SQL: Assembling Non-Overlapping Sets

Question

I have sets of consecutive integers, organized by type, in table1. All values are between 1 and 10, inclusive.

table1:
row_id  set_id  type    min_value   max_value
1       1       a       1           3
2       2       a       4           10
3       3       a       6           10
4       4       a       2           5
5       5       b       1           9
6       6       c       1           7
7       7       c       3           10
8       8       d       1           2
9       9       d       3           3
10      10      d       4           5
11      11      d       7           10

In table2, within each type, I want to assemble all possible maximal, non-overlapping sets (though gaps that cannot be filled by any sets of the correct type are okay). Desired output:

table2:
row_id  type    group_id    set_id
1       a       1           1
2       a       1           2
3       a       2           1
4       a       2           3
5       a       3           3
6       a       3           4
7       b       4           5
8       c       5           6
9       c       6           7
10      d       7           8
11      d       7           9
12      d       7           10
13      d       7           11

My current idea is to use the fact that there is a limited number of possible values. Steps:

1. Find all sets in table1 containing value 1. Copy them into table2.

2. Find all sets in table1 containing value 2 and not already in table2.

3. Join the sets from (2) with table1 on type, set_id, and having min_value greater than the group's greatest max_value.

4. For the sets from (2) that did not join in (3), insert them into table2. These start new groups that may be extended later.

5. Repeat steps (2) through (4) for values 3 through 10.

I think this will work, but it has a lot of pain-in-the-butt steps, especially for (2)--finding the sets not in table2, and (4)--finding the sets that did not join.

Do you know a faster, more efficient method? My real data has millions of sets, thousands of types, and hundreds of values (though fortunately, as in the example, the values are bounded), so scalability is essential.

I'm using PLSQL Developer with Oracle 10g (not 11g as I stated before--thanks, IT department). Thanks!

Answer 1

For Oracle 10g you can't use recursive CTEs, but with a bit of work you can do something similar with the connect by syntax. First you need to generate a CTE or in-line view which has all the non-overlapping links, which you can do with:

select t1.type, t1.set_id, t1.min_value, t1.max_value,
  t2.set_id as next_set_id, t2.min_value as next_min_value,
  t2.max_value as next_max_value,
  row_number() over (order by t1.type, t1.set_id, t2.set_id) as group_id
from table1 t1
left join table1 t2 on t2.type = t1.type
and t2.min_value > t1.max_value
where not exists (
  select 1
  from table1 t4
  where t4.type = t1.type
  and t4.min_value > t1.max_value
  and t4.max_value < t2.min_value
)
order by t1.type, group_id, t1.set_id, t2.set_id;

This took a bit of experimentation and it's certainly possible I've missed or lost something about the rules in the process; but that gives you 12 pseudo-rows, and is in my previous answer this allows the two separate chains starting with a/1 to be followed while constraining the d values to a single chain:

TYPE SET_ID  MIN_VALUE  MAX_VALUE NEXT_SET_ID NEXT_MIN_VALUE NEXT_MAX_VALUE GROUP_ID
---- ------ ---------- ---------- ----------- -------------- -------------- --------
a         1          1          3           2              4             10        1 
a         1          1          3           3              6             10        2 
a         2          4         10                                                  3 
a         3          6         10                                                  4 
a         4          2          5           3              6             10        5 
b         5          1          9                                                  6 
c         6          1          7                                                  7 
c         7          3         10                                                  8 
d         8          1          2           9              3              3        9 
d         9          3          3          10              4              5       10 
d        10          4          5          11              7             10       11 
d        11          7         10                                                 12 

And that can be used as a CTE; querying that with a connect-by loop:

with t as (
   ... -- same as above query
)
select t1.type,
  dense_rank() over (partition by null
    order by connect_by_root group_id) as group_id,
  t1.set_id
from t t1
connect by type = prior type
and set_id = prior next_set_id
start with not exists (
  select 1 from table1 t2
  where t2.type = t1.type
  and t2.max_value < t1.min_value
)
and not exists (
  select 1 from t t3
  where t3.type = t1.type
  and t3.next_max_value < t1.next_min_value
)
order by t1.type, group_id, t1.min_value;

The dense_rank() makes the group IDs contiguous; not sure if you actually need those at all, or if their sequence matters, so it's optional really. connect_by_root gives the group ID for the start of the chain, so although there were 12 rows and 12 group_id values in the initial query, they don't all appear in the final result.

The connection is via two prior values, type and the next set ID found in the initial query. That creates all the chains, but own its own would also include shorter chains - for d you'd see 8,9,10,11 but also 9,10,11 and 10,11, which you don't want as separate groups. Those are eliminated by the start with conditions, which could maybe be simplified.

That gives:

TYPE GROUP_ID SET_ID
---- -------- ------
a           1      1 
a           1      2 
a           2      1 
a           2      3 
a           3      4 
a           3      3 
b           4      5 
c           5      6 
c           6      7 
d           7      8 
d           7      9 
d           7     10 
d           7     11 

SQL Fiddle demo.

Answer 2

If you can identify all the groups and their starting set_id then you can use a recursive approach and do this all in a single statement, rather than needing to populate a table iteratively. However you'd need to benchmark both approaches both for speed/efficiency and resource consumption - whether it will scale for your data volumes and within your system's available resources would need to be verified.

If I understand when you decide to start a new group you can identify them all at once with a query like:

with t as (
  select t1.type, t1.set_id, t1.min_value, t1.max_value,
    t2.set_id as next_set_id, t2.min_value as next_min_value,
    t2.max_value as next_max_value
  from table1 t1
  left join table1 t2 on t2.type = t1.type and t2.min_value > t1.max_value
  where not exists (
    select 1
    from table1 t3
    where t3.type = t1.type
    and t3.max_value < t1.min_value
  )
)
select t.type, t.set_id, t.min_value, t.max_value,
  t.next_set_id, t.next_min_value, t.next_max_value,
  row_number() over (order by t.type, t.min_value, t.next_min_value) as grp_id
from t
where not exists (
  select 1 from t t2
  where t2.type = t.type
  and t2.next_max_value < t.next_min_value
)
order by grp_id;

The tricky bit here is getting all three groups for a, specifically the two groups that start with set_id = 1, but only one group for d. The inner select (in the CTE) looks for sets that don't have a lower non-overlapping range via the not exists clause, and outer-joins to the same table to get the next set(s) that don't overlap, which gives you two groups that start with set_id = 1, but also four that start with set_id = 9. The outer select then ignores everything but the lowest non-overlapping with a second not exists clause - but doesn't have to hit the real table again.

So that gives you:

TYPE SET_ID  MIN_VALUE  MAX_VALUE NEXT_SET_ID NEXT_MIN_VALUE NEXT_MAX_VALUE GRP_ID
---- ------ ---------- ---------- ----------- -------------- -------------- ------
a         1          1          3           2              4             10      1 
a         1          1          3           3              6             10      2 
a         4          2          5           3              6             10      3 
b         5          1          9                                                4 
c         6          1          7                                                5 
c         7          3         10                                                6 
d         8          1          2           9              3              3      7 

You can then use that as the anchor member in a recursive subquery factoring clause:

with t as (
  ...
),
r (type, set_id, min_value, max_value,
    next_set_id, next_min_value, next_max_value, grp_id) as (
  select t.type, t.set_id, t.min_value, t.max_value,
    t.next_set_id, t.next_min_value, t.next_max_value,
    row_number() over (order by t.type, t.min_value, t.next_min_value)
  from t
  where not exists (
    select 1 from t t2
    where t2.type = t.type
    and t2.next_max_value < t.next_min_value
  )
  ...

If you left the r CTE with that and just did sleect * from r you'd get the same seven groups.

The recursive member then uses the next set_id and its range from that query as the next member of each group, and repeats the outer join/not-exists look up to find the next set(s) again; stopping when there is no next non-overlapping set:

  ...
  union all
  select r.type, r.next_set_id, r.next_min_value, r.next_max_value,
    t.set_id, t.min_value, t.max_value, r.grp_id
  from r
  left join table1 t
  on t.type = r.type
  and t.min_value > r.next_max_value
  and not exists (
    select 1 from table1 t2
    where t2.type = r.type
    and t2.min_value > r.next_max_value
    and t2.max_value < t.min_value
  )
  where r.next_set_id is not null -- to stop looking when you reach a leaf node
)
...

Finally you have a query based on the recursive CTE to get the columns you want and to specify the order:

...
select r.type, r.grp_id, r.set_id
from r
order by r.type, r.grp_id, r.min_value;

Which gets:

TYPE     GRP_ID     SET_ID
---- ---------- ----------
a             1          1 
a             1          2 
a             2          1 
a             2          3 
a             3          4 
a             3          3 
b             4          5 
c             5          6 
c             6          7 
d             7          8 
d             7          9 
d             7         10 
d             7         11 

SQL Fiddle demo.

If you wanted to you could show the min/max values for each set, and could track and show the min/max value for each group. I've just show then columns from the question though.