ideas for algorithm? sorting a list randomly with emphasis on variety

Question

I have a table of items with [ID, ATTR1, ATTR2, ATTR3]. I'd like to select about half of the items, but try to get a random result set that is NOT clustered. In other words, there's a fairly even spread of ATTR1 values, ATTR2 values, and ATTR3 values. This does NOT necessarily represent the data as a whole, in other words, the total table may be generally concentrated on certain attribute values, but I'd like to select a subset with more variety. The attributes are not inter-related, so there's not really a correlation between ATTR1 and ATTR2.

As an example, imagine ATTR1 = "State". I'd like each line item in my subset to be from a different state, even if in the whole set, most of my data is concentrated on a few states. And for this to simultaneously be true of the other 2 attributes, too. (I realize that some tables might not make this possible, but there's enough data that it's unlikely to have no solution)

Any ideas for an efficient algorithm? Thanks! I don't really even know how to search for this :)

(by the way, it's OK if this requires pre-calculation or -indexing on the whole set, so long as I can draw out random varied subsets quickly)

Answer 1

I don't know (and I hope someone who does will answer). Here's what comes to mind: make up a distribution for MCMC putting the most weight on the subsets with 'variety'.

Answer 2

Naive solution.

If you have M items I and want to select K of them.
Divide the I's into M/K chunks then use a standard rand to select one I from each chunk.

Now you have a not-random-but-sort-of sample of I's spread out over all M I's if you get what I mean.

Answer 3

Assuming the items in your table are indexed by some form of id, I would in a loop, iterate through half of the items in your table, and use a random number generator to get the number.

Answer 4

IMHO

Finding variety is difficult but generating it is easy.

So we can generate variety of combinations and then seach the table for records with those combinations.

If the table is sorted then searching also becomes easy.

Sample python code:

d = {}
d[('a',0,'A')]=0
d[('a',1,'A')]=1
d[('a',0,'A')]=2
d[('b',1,'B')]=3
d[('b',0,'C')]=4
d[('c',1,'C')]=5
d[('c',0,'D')]=6
d[('a',0,'A')]=7

print d

attr1 = ['a','b','c']
attr2 = [0,1]
attr3 = ['A','B','C','D']

# no of items in
# attr2 < attr1 < attr3

# ;) reason for strange nesting of loops

for z in attr3:
    for x in attr1:
        for y in attr2:
            k = (x,y,z)
            if d.has_key(k):
                print '%s->%s'%(k,d[k])
            else:
                print k

Output:

('a', 0, 'A')->7
('a', 1, 'A')->1
('b', 0, 'A')
('b', 1, 'A')
('c', 0, 'A')
('c', 1, 'A')
('a', 0, 'B')
('a', 1, 'B')
('b', 0, 'B')
('b', 1, 'B')->3
('c', 0, 'B')
('c', 1, 'B')
('a', 0, 'C')
('a', 1, 'C')
('b', 0, 'C')->4
('b', 1, 'C')
('c', 0, 'C')
('c', 1, 'C')->5
('a', 0, 'D')
('a', 1, 'D')
('b', 0, 'D')
('b', 1, 'D')
('c', 0, 'D')->6
('c', 1, 'D')

But assuming your table is very big (otherwise why would you need algorithm ;) and data is fairly uniformly distributed there will be more hits in actual scenario. In this dummy case there are too many misses which makes algorithm look inefficient.

Answer 5

Let's assume that ATTR1, ATTR2, and ATTR3 are independent random variables (over a uniform random item). (If ATTR1, ATTR2, and ATTR3 are only approximately independent, then this sample should be approximately uniform in each attribute.) To sample one item (VAL1, VAL2, VAL3) whose attributes are uniformly distributed, choose VAL1 uniformly at random from the set of values for ATTR1, choose VAL2 uniformly at random from the set of values for ATTR2 over items with ATTR1 = VAL1, choose VAL3 uniformly at random from the set of values for ATTR3 over items with ATTR1 = VAL1 and ATTR2 = VAL2.

To get a sample of distinct items, apply the above procedure repeatedly, deleting each item after it is chosen. Probably the best way to implement this would be a tree. For example, if we have

ID    ATTR1 ATTR2 ATTR3
1     a     c     e
2     a     c     f
3     a     d     e
4     a     d     f
5     b     c     e
6     b     c     f
7     b     d     e
8     b     d     f
9     a     c     e

then the tree is, in JavaScript object notation,

{"a": {"c": {"e": [1, 9], "f": [2]},
       "d": {"e": [3], "f": [4]}},
 "b": {"c": {"e": [5], "f": [6]},
       "d": {"e": [7], "f": [8]}}}

Deletion is accomplished recursively. If we sample id 4, then we delete it from its list at the leaf level. This list empties, so we delete the entry "f": [] from tree["a"]["d"]. If we now delete 3, then we delete 3 from its list, which empties, so we delete the entry "e": [] from tree["a"]["d"], which empties tree["a"]["d"], so we delete it in turn. In a good implementation, each item should take time O(# of attributes).

EDIT: For repeated use, reinsert the items into the tree after the whole sample is collected. This doesn't affect the asymptotic running time.

Answer 6

Interesting problem. Since you want about half of the list, how about this:-

Create a list of half the values chosen entirely at random. Compute histograms for the value of ATTR1, ATTR2, ATTR3 for each of the chosen items.

:loop

Now randomly pick an item that's in the current list and an item that isn't.

If the new item increases the 'entropy' of the number of unique attributes in the histograms, keep it and update the histograms to reflect the change you just made.

Repeat N/2 times, or more depending on how much you want to force it to move towards covering every value rather than being random. You could also use 'simulated annealing' and gradually change the probability to accepting the swap - starting with 'sometimes allow a swap even if it makes it worse' down to 'only swap if it increases variety'.

Answer 7

Idea #2.

Compute histograms for each attribute on the original table.

For each item compute it's uniqueness score = p(ATTR1) x p(ATTR2) x p(ATTR3) (multiply the probabilities for each attribute it has).

Sort by uniqueness.

Chose a probability distribution curve for your random numbers ranging from picking only values in the first half of the set (a step function) to picking values evenly over the entire set (a flat line). Maybe a 1/x curve might work well for you in this case.

Pick values from the sorted list using your chosen probability curve.

This allows you to bias it towards more unique values or towards more evenness just by adjusting the probability curve you use to generate the random numbers.

Answer 8

Taking over your example, assign every possible 'State' a numeric value (say, between 1 and 9). Do the same for the other attributes.

Now, assuming you don't have more than 10 possible values for each attribute, multiply the values for ATTR3 for 100, ATTR2 for 1000, ATTR1 for 10000. Add up the results, you will end up with what can resemble a vague hash of the item. Something like

10,000 * |ATTR1| + 1000 * |ATTR2| + 100 * |ATTR3|

the advantage here is that you know that values between 10000 and 19000 have the same 'State' value; in other words, the first digit represents ATTR1. Same for ATTR2 and the other attributes.

You can sort all values and using something like bucket-sort pick one for each type, checking that the digit you're considering hasn't been picked already.

An example: if your end values are

A: 15,700 = 10,000 * 1 + 1,000 * 5 + 100 * 7 B: 13,400 = 10,000 * 1 + 1,000 * 3 + 100 * 4 C: 13,200 = ... D: 12,300 E: 11,400 F: 10,900

you know that all your values have the same ATTR1; 2 have the same ATTR2 (that being B and C); and 2 have the same ATTR3 (B, E).

This, of course, assuming I understood correctly what you want to do. It's saturday night, afterall.

ps: yes, I could have used '10' as the first multiplier but the example would have been messier; and yes, it's clearly a naive example and there are lots of possible optimizations here, which are left as an exercise to the reader

Answer 9

It's a very interesting problem, for which I can see a number of applications. Notably for testing software: you get many 'main-flow' transactions, but only one is necessary to test that it works and you would prefer when selecting to get an extremely varied sample.

I don't think you really need a histogram structure, or at least only a binary one (absent/present).

{ ATTR1: [val1, val2], ATTR2: [i,j,k], ATTR3: [1,2,3] }

This is used in fact to generate a list of predicates:

Predicates = [ lambda x: x.attr1 == val1, lambda x: x.attr1 == val2,
               lambda x: x.attr2 == i, ...]

This list will contain say N elements.

Now you wish to select K elements from this list. If K is less than N it's fine, otherwise we will duplicate the list i times, so that K <= N*i and with i minimal of course, so i = ceil(K/N) (note that it works although if K <= N, with i == 1).

i = ceil(K/N)
Predz = Predicates * i # python's wonderful

And finally, pick up a predicate there, and look for an element that satisfies it... that's where randomness actually hits and I am less than adequate here.

Two remarks:

• if K > N you may be willing to actually select i-1 times each predicate and then select randomly from the list of predicates only to top off your selection. Thus ensuring the over representation of even the least common elements.
• the attributes are completely uncorrelated this way, you may be willing to select patterns as you could never get the tuple (1,2,3) by selecting on the third element being 3, so perhaps a refinement would be to group some related attributes together, though it would probably increase the number of predicates generated
• for efficiency reasons, you should have the table by the predicate category if you wish to have an efficient select.