Data structure for matching sets

Question

I have an application where I have a number of sets. A set might be
{4, 7, 12, 18}
unique numbers and all less than 50.

I then have several data items:
1 {1, 2, 4, 7, 8, 12, 18, 23, 29}
2 {3, 4, 6, 7, 15, 23, 34, 38}
3 {4, 7, 12, 18}
4 {1, 4, 7, 12, 13, 14, 15, 16, 17, 18}
5 {2, 4, 6, 7, 13, 15}

Data items 1, 3 and 4 match the set because they contain all items in the set.

I need to design a data structure that is super fast at identifying whether a data item is a member of a set includes all the members that are part of the set (so the data item is a superset of the set). My best estimates at the moment suggest that there will be fewer than 50,000 sets.

My current implementation has my sets and data as unsigned 64 bit integers and the sets stored in a list. Then to check a data item I iterate through the list doing a ((set & data) == set) comparison. It works and it's space efficient but it's slow (O(n)) and I'd be happy to trade some memory for some performance. Does anyone have any better ideas about how to organize this?

Edit: Thanks very much for all the answers. It looks like I need to provide some more information about the problem. I get the sets first and I then get the data items one by one. I need to check whether the data item is matches one of the sets.
The sets are very likely to be 'clumpy' for example for a given problem 1, 3 and 9 might be contained in 95% of sets; I can predict this to some degree in advance (but not well).

For those suggesting memoization: this is this the data structure for a memoized function. The sets represent general solutions that have already been computed and the data items are new inputs to the function. By matching a data item to a general solution I can avoid a whole lot of processing.

Answer 1

I can't prove it, but I'm fairly certain that there is no solution that can easily beat the O(n) bound. Your problem is "too general": every set has m = 50 properties (namely, property k is that it contains the number k) and the point is that all these properties are independent of each other. There aren't any clever combinations of properties that can predict the presence of other properties. Sorting doesn't work because the problem is very symmetric, any permutation of your 50 numbers will give the same problem but screw up any kind of ordering. Unless your input has a hidden structure, you're out of luck.

However, there is some room for speed / memory tradeoffs. Namely, you can precompute the answers for small queries. Let Q be a query set, and supersets(Q) be the collection of sets that contain Q, i.e. the solution to your problem. Then, your problem has the following key property

Q ⊆ P  =>  supersets(Q) ⊇ supersets(P)

In other words, the results for P = {1,3,4} are a subcollection of the results for Q = {1,3}.

Now, precompute all answers for small queries. For demonstration, let's take all queries of size <= 3. You'll get a table

supersets({1})
supersets({2})
...
supersets({50})
supersets({1,2})
supersets({2,3})
...
supersets({1,2,3})
supersets({1,2,4})
...

supersets({48,49,50})

with O(m^3) entries. To compute, say, supersets({1,2,3,4}), you look up superset({1,2,3}) and run your linear algorithm on this collection. The point is that on average, superset({1,2,3}) will not contain the full n = 50,000 elements, but only a fraction n/2^3 = 6250 of those, giving an 8-fold increase in speed.

(This is a generalization of the "reverse index" method that other answers suggested.)

Depending on your data set, memory use will be rather terrible, though. But you might be able to omit some rows or speed up the algorithm by noting that a query like {1,2,3,4} can be calculated from several different precomputed answers, like supersets({1,2,3}) and supersets({1,2,4}), and you'll use the smallest of these.

Answer 2

If you're going to improve performance, you're going to have to do something fancy to reduce the number of set comparisons you make.

Maybe you can partition the data items so that you have all those where 1 is the smallest element in one group, and all those where 2 is the smallest item in another group, and so on.

When it comes to searching, you find the smallest value in the search set, and look at the group where that value is present.

Or, perhaps, group them into 50 groups by 'this data item contains N' for N = 1..50.

When it comes to searching, you find the size of each group that holds each element of the set, and then search just the smallest group.

The concern with this - especially the latter - is that the overhead of reducing the search time might outweigh the performance benefit from the reduced search space.

Answer 3

Another idea is to completely prehunt your elephants.

Setup

Create a 64 bit X 50,000 element bit array.

Analyze your search set, and set the corresponding bits in each row.

Save the bit map to disk, so it can be reloaded as needed.

Searching

Load the element bit array into memory.

Create a bit map array, 1 X 50000. Set all of the values to 1. This is the search bit array

Take your needle, and walk though each value. Use it as a subscript into the element bit array. Take the corresponding bit array, then AND it into the search array.

Do that for all values in your needle, and your search bit array, will hold a 1, for every matching element.

Reconstruct

Walk through the search bit array, and for each 1, you can use the element bit array, to reconstruct the original values.

Answer 4

Put your sets into an array (not a linked list) and SORT THEM. The sorting criteria can be either 1) the number of elements in the set (number of 1-bits in the set representation), or 2) the lowest element in the set. For example, let A={7, 10, 16} and B={11, 17}. Then B<A under criterion 1), and A<B under criterion 2). Sorting is O(n log n), but I assume that you can afford some preprocessing time, i.e., that the search structure is static.

When a new data item arrives, you can use binary search (logarithmic time) to find the starting candidate set in the array. Then you search linearly through the array and test the data item against the set in the array until the data item becomes "greater" than the set.

You should choose your sorting criterion based on the spread of your sets. If all sets have 0 as their lowest element, you shouldn't choose criterion 2). Vice-versa, if the distribution of set cardinalities is not uniform, you shouldn't choose criterion 1).

Yet another, more robust, sorting criterion would be to compute the span of elements in each set, and sort them according to that. For example, the lowest element in set A is 7, and highest is 16, so you would encode its span as 0x1007; similarly the B's span would be 0x110B. Sort the sets according to the "span code" and again use binary search to find all sets with the same "span code" as your data item.

Computing the "span code" is slow in ordinary C, but it can be done fast if you resort to assembly -- most CPUs have instructions that find the most/least significant set bit.

Answer 5

A possible way to divvy up the list of bitmaps, would be to create an array of (Compiled Nibble Indicators)

Let's say one of your 64 bit bitmaps has the bit 0 to bit 8 set.
In hex we can look at it as 0x000000000000001F

Now, let's transform that into a simpler and smaller representation. Each 4 bit Nibble, either has at least one bit set, or not. If it does, we represent it as a 1, if not we represent it as a 0.

So the hex value reduces to bit pattern 0000000000000011, as the right hand 2 nibbles have are the only ones that have bits in them. Create an array, that holds 65536 values, and use them as a head of linked lists, or set of large arrays....

Compile each of your bit maps, into it's compact CNI. Add it to the correct list, until all of the lists have been compiled.

Then take your needle. Compile it into its CNI form. Use that to value, to subscript to the head of the list. All bitmaps in that list have a possibility of being a match. All bitmaps in the other lists can not match.

That is a way to divvy them up.

Now in practice, I doubt a linked list would meet your performance requirements.

If you write a function to compile a bit map to CNI, you could use it as a basis to sort your array by the CNI. Then have your array of 65536 heads, simply subscript into the original array as the start of a range.

Another technique would be to just compile a part of the 64 bit bitmap, so you have fewer heads. Analysis of your patterns should give you an idea of what nibbles are most effective in partitioning them up.

Good luck to you, and please let us know what you finally end up doing.

Evil.

Answer 6

How many data items do you have? Are they really all unique? Could you cache popular data items, or use a bucket/radix sort before the run to group repeated items together?

Here is an indexing approach:

1) Divide the 50-bit field into e.g. 10 5-bit sub-fields. If you really have 50K sets then 3 17-bit chunks might be nearer the mark.

2) For each set, choose a single subfield. A good choice is the sub-field where that set has the most bits set, with ties broken almost arbitrarily - e.g. use the leftmost such sub-field.

3) For each possible bit-pattern in each sub-field note down the list of sets which are allocated to that sub-field and match that pattern, considering only the sub-field.

4) Given a new data item, divide it into its 5-bit chunks and look each up in its own lookup table to get a list of sets to test against. If your data is completely random you get a factor of two speedup or more, depending on how many bits are set in the densest sub-field of each set. If an adversary gets to make up random data for you, perhaps they find data items that almost but not quite match loads of sets and you don't do very well at all.

Possibly there is scope for taking advantage of any structure in your sets, by numbering bits so that sets tend to have two or more bits in their best sub-field - e.g. do cluster analysis on the bits, treating them as similar if they tend to appear together in sets. Or if you can predict patterns in the data items, alter the allocation of sets to sub-fields in step(2) to reduce the number of expected false matches.

Addition: How many tables would need to have to guarantee that any 2 bits always fall into the same table? If you look at the combinatorial definition in http://en.wikipedia.org/wiki/Projective_plane, you can see that there is a way to extract collections of 7 bits from 57 (=1 + 7 + 49) bits in 57 different ways so that for any two bits at least one collection contains both of them. Probably not very useful, but it's still an answer.