Optimizing Hash Table For N_is_prime

Optimizing Hash Table Construction for n_is_prime

Hey guys, let's dive into the fascinating world of prime number testing and hash tables! This article explores the optimization of a hash table used in the n_is_prime function within the FLINT library. We'll look at the challenges, the code, and how we can potentially speed things up. It's a bit technical, but I'll try to keep it friendly and easy to follow. We're gonna see how we can build a smaller hash table that is designed to help us with prime number determination. Let's get started!

The Basics: What is n_is_prime and Why a Hash Table?

First things first, what is n_is_prime? It's a function designed to check if a given number n is prime. A key part of this function involves a primality test, often using probabilistic methods. These methods check if n passes certain tests (like the Miller-Rabin test) with respect to a set of bases. The more bases n passes, the higher the probability that n is actually prime. The hash table comes into play to store precomputed data about these bases, allowing for quick lookups and efficient primality testing.

Now, why a hash table? Because it offers near-constant time (O(1)) for looking up data. This is super important! When you're dealing with primality tests, you're often checking the same bases repeatedly. Having a hash table lets you quickly retrieve precomputed information for each base, speeding up the entire process. This is the core principle behind making primality testing efficient.

Generating the Hash Table Bases

The code used to generate the bases for the hash table is located at the provided link (https://gist.github.com/fredrik-johansson/2c568b0b20675a22d77f89b9abdf3215). This script is super important as it is the foundation of our hash table.

The Data Source: Strong Base-2 Pseudoprimes

To build this hash table effectively, we need a reliable source of pseudoprimes. The resource used is the list of strong base-2 pseudoprimes up to 2^64, available at https://flintlib.org/data/psp2.txt.xz. This data is the backbone of our table, containing crucial information for our prime testing.

Building the Hash Table: Challenges and Optimization

Building the hash table is where the fun begins, but also where we encounter the challenges! The initial approach involved generating a hash table with 131072 buckets, which was relatively quick. However, the goal is to make it even smaller and more efficient. The author suggests creating a smaller table with, say, 98304 buckets (3 * 2^15). The problem is that creating a table of this size takes a long time. It can take several days even on a machine with multiple cores! This involves testing various numbers against different bases and storing the results.

As the author mentions, after running the process for eight hours on an 8-core machine, only about 10% of the buckets get completed. This highlights the time-consuming nature of the task. The output logs give us some insight into the process: the max value represents the maximum count of bases hitting a particular bucket, the average is the average count across all buckets, and the histogram provides a distribution of how many bases land in each bucket. The challenge lies in distributing the bases across the buckets evenly.

The Goal: Smaller Tables, Faster Primality Tests

To make things super efficient, we want even smaller tables. The author suggests a target of 2^16 buckets. Here's where the real optimization kicks in. This approach would potentially allow the data to fit comfortably into less than 256KB, making lookups super fast. But the calculations involved are still extensive and require many CPU years to complete.

The Key Question: How to Speed Up the Search?

So, the million-dollar question: How do we significantly speed up the construction of this hash table? Here are some ideas and potential solutions:

• SIMD Vectorization: One major idea is to utilize Single Instruction, Multiple Data (SIMD) to run the probable prime tests for a vector of bases simultaneously. SIMD allows you to perform the same operation on multiple pieces of data at once, dramatically increasing the throughput. This can be a huge win here.
• GPU Parallelization: GPUs, with their massive parallelism, are another good way to go. Even if the 64-bit integer operations on the GPU are slower, the sheer number of cores available can make up for it. The high degree of parallelism makes GPUs well-suited for this kind of computational task. Think about it: a ton of cores all working in parallel to check primality—super efficient!
• Pruning Strategies: Can we eliminate some candidates early on to reduce the workload? Pruning involves identifying and discarding bases that are likely to fail the primality test quickly, avoiding unnecessary computations. This is super important to help make the process faster.
• Multipoint Evaluation: Consider techniques like multipoint evaluation to assess multiple bases simultaneously. This strategy could allow us to determine the values of multiple bases with reduced computational overhead.
• Meet-in-the-Middle Techniques: Another intriguing option is to explore meet-in-the-middle approaches. These techniques divide the problem into smaller subproblems and try to merge the solutions. This could potentially reduce the overall processing time.

Conclusion: The Road Ahead

Building a hash table for efficient primality testing is a complex but rewarding task. The original approach, while functional, can be improved. By optimizing the hash table construction and leveraging techniques such as SIMD vectorization and GPU parallelism, we can significantly speed up the process and ultimately improve the n_is_prime function. The key lies in balancing computational resources, data storage, and algorithmic efficiency. The path to faster primality testing is paved with a lot of coding and testing, but it is super exciting. Keep in mind that as the computation progresses, we are bound to have more efficient primality testing methods, that help with the speed and efficiency of code.