Simple algebra with recursive SQL - sql

Question

The following schema is used to create simple algebraic formulas. variables is used to create formulas such as x=3+4y. variables_has_sub_variables is used to combine the previous mentioned formulas and uses the sign column (will be +1 or -1 only) to determine whether the formula should be added or subtracted to the combination.
For instance, variables table might have the following data where the Implied Formulas column is not really in the table but just for illustrative purposes only.
variables table
+-----------+-----------+-------+------------------+
| variables | intercept | slope | Implied Formula |
+-----------+-----------+-------+------------------+
| 1 | 2.86 | -0.82 | Y1=+2.86-0.82*X1 |
| 2 | 2.96 | -3.49 | Y2=+2.96-3.49*X2 |
| 3 | 2.56 | 2.81 | Y3=+2.56+2.81*X3 |
| 4 | 3.04 | -3.43 | Y4=+3.04-3.43*X4 |
| 5 | -1.94 | 4.11 | Y5=-1.94+4.11*X5 |
| 6 | -1.21 | -0.62 | Y6=-1.21-0.62*X6 |
| 7 | 0.88 | -0.61 | Y7=+0.88-0.61*X7 |
| 8 | -2.77 | -0.34 | Y8=-2.77-0.34*X8 |
| 9 | 1.81 | 1.65 | Y9=+1.81+1.65*X9 |
+-----------+-----------+-------+------------------+
Then, given the below variables_has_sub_variables data, the variables combined resulting in X7=+Y1-Y2+Y3, X8=+Y4+Y5-Y7, and X9=+Y6-Y7+Y8. Next Y7, Y8, and Y9 can be derived using the variables table resulting in Y7=+0.88-0.61*X7, etc. Note that the application will prevent an endless loop such as inserting a record where variables equals 7 and sub_variables equals 9 as variable 9 is based on variable 7.
variables_has_sub_variables table
+-----------+---------------+------+
| variables | sub_variables | sign |
+-----------+---------------+------+
| 7 | 1 | 1 |
| 7 | 2 | -1 |
| 7 | 3 | 1 |
| 8 | 4 | 1 |
| 8 | 5 | 1 |
| 8 | 7 | -1 |
| 9 | 6 | 1 |
| 9 | 7 | -1 |
| 9 | 8 | 1 |
+-----------+---------------+------+
My objective is given any variable (i.e. 1 to 9), determine the constants and root variables where a root variable is defined as not being in variables_has_sub_variables.variables (I can also easily a root column to variables if needed), and these root variables includes 1 through 6 using my above example data.
Doing so for a root variable is easier as there are no sub_variables and is simply Y1=+2.86-0.82*X1.
Doing so for variable 7 is a little trickier:
Y7=+0.88-0.61*X7
=+0.88-0.61*(+Y1-Y2+Y3)
=+0.88-0.61*(+(+2.86-0.82*X1)-(+2.96-3.49*X2)+( +2.56+2.81*X3))
= -0.62 + 0.50*X1 - 2.13*X2 - 1.71*X3
Now the SQL. Below is how I created the tables:
CREATE DATABASE algebra;
USE algebra;
CREATE TABLE `variables` (
`variables` INT NOT NULL,
`slope` DECIMAL(6,2) NOT NULL DEFAULT 1,
`intercept` DECIMAL(6,2) NOT NULL DEFAULT 0,
PRIMARY KEY (`variables`))
ENGINE = InnoDB;
CREATE TABLE `variables_has_sub_variables` (
`variables` INT NOT NULL,
`sub_variables` INT NOT NULL,
`sign` TINYINT NOT NULL,
PRIMARY KEY (`variables`, `sub_variables`),
INDEX `fk_variables_has_variables_variables1_idx` (`sub_variables` ASC),
INDEX `fk_variables_has_variables_variables_idx` (`variables` ASC),
CONSTRAINT `fk_variables_has_variables_variables`
FOREIGN KEY (`variables`)
REFERENCES `variables` (`variables`)
ON DELETE NO ACTION
ON UPDATE NO ACTION,
CONSTRAINT `fk_variables_has_variables_variables1`
FOREIGN KEY (`sub_variables`)
REFERENCES `variables` (`variables`)
ON DELETE NO ACTION
ON UPDATE NO ACTION)
ENGINE = InnoDB;
INSERT INTO variables(variables,intercept,slope) VALUES (1,2.86,-0.82),(2,2.96,-3.49),(3,2.56,2.81),(4,3.04,-3.43),(5,-1.94,4.11),(6,-1.21,-0.62),(7,0.88,-0.61),(8,-2.77,-0.34),(9,1.81,1.65);
INSERT INTO variables_has_sub_variables(variables,sub_variables,sign) VALUES (7,1,1),(7,2,-1),(7,3,1),(8,4,1),(8,5,1),(8,7,-1),(9,6,1),(9,7,-1),(9,8,1);
And now the queries. XXXX is 7, 8, and 9 for the following results. Before each query, I show my expected results.
WITH RECURSIVE t AS (
SELECT v.variables, v.slope, v.intercept
FROM variables v
WHERE v.variables=XXXX
UNION ALL
SELECT v.variables, vhsv.sign*t.slope*v.slope slope, vhsv.sign*t.slope*v.intercept intercept
FROM t
INNER JOIN variables_has_sub_variables vhsv ON vhsv.variables=t.variables
INNER JOIN variables v ON v.variables=vhsv.sub_variables
)
SELECT variables, SUM(slope) constant FROM t GROUP BY variables
UNION SELECT 'intercept' variables, SUM(intercept) intercept FROM t;
Variable 7 Desired
+-----------+----------+
| variables | constant |
+-----------+----------+
| 1 | 0.50 |
| 2 | -2.13 |
| 3 | -1.71 |
| intercept | -0.6206 |
+-----------+----------+
Variable 7 Actual
+-----------+----------+
| variables | constant |
+-----------+----------+
| 1 | 0.50 |
| 2 | -2.13 |
| 3 | -1.71 |
| 7 | -0.61 |
| intercept | -0.61 |
+-----------+----------+
5 rows in set (0.00 sec)
Variable 8 Desired
+-----------+-----------+
| variables | constant |
+-----------+-----------+
| 1 | 0.17 |
| 2 | -0.72 |
| 3 | -0.58 |
| 4 | 1.17 |
| 5 | -1.40 |
| intercept | -3.355004 |
+-----------+-----------+
Variable 8 Actual
+-----------+----------+
| variables | constant |
+-----------+----------+
| 1 | 0.17 |
| 2 | -0.73 |
| 3 | -0.59 |
| 4 | 1.17 |
| 5 | -1.40 |
| 7 | -0.21 |
| 8 | -0.34 |
| intercept | -3.36 |
+-----------+----------+
8 rows in set (0.00 sec)
Variable 9 Desired
+-----------+------------+
| variables | constant |
+-----------+------------+
| 1 | -0.54 |
| 2 | 2.32 |
| 3 | 1.87 |
| 4 | 1.92 |
| 5 | -2.31 |
| 6 | -1.02 |
| intercept | -4.6982666 |
+-----------+------------+
Variable 9 Actual
+-----------+----------+
| variables | constant |
+-----------+----------+
| 1 | -0.55 |
| 2 | 2.33 |
| 3 | 1.88 |
| 4 | 1.92 |
| 5 | -2.30 |
| 6 | -1.02 |
| 7 | 0.67 |
| 8 | -0.56 |
| 9 | 1.65 |
| intercept | -4.67 |
+-----------+----------+
10 rows in set (0.00 sec)
All I need to do is detect which variables are not the root variables and filter them out. How should this be accomplished?
In response to JNevill's answer:
For v.variables of 9
+-----------+-------+-------+----------+
| variables | depth | path | constant |
+-----------+-------+-------+----------+
| 1 | 3 | 9>7>1 | -0.55 |
| 2 | 3 | 9>7>2 | 2.33 |
| 3 | 3 | 9>7>3 | 1.88 |
| 4 | 3 | 9>8>4 | 1.92 |
| 5 | 3 | 9>8>5 | -2.30 |
| 6 | 2 | 9>6 | -1.02 |
| 7 | 2 | 9>7 | 0.67 |
| 8 | 2 | 9>8 | -0.56 |
| 9 | 1 | 9 | 1.65 |
| intercept | 1 | 9 | -4.67 |
+-----------+-------+-------+----------+
10 rows in set (0.00 sec)

Answer 1

I'm not going to attempt to fully wrap my head around what you are doing, and I would agree with #RickJames up in the comments that this feels like maybe not the best use-case for a database. I too am a little obsessive though. I get it.
There are couple of things that I almost always track in a recursive CTE.
The "Path". If I'm going to let a query head down a rabbit hole, I want to know how it got to the end point. So I track a path so I know which primary key was selected through each iteration. In the recursive seed (top portion) I use something like SELECT CAST(id as varchar(500)) as path... and in the recursive member (bottom portion) I do something like recursiveCTE.path + '>' + id as path...
The "Depth". I want to know how deep the iterations went to get to the resulting record. This is tracked by adding SELECT 1 as depth to the recursive seed and recursiveCTE + 1 as depth to the recursive member. Now I know how deep each record is.
I believe number 2 will solve your issue:
WITH RECURSIVE t
AS (
SELECT v.variables,
v.slope,
v.intercept,
1 as depth
FROM variables v
WHERE v.variables = XXXX
UNION ALL
SELECT v.variables,
vhsv.sign * t.slope * v.slope slope,
vhsv.sign * t.slope * v.intercept intercept,
t.depth + 1
FROM t
INNER JOIN variables_has_sub_variables vhsv ON vhsv.variables = t.variables
INNER JOIN variables v ON v.variables = vhsv.sub_variables
)
SELECT variables,
SUM(slope) constant
FROM t
WHERE depth > 1
GROUP BY variables
UNION
SELECT 'intercept' variables,
SUM(intercept) intercept
FROM t;
The WHERE clause here will restrict records in your recursive result set that have a depth of 1, meaning they were brought in from the recursive seed portion of the recursive CTE (That they are a root).
It wasn't clear if you required that the root be removed from your second UNION of your t CTE. If so, the same logic applies; just toss that WHERE clause on to restrict depth records of 1
While it may not be helpful here, an example of your recursive cte with PATH would be:
WITH RECURSIVE t
AS (
SELECT v.variables,
v.slope,
v.intercept,
1 as depth,
CAST(v.variables as CHAR(30)) as path
FROM variables v
WHERE v.variables = XXXX
UNION ALL
SELECT v.variables,
vhsv.sign * t.slope * v.slope slope,
vhsv.sign * t.slope * v.intercept intercept,
t.depth + 1,
CONCAT(t.path,'>', v.variables)
FROM t
INNER JOIN variables_has_sub_variables vhsv ON vhsv.variables = t.variables
INNER JOIN variables v ON v.variables = vhsv.sub_variables
)
SELECT variables,
SUM(slope) constant
FROM t
WHERE depth > 1
GROUP BY variables
UNION
SELECT 'intercept' variables,
SUM(intercept) intercept
FROM t;