Recursive SQL statement (Postgresql) - simplified version

Question

This is simplified question for more complicated one posted here:

Recursive SQL statement (PostgreSQL 9.1.4)

Simplified question

Given you have upper triangular matrix stored in 3 columns (RowIndex, ColumnIndex, MatrixValue):

   ColumnIndex       
    1   2   3   4   5
1   2   2   3   3   4
2   4   4   5   6   X
3   3   2   2   X   X
4   2   1   X   X   X
5   1   X   X   X   X

X values are to be calculated using the following algorithm:

M[i,j] = (M[i-1,j]+M[i,j-1])/2
(i= rows, j = columns, M=matrix)

Example:
M[3,4] = (M[2,4]+M[3,3])/2
M[3,5] = (m[2,5]+M[3,4])/2

The full required result is:

   ColumnIndex       
    1   2   3    4      5
1   2   2   3    3      4
2   4   4   5    6      5
3   3   2   2    4      4.5
4   2   1   1.5  2.75   3.625
5   1   1   1.25 2.00   2.8125

Sample data:

create table matrix_data (
    RowIndex integer,
    ColumnIndex integer,
    MatrixValue numeric);

    insert into matrix_data values (1,1,2);
    insert into matrix_data values (1,2,2);
    insert into matrix_data values (1,3,3);
    insert into matrix_data values (1,4,3);
    insert into matrix_data values (1,5,4);
    insert into matrix_data values (2,1,4);
    insert into matrix_data values (2,2,4);
    insert into matrix_data values (2,3,5);
    insert into matrix_data values (2,4,6);
    insert into matrix_data values (3,1,3);
    insert into matrix_data values (3,2,2);
    insert into matrix_data values (3,3,2);
    insert into matrix_data values (4,1,2);
    insert into matrix_data values (4,2,1);
    insert into matrix_data values (5,1,1);

Can this be done?

Answer 1

... and here comes the recursive CTE with multiple embedded CTE's (tm):

DROP SCHEMA tmp CASCADE;
CREATE SCHEMA tmp ;
SET search_path=tmp;

CREATE TABLE matrix_data (
    yyy integer,
    xxx integer,
    val numeric);

    insert into matrix_data (yyy,xxx,val) values
      (1,1,2) , (1,2,2) , (1,3,3) , (1,4,3) , (1,5,4)
    , (2,1,4) , (2,2,4) , (2,3,5) , (2,4,6)
    , (3,1,3) , (3,2,2) , (3,3,2)
    , (4,1,2) , (4,2,1)
    , (5,1,1)
        ;

WITH RECURSIVE rr AS (
        WITH xx AS (
                SELECT MIN(xxx) AS x0
                , MAX(xxx) AS x1
                FROM matrix_data
                )
        , mimax AS (
                SELECT generate_series(xx.x0,xx.x1) AS xxx
                FROM xx
                )
        , yy AS (
                SELECT MIN(yyy) AS y0
                , MAX(yyy) AS y1
                FROM matrix_data
                )
        , mimay AS (
                SELECT generate_series(yy.y0,yy.y1) AS yyy
                FROM yy
                )
        , cart AS (
                SELECT * FROM mimax mm
                JOIN mimay my ON (1=1)
                )
        , empty AS (
                SELECT * FROM cart ca
                WHERE NOT EXISTS (
                        SELECT *
                        FROM matrix_data nx
                        WHERE nx.xxx = ca.xxx
                        AND nx.yyy = ca.yyy
                        )
                )
        , hot AS (
                SELECT * FROM empty emp
                WHERE EXISTS (
                        SELECT *
                        FROM matrix_data ex
                        WHERE ex.xxx = emp.xxx -1
                        AND ex.yyy = emp.yyy
                        )
                AND EXISTS (
                        SELECT *
                        FROM matrix_data ex
                        WHERE ex.xxx = emp.xxx
                        AND ex.yyy = emp.yyy -1
                        )
                    )
        -- UPDATE from here:
        SELECT h.xxx,h.yyy, md.val / 2 AS val
        FROM hot h
        JOIN matrix_data md ON
                (md.yyy = h.yyy AND md.xxx = h.xxx-1)
                OR (md.yyy = h.yyy-1 AND md.xxx = h.xxx)
        UNION ALL
        SELECT e.xxx,e.yyy, r.val / 2 AS val
        FROM empty e
        JOIN rr r ON ( e.xxx = r.xxx+1 AND e.yyy = r.yyy)
                OR ( e.xxx = r.xxx AND e.yyy = r.yyy+1 )
        )
INSERT INTO matrix_data(yyy,xxx,val)
SELECT DISTINCT yyy,xxx
        ,SUM(val)
FROM rr
GROUP BY yyy,xxx
        ;

SELECT * FROM matrix_data
        ;

New result:

NOTICE:  drop cascades to table tmp.matrix_data
DROP SCHEMA
CREATE SCHEMA
SET
CREATE TABLE
INSERT 0 15
INSERT 0 10
 yyy | xxx |          val           
-----+-----+------------------------
   1 |   1 |                      2
   1 |   2 |                      2
   1 |   3 |                      3
   1 |   4 |                      3
   1 |   5 |                      4
   2 |   1 |                      4
   2 |   2 |                      4
   2 |   3 |                      5
   2 |   4 |                      6
   3 |   1 |                      3
   3 |   2 |                      2
   3 |   3 |                      2
   4 |   1 |                      2
   4 |   2 |                      1
   5 |   1 |                      1
   2 |   5 |     5.0000000000000000
   5 |   5 | 2.81250000000000000000
   4 |   3 | 1.50000000000000000000
   3 |   5 | 4.50000000000000000000
   5 |   2 | 1.00000000000000000000
   3 |   4 | 4.00000000000000000000
   5 |   3 | 1.25000000000000000000
   4 |   5 | 3.62500000000000000000
   4 |   4 | 2.75000000000000000000
   5 |   4 | 2.00000000000000000000
(25 rows)

Answer 2

 while (select max(ColumnIndex+RowIndex) from matrix_data)<10
 begin
      insert matrix_data
      select c1.RowIndex, c1.ColumnIndex+1, (c1.MatrixValue+c2.MatrixValue)/2 
      from matrix_data c1
           inner join
           matrix_data c2 
           on c1.ColumnIndex+1=c2.ColumnIndex and c1.RowIndex-1 = c2.RowIndex   
      where c1.RowIndex+c1.ColumnIndex=(select max(RowIndex+ColumnIndex) from matrix_data) 
      and c1.ColumnIndex<5
 end

Answer 3

Test setup:

CREATE TEMP TABLE matrix (
    rowindex integer,
    columnindex integer,
    matrixvalue numeric);

INSERT INTO matrix VALUES
 (1,1,2),(1,2,2),(1,3,3),(1,4,3),(1,5,4)
,(2,1,4),(2,2,4),(2,3,5),(2,4,6)
,(3,1,3),(3,2,2),(3,3,2)
,(4,1,2),(4,2,1)
,(5,1,1);

Run INSERTs in a LOOP with DO:

DO $$
BEGIN

FOR i IN 2 .. 5 LOOP
   FOR j IN 7-i .. 5 LOOP
      INSERT INTO matrix
      VALUES (i,j, (
         SELECT sum(matrixvalue)/2
         FROM   matrix
         WHERE  (rowindex, columnindex) IN ((i-1, j),(i, j-1))
         ));
   END LOOP;
END LOOP;

END;
$$

See result:

SELECT * FROM matrix order BY 1,2;

Answer 4

This can be done in a single SQL select statement, but only because recursion is not necessary. I'll outline the solution. If you actually want the SQL code, let me know.

First, notice that the only items that contribute to the sums are along the diagonal. Now, if we follow the contribution of the value "4" in (1, 5), it contributes 4/2 to (2,5) and 4/4 to (3,5) and 4/8 to (4,5). Each time, the contribution is cut in half, because (a+b)/2 is (a/2 + b/2).

When we extend this, we start to see a pattern similar to Pascal's triangle. In fact, for any given point in the lower triangular matrix (below where you have values), you can find the diagonal elements that contribute to the value. Extend a vertical line up to hit the diagonal and a horizontal line to hit the diagonal. Those are the contributors from the diagonal row.

How much do they contribute? Well, for that we can go to Pascal's triangle. For the first diagonal below where we have values, the contributions are (1,1)/2. For the second diagonal, (1,2,1)/4. For the third, (1,3,3,1)/8 . . . and so on.

Fortunately, we can calculate the contributions for each value using a formula (the "choose" function from combinatorics). The power of 2 is easy. And, determining how far a given cell is from the diagonal is not too hard.

All of this can be combined into a single Postgres SQL statement. However, @Erwin's solution also works. I only want to put the effort into debugging the statement if his solution doesn't meet your needs.