sql/PLSQL algorithm: is a rectangle filled with the other rectangles

Hello
I have 2 tables: T1 and T2:
with T1 as
(select 2 FromPeriod, 4 ToPeriod, 10 FromSum, 12 ToSum  from dual)
select * from t1;


with T2 as
(select 1 FromPeriod, 2 ToPeriod, 9 FromSum, 5 ToSum  from dual
union all
select 1 FromPeriod, 3 ToPeriod, 9 FromSum, 4 ToSum  from dual
union all
select 2 FromPeriod, 5 ToPeriod, 9 FromSum, 11 ToSum  from dual
union all
select 2 FromPeriod, 5 ToPeriod, 11 FromSum, 12 ToSum  from dual)
select * from t2;
FromPeriod, ToPeriod are positive whole number without decimal point.
FromSum, ToSum are positive numbers can have the comma.

Table T1 contains unique in the columns 'FromPeriod-ToPeriod' points, that means that is there is a record where "FromPeriod-ToPeriod" = [2-4], then it is certainly not rendered as "FromPeriod-ToPeriod" = [1-4] keys wich [2-4] period, such a thing is not possible in table P1, it contains only unique periods that do not touch. Then the records with periods [2-4], [5-6] are ok in there.
But table T2 allows periods affecting somehow. You can see that T2 contains "FromPeriod-ToPeriod" = [1-2] and also [1-3] that touch each other.

We can look at the two tables T1 and T2 as rectangles where X - axis is the values of the axes and the "FromPeriod-ToPeriod" is [FromSum, ToSum].
I need sql/plsql algorithm that returns true if the rectangle of table T1 is discovered/filled with rectangles in table T2. If the rectangle in T1 is completely filled or even more filled than necessary, then algorithm should return true. If even a small neighborhood in rectangle T1 is not filled by rectangles in T2, then the algorithm must return false.

As see you in the sample data, T1 has a rectangle: [2-4; 10-12] T2 contains 4 mentions that fill/overdraw this area defined by T1. So for the moment the algorithm must return true.

I do not know how to write effective elsewhere such algorithm, can you help me?

--
At the moment I only came with such algorithm/query:

Test1: this test should return true:
--query: is rectangle [2-4;10-12] filled? -Yes.
with T2 as
(select 1 FromPeriod, 2 ToPeriod, 9 FromSum, 5 ToSum  from dual
union all
select 1 FromPeriod, 3 ToPeriod, 9 FromSum, 4 ToSum  from dual
union all
select 2 FromPeriod, 5 ToPeriod, 9 FromSum, 11 ToSum  from dual
union all
select 2 FromPeriod, 5 ToPeriod, 11 FromSum, 12 ToSum  from dual)
select * from t2
where (2 <= ToPeriod and 4 >= FromPeriod) -- x-axes overlapping
and (10 <= ToSum and 12 >= FromSum)--y-axes overlapping
;--2 rows

--

2     5     9     11
2     5     11     12
I takes as input the T1 table horizontal axes (2, [4]) and research data table T2 that affect this horizontal axes, also the value axes is compared. But how then? There are 2 files, but it can give 10 stores overlap in the periods and amounts. How to determine now that rectange T1 is filled?

Test2: This case should return false:
--query: is rectangle [2-4;10-12] filled? -No.
with T2 as
(select 1 FromPeriod, 2 ToPeriod, 9 FromSum, 4 ToSum  from dual
union all
select 1 FromPeriod, 3 ToPeriod, 9 FromSum, 4 ToSum  from dual
union all
select 2 FromPeriod, 4 ToPeriod, 9 FromSum, 11 ToSum  from dual
union all
select 2 FromPeriod, 5 ToPeriod, 13 FromSum, 13 ToSum  from dual)
select * from t2
where (2 <= ToPeriod and 4 >= FromPeriod) -- x-axes overlapping
and (10 <= ToSum and 12 >= FromSum)--y-axes overlapping
;--1 row.

2     4     9     11

Charles,

I see 2 choices, but none is likely to evolve very well. The first can be with SQL - subdivision T1 into unit squares and see if everything not covered by T2:

with T1 as
(select 2 FromPeriod, 4 ToPeriod, 10.1 FromSum, 12 ToSum  from dual)
, T1X as
(select (FromPeriod+level)-1 as Period from T1 connect by level <= (ToPeriod-FromPeriod)+1)
, T1Y as
(select (FromSum+level*.01)-.01 as Summ from T1 connect by level <= (ToSum-FromSum+.01)/.01)
, T1XY as (select Period,Summ from T1X,T1Y)
,T2 as
(select 1 FromPeriod, 2 ToPeriod, 9.99 FromSum, 10.1 ToSum  from dual--satisfies sub-region [(2,2);(10.1,10.1)]
union all
select 1 FromPeriod, 2 ToPeriod, 9.99 FromSum, 10.11 ToSum  from dual--now satisfied sub-regions: [(2,2);(10.1,10.11)
union all
select 2 FromPeriod, 2 ToPeriod, 10 FromSum, 12.01 ToSum  from dual--now satisfied sub-regions: [(2,2);(10.1,12)
union all
select 3 FromPeriod, 4 ToPeriod, 10 FromSum, 11.99 ToSum  from dual--now satisfied sub-regions: [(2,2);(10.1,12)],[(3,4);(10,11.99)]
union all
select 4 FromPeriod, 4 ToPeriod, 10 FromSum, 13 ToSum  from dual--now satisfied sub-regions: all filled, return true
)
select Period,Summ from T1XY
where not exists (
select 1
from T2
where T2.FromPeriod<=T1XY.Period
and   T2.ToPeriod>=T1XY.Period
and   T2.FromSum<=T1XY.Summ
and   T2.ToSum>=T1XY.Summ
)
;

The second option is more inspiring you described, using a recursive algorithm to subdivide the T1 into small rectangles for each rectangle in T2. For example:

X(1)={[a,b;c,d]}

Y={[e,f;g,h];[i,j;k,l]}

X(2)={[a,e;c,g];[e,f;c,g];[f,b;c,g];[a,e;g,h];[f,b;g,h];[a,e;h,d];[e,f;h,d];[f,b;h,d]}
or
X(2)={[a,b;c,g];[a,b;h,d];[a,e;c,d];[f,b;c,d]}  -- this gives some overlap, but fewer rectangles and the algorithm still works.

You would have to refine it to handle partial overlaps correctly, then drop areas that don't have a positive area. If
X(2) becomes empty then it is fully covered. It might be most efficient to sort T2 in order of descending area.

Kind regards
Bob

Tags: Database

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