this with multiplication by a rational number, which would lead to points eventually aligning and creating clustering.
The post further emphasizes the performance benefits of Fibonacci hashing. Since multiplication and bitwise shifts are typically faster operations than the modulo operation, especially with modern processors, Fibonacci hashing often leads to a noticeable speedup in hash table operations. This is particularly pronounced when the table size is a power of two, as the bitwise shift becomes highly optimized. The author provides some benchmark results showcasing these performance gains.
Finally, the post acknowledges some potential drawbacks of Fibonacci hashing, such as the need for a large multiplier and the potential for bias if the initial hash function is poorly designed. However, it concludes by asserting that for the majority of use cases, Fibonacci hashing provides a superior alternative to integer modulo, especially when the hash table size is a power of two, offering improved performance and more robust hash distribution even with non-ideal hash functions. The simplicity of implementing Fibonacci hashing, requiring only multiplication and a bit shift, further strengthens its case as a powerful optimization technique.
Summary of Comments ( 10 )
https://news.ycombinator.com/item?id=43677122
HN commenters generally praise the article for clearly explaining Fibonacci hashing and its benefits over modulo. Some point out that the technique is not forgotten, being used in game development and hash table implementations within popular languages like Java. A few commenters discuss the nuances of the golden ratio's properties and its suitability for hashing, with one noting the importance of good hash functions over minor speed differences in the hashing algorithm itself. Others shared alternative hashing methods like "Multiply-with-carry" and "SplitMix64", along with links to resources on hash table performance testing. A recurring theme is that Fibonacci hashing shines with power-of-two table sizes, losing its advantages (and potentially becoming worse) with prime table sizes.
The Hacker News post titled "Fibonacci Hashing: The Optimization That the World Forgot" (https://news.ycombinator.com/item?id=43677122) has a moderate number of comments, generating a discussion around the merits and applicability of Fibonacci hashing.
Several commenters delve into the practicalities of Fibonacci hashing, questioning its supposed superiority over simpler modulo methods. One recurring point is the potential performance impact of multiplication on various architectures. While the article champions multiplication as faster than modulo, some commenters argue that this isn't universally true. Modern CPUs, they point out, often have efficient modulo instructions, especially when dealing with powers of two. One commenter specifically mentions that modulo by a power of two can be as simple as a bitwise AND operation, which is extremely fast. Therefore, the supposed speed advantage of Fibonacci hashing becomes less clear-cut and highly dependent on the specific hardware.
Another key discussion thread centers around the quality of hash distribution. Some commenters express skepticism about Fibonacci hashing consistently outperforming modulo, especially when dealing with real-world data that might not be uniformly distributed. Concerns are raised about potential clustering or patterns in the hashed values that could negatively impact performance. One commenter highlights the importance of benchmarking with realistic datasets to demonstrate any tangible benefits over traditional methods. They also mention Knuth's multiplicative hashing method as a strong contender, suggesting it often provides a good balance between speed and distribution quality.
A few commenters provide valuable context by linking to related resources and discussions. One link points to a Stack Overflow post discussing the choice of the multiplier in multiplicative hashing. Another commenter shares a link to a paper analyzing different hashing methods. These external resources add depth to the conversation and provide alternative perspectives on the topic.
Finally, some commenters offer practical advice and considerations. One commenter suggests that the choice of hashing method should depend on the specific application and its performance requirements. They emphasize the need to profile and measure the impact of different hashing strategies rather than relying on theoretical assumptions. Another commenter points out the potential complexity of implementing Fibonacci hashing correctly, which could outweigh its theoretical benefits in some cases.
In summary, the comments section provides a balanced perspective on Fibonacci hashing, challenging the article's claim of it being a forgotten optimization. The discussion highlights the importance of considering hardware specifics, data distribution, and practical implementation challenges when evaluating any hashing method.