Over the past 24 hours, we’ve made significant progress in developing and optimizing an algorithm to find large Salem-Spencer sets within a given range. Here’s a recap of the key developments:
- Algorithm Development: We started by conceptualizing and implementing an algorithm to build Salem-Spencer sets. The algorithm involves filling a set to capacity without forming any length-3 arithmetic progressions, backtracking when necessary, and then trying to expand again. This process repeats for a specified number of attempts.
- Optimization and Debugging: We’ve gone through several iterations of the algorithm, each time refining and debugging the code. This included optimizing functions like
fill_to_capacity,backtrack, and the check for arithmetic progressions. - Memory Management: A significant part of our work involved improving memory management. This included introducing proper node allocation and deallocation to prevent memory leaks and ensure efficient use of resources.
- Performance Enhancement: We implemented performance enhancements like random shuffling of numbers before adding them to the set and tweaks in the backtracking logic.
- Record Tracking: The algorithm was modified to keep track of and report new record sizes of Salem-Spencer sets found during the execution.
- Output and Diagnostics: The code was adjusted to provide meaningful output and diagnostics, including hexadecimal representation of the sets and tracking of memory allocations and deallocations.
- Handling Large Data Sets: Modifications were made to handle larger sets (up to 192 elements) using bitwise operations and efficient data structures like bitsets.
- Exploring Further Optimizations: We discussed potential strategies for further optimizations, such as parallel processing, heuristic enhancements, and leveraging hardware capabilities.
As a result of these efforts, the program has successfully found Salem-Spencer sets of significant sizes, with the largest found being a set of 29 elements. This is a notable achievement, considering the complexity of avoiding length-3 arithmetic progressions.
The work done so far provides a solid foundation for further exploration and potential breakthroughs in finding even larger Salem-Spencer sets.
