JOIN each row only once — arrange (distribute) rows 1 to 1 - mysql

Question

I have two tables to join, and I want each row to be joined only once. Here is sample data:
CREATE TABLE A (id smallint, val varchar(1) );
CREATE TABLE B (id smallint, val varchar(1) );
INSERT INTO A VALUES (1, 'a'), (2, 'b'), (3, 'c'), (3, 'd');
INSERT INTO B VALUES (2, 'x'), (3, 'y'), (4, 'z'), (3, 'k');
When we join on id we obtain:
mysql> SELECT * FROM A JOIN B ON A.id = B.id;
+------+------+------+------+
| id | val | id | val |
+------+------+------+------+
| 2 | b | 2 | x |
| 3 | c | 3 | y |
| 3 | d | 3 | y |
| 3 | c | 3 | k |
| 3 | d | 3 | k |
+------+------+------+------+
What I want is either:
+------+------+------+------+ +------+------+------+------+
| id | val | id | val | | id | val | id | val |
+------+------+------+------+ or +------+------+------+------+
| 2 | b | 2 | x | | 2 | b | 2 | x |
| 3 | c | 3 | y | | 3 | d | 3 | y |
| 3 | d | 3 | k | | 3 | c | 3 | k |
+------+------+------+------+ +------+------+------+------+
The order and arrangement don’t matter.
Is it possible? How?
According to this answer I need to specify how to select the matching row. In that case I guess would need to check in a subquery if the row of the joined table has already been used; or a kind of counter related to the id... but I do not know how to write this.
Edit:
To clarify I want each row with id 3 to be mapped with another one in the joined table, such as each row is mapped only once (I am also interested to know what happens when the number of rows with same id is different in the two tables):
(3, c) -> (3, y) [join only with the first row such as B.id = 3]
(3, d) -> (3, k) [the first row has been used, so map with (and only with) the second row such as B.id = 3]
But as I said mapping may be in any other order (e.g. mapping rows in reverse order).

Answer 1

SQL Fiddle
MySQL 5.6 Schema Setup:
CREATE TABLE A (id smallint, val varchar(1) );
CREATE TABLE B (id smallint, val varchar(1) );
INSERT INTO A VALUES (1, 'a'), (2, 'b'), (3, 'c'), (3, 'd');
INSERT INTO B VALUES (2, 'x'), (3, 'y'), (4, 'z'), (3, 'k');
Query 1:
select
aa.id as aid
, aa.val as aval
, bb.id as bid
, bb.val as bval
from (
select
#row_num :=IF(#prev_value=a.id,#row_num+1,1)AS RowInGroup
, a.id
, a.val
, #prev_value := a.id
from (
SELECT id, val
FROM A
group by id, val
/* order by ?? */
) a
CROSS JOIN (
SELECT #row_num :=1, #prev_value :=''
) vars
) aa
INNER JOIN (
select
#row_num :=IF(#prev_value=b.id,#row_num+1,1)AS RowInGroup
, b.id
, b.val
, #prev_value := b.id
from (
SELECT id, val
FROM B
group by id, val
/* order by ?? */
) b
CROSS JOIN (
SELECT #row_num :=1, #prev_value :=''
) vars
) bb on aa.id = bb.id and aa.RowInGroup = bb.RowInGroup
order by
aa.id
, aa.val
Results:
| id | val | id | val |
|----|-----|----|-----|
| 2 | b | 2 | x |
| 3 | c | 3 | k |
| 3 | d | 3 | y |
nb: you could influence the final result by introducing order by in the subqueries that group by id, val where the sequence RowInGroup is calculated.

Answer 2

Finally I did it!
SELECT T.ID_A,
T.VAL_A,
T.XXXX,
T.ID_B,
T.VAL_B,
T.YYYY
FROM (
SELECT A.id AS ID_A,
A.VAL AS VAL_A,
ROW_NUMBER() OVER (PARTITION BY A.ID, A.VAL ORDER BY A.ID, A.VAL) AS XXXX,
B.ID AS ID_B,
B.VAL AS VAL_B,
ROW_NUMBER() OVER (PARTITION BY B.ID, B.VAL ORDER BY B.ID DESC, B.VAL) AS YYYY
FROM A INNER JOIN B ON A.id = B.id) AS T
WHERE T.YYYY = 1

Answer 3

I could make it thanks to this blog post:
SELECT A2.id, A2.val, B2.val FROM (
SELECT l.id, l.val, COUNT(*) AS n1 FROM A AS l JOIN A AS r ON l.id = r.id AND l.val >= r.val GROUP BY l.id, l.val
) AS A2 JOIN (
SELECT l.id, l.val, COUNT(*) AS n2 FROM B AS l JOIN B AS r ON l.id = r.id AND l.val >= r.val GROUP BY l.id, l.val
) AS B2 ON
A2.id = B2.id AND n1 = n2;
The result is:
+------+------+------+
| id | val | val |
+------+------+------+
| 2 | b | x |
| 3 | c | k |
| 3 | d | y |
+------+------+------+