Implementing Disjoint Set Approximation (Union Fin

问题:

What would be the best way to implement approximate Disjoint Sets using SQL?

Details

I have a table of edges, stored as a two-column table of [vertex_a, vertex_b].

I need a table of distinct sets, stored as [vertex, set_id] with one row per vertex, labeling each vertex with a disjoint set_id.

Constraints

• Must be a purely SQL implementation. It can leverage Postgres-specific functions, but pure ANSI SQL highly preferred.
• The result can be approximate- it's acceptable to label a few sets as disjoint when they're actually connected. It's even better if the approximation bounds can be adjusted- by increasing iterations for example.
• Libraries are out (no Boost, Numpy, Scipy). Must be SQL.
• Most sets will contain 1 to 3 vertices. Very few large sets, expected max to be 10 vertices.

Related

• Related to: Implementing Disjoint Sets (Union Find) in C++
• This would be an approximate implementation of Disjoint-set (Union Find) - Wikipedia

回答1:

I'm actually working on the same problem. Unfortunately, I don't think a very efficient solution can be found - at least not easily using just SQL. Just remove the 'distinct' and the self-eliminating query to observe how large the working sets become. That said, the following solution will work.

drop table if exists foobar;
drop function if exists addset(v int[], a int);
/* our vertices table */
create table foobar (
   src int,
   dst int
);

/* Create a small function to treat an array like a set, 
   not strictly necessary but convenient */
create function addset(v int[], a int) returns int[]
as $$
begin
    return (select array_agg(e order by e) 
                   from (select unnest(v || a) as e) f);
end
$$ language plpgsql;

/* fill our table with vertices, note the ordering of each value */
insert into foobar (src, dst) 
     values (1,2), (1,3), (2,3),  (3,4), (4,5), (6,7), (6,8);
/* use a recursive query to extend the sets */
with recursive foo_union (v) as (
    select array[src, dst] as v from foobar /* starter sets */
    union all
    /* join self to original array; i can use a CTE as a 'starter',
       but that is not necessary here */
    select addset(v, dst) as v from foo_union u, foobar f
        where src = any(v) and not dst = any(v)
 ) select distinct v from foo_union a where not exists (
     /* eliminate the many overlapping results */
     select * from foo_union b where b.v @> a.v and b.v != a.v
 );

But again, this is completely impractical on larger data sets; any other solution would require iteration.

回答2:

This pure sql code grouped about 35000 records in 5 minutes (8 cores/32 gb ram). Enjoy.

--table with RELATIONS, idea is to place every related item in a bucket
create table RELATIONS
(
    bucket int,        -- initially 0
    bucketsub int,    -- initially 0
    relnr1 float,    
    relnr2 float    -- relnr1 = a, relnr2 = b means a and b are related
)

create table ##BucketRelnrs ( relnr float ); --table functions as temp list
declare @bucket int; 
declare @bucketsub int;
declare @nrOfUpdates int;
declare @relnr float;

set @bucket=0;
set @relnr=0;
WHILE @relnr>=0 and @bucket<50000 --to prevent the while loop from hanging.
BEGIN
    set @bucket = @bucket+1
    set @bucketsub=1;

    set @relnr = (select isnull(min(relnr1),-1) from RELATIONS where bucket=0); --pick the smallest relnr that has not been assigned a bucket yet
    set @nrOfUpdates = (select count(*) from RELATIONS where bucket=0 and (relnr1=@relnr or relnr2=@relnr));
    update RELATIONS set bucket=@bucket, bucketsub=@bucketsub where bucket=0 and (relnr1=@relnr or relnr2=@relnr);
    set @bucketsub = @bucketsub+1;

    WHILE @nrOfUpdates>0 and @bucketsub<=10    --to prevent the inner while loop from hanging, actually determines the number of iterations
    BEGIN
        --refill temp list with newly found related relnrs
        truncate table ##BucketRelnrs;
        insert into ##BucketRelnrs
        select distinct relnr1 from RELATIONS where bucket=@bucket
        union select distinct relnr2 from RELATIONS where bucket=@bucket;

        --calculate the number of relations that will be updated next, if zero then stop iteration
        set @nrOfUpdates =
        (
            select count(*) from RELATIONS where bucket=0
            and (relnr1 in (select relnr from ##BucketRelnrs) or relnr2 in (select relnr from ##BucketRelnrs))
        );

        --update the RELATIONS table
        update RELATIONS set bucket=@bucket, bucketsub=@bucketsub where bucket=0
        and (relnr1 in (select relnr from ##BucketRelnrs) or relnr2 in (select relnr from ##BucketRelnrs));

        set @bucketsub = @bucketsub+1;
    END
END

drop table ##BucketRelnrs; --clean temp table