Optimizing Set Theory Hash Tables for Efficient Data Representation

[tickle_monste]

1. Homework Statement
I posted this on Calculus & Beyond because all of this came from a math idea, but I realize now that it belongs in this section. Given that I have a set W, with a multitude of subsets w1...wn, with arbitrary intersections, worst-case-scenario-unordered, I want to know what would be a good representation in a hash table. Basically I want to have things like A$$\cup$$B, A$$\cap$$ B, A - B, etc (the basic set theory operations) have an associated hash-table address where a pointer to exactly that set is stored, and want to know how to quantify the limitations of such a data-structure.

2. Homework Equations

3. The Attempt at a Solution
I'll give an example to motivate my solution, with just two subsets of W: A and B, which intersect.
W can be divided into the following partitions:
W - A - B
A-B
B-A
A$$\cap$$B

None of these subsets intersect with each other, and there's no redundant data. Now only metadata can be redundant. What I mean is that when I specify the set A, this would translate to the hash-table as [A-B]$$\cup$$[A$$\cap$$B], and B would translate to [B-A]$$\cup$$[A$$\cap$$B] i.e. meta-data is references to non-redundant partitions.

Given that there are N objects in W, with enough subsets W could be divided into a maximum of N partitions, so while there is no redundant data, the metadata becomes increasingly bulky and redundant.

I could reverse this and have 0-redundancy metadata, with redundant data. I am wondering what are the limitations of both, and what compromises between the two can be made. Mixing the representations would add another layer of metadata, and I am not quite sure how to go about that, or if this if I'm going down the right path with this.

 

[KataKoniK]

Thank you for your question. I can offer some insights and suggestions on how to approach this problem.

Firstly, I would recommend considering the purpose and usage of your hash table. Are you looking to optimize for storage space, retrieval speed, or a balance between the two? This will help guide your decisions on the representation and structure of your hash table.

Secondly, it may be helpful to consider the types of operations and queries that will be performed on your hash table. For example, if you anticipate a lot of set intersections, it may be beneficial to have a representation that allows for efficient intersection operations.

In terms of limitations, it is important to consider the size and complexity of your sets and subsets. As you mentioned, with a large number of objects and subsets, the metadata can become quite bulky and potentially slow down retrieval times. Therefore, it may be necessary to find a balance between redundancy and non-redundancy in your metadata.

One approach to consider is using a hybrid representation, where some operations are optimized for non-redundancy while others are optimized for redundancy. This can help reduce the overall size of your metadata while still allowing for efficient operations.

In addition, it may be beneficial to explore different data structures and algorithms that are specifically designed for set operations, such as binary decision diagrams or compressed binary decision diagrams.

Overall, there are many factors to consider when designing a hash table for sets and subsets. I would recommend experimenting with different representations and structures to find the best solution for your specific needs. I hope this helps guide you in the right direction. Good luck!