How to Efficiently Represent Set Operations in a Hash Table?

[tickle_monste]

Homework Statement

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.

Homework Equations

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.

 

[mighty2000]

I appreciate your inquiry into finding a good representation for your set W in a hash table. There are a few key points to consider when determining the best approach for this problem.

Firstly, it is important to understand the limitations of a hash table. Hash tables are efficient for data retrieval and storage, but they may not be the best choice for representing sets with arbitrary intersections. This is because hash tables are designed for fast access to specific elements, rather than the efficient storage and retrieval of subsets.

One possible solution to this problem could be to use a tree data structure instead of a hash table. Trees are better suited for representing hierarchical data, such as the subsets of W with arbitrary intersections. This would allow for efficient storage and retrieval of subsets, as well as the set operations you mentioned (A∪B, A∩B, A-B, etc.).

Another consideration is the size and complexity of your data. If W is a large set with many subsets, it may be more efficient to use a hybrid approach, as you suggested. This could involve a combination of a hash table for fast access to individual elements, and a tree structure for efficient storage and retrieval of subsets.

Ultimately, the best approach will depend on your specific needs and the characteristics of your data. I would recommend experimenting with different data structures and measuring their performance to determine the most suitable solution for your problem.

I hope this helps in your search for a good representation of W in a hash table. Best of luck in your research!