Need algorithm for fast storage and retrieval (search) of sets and subsets

Question

I need a way of storing sets of arbitrary size for fast query later on. I\'ll be needing to query the resulting data structure for subsets or sets that are already stored.

Answer 1

I'm confident that I can now contribute to the solution. One possible quite efficient way is a:

Trie invented by Frankling Mark Liang

Such a special tree is used for example in spell checking or autocompletion and that actually comes close to your desired behavior, especially allowing to search for subsets quite conveniently.

The difference in your case is that you're not interested in the order of your attributes/features. For your case a Set-Trie was invented by Iztok Savnik.

What is a Set-Tree? A tree where each node except the root contains a single attribute value (number) and a marker (bool) if at this node there is a data entry. Each subtree contains only attributes whose values are larger than the attribute value of the parent node. The root of the Set-Tree is empty. The search key is the path from the root to a certain node of the tree. The search result is the set of paths from the root to all nodes containing a marker that you reach when you go down the tree and up the search key simultaneously (see below).

But first a drawing by me:

The attributes are {1,2,3,4,5} which can be anything really but we just enumerate them and therefore naturally obtain an order. The data is {{1,2,4}, {1,3}, {1,4}, {2,3,5}, {2,4}} which in the picture is the set of paths from the root to any circle. The circles are the markers for the data in the picture.

Please note that the right subtree from root does not contain attribute 1 at all. That's the clue.

Searching including subsets Say you want to search for attributes 4 and 1. First you order them, the search key is {1,4}. Now startin from root you go simultaneously up the search key and down the tree. This means you take the first attribute in the key (1) and go through all child nodes whose attribute is smaller or equal to 1. There is only one, namely 1. Inside you take the next attribute in the key (4) and visit all child nodes whose attribute value is smaller than 4, that are all. You continue until there is nothing left to do and collect all circles (data entries) that have the attribute value exactly 4 (or the last attribute in the key). These are {1,2,4} and {1,4} but not {1,3} (no 4) or {2,4} (no 1).

Insertion Is very easy. Go down the tree and store a data entry at the appropriate position. For example data entry {2.5} would be stored as child of {2}.

Add attributes dynamically Is naturally ready, you could immediately insert {1,4,6}. It would come below {1,4} of course.

I hope you understand what I want to say about Set-Tries. In the paper by Iztok Savnik it's explained in much more detail. They probably are very efficient.

I don't know if you still want to store the data in a database. I think this would complicate things further and I don't know what is the best to do then.