Optimize query to GROUP BY uid with UNION - mysql

Question

When GROUP BY uid and GROUP BY anonym_id are placed into queries the execution time problem appears > 2sec.
The question is - how to optimize/override this query with effect like GROUP BY statement?
( SELECT u.id as uid, u.name, u.avatar, u.avatar_date, u.driver,
u.vehicle, m.msg, m.removed, (m.date DIV 1000) AS date, m.from_id = 1 AS outbox,
CASE WHEN m.read_state IS NULL THEN 1 ELSE 0 END AS read_state, d.anonym_id
FROM dialog as d CROSS JOIN messages AS m CROSS JOIN users AS u
WHERE (m.id=d.mid AND ((d.uid1 = 1 and u.id = d.uid2) or (d.uid2 = 1 and u.id = d.uid1)))
GROUP BY uid
ORDER BY d.id DESC )
UNION
( SELECT u.id as uid, u.name, u.avatar, u.avatar_date, u.driver, u.vehicle,
m.msg, m.removed, (m.date DIV 1000) AS date, m.from_id = 1 AS outbox,
CASE WHEN m.read_state IS NULL THEN 1 ELSE 0 END AS read_state,
d.anonym_id
FROM dialog as d FORCE KEY(anonym_id) CROSS JOIN messages AS m ON d.mid=m.id
CROSS JOIN users AS u ON u.id=d.uid2
WHERE d.anonym_id=100001
GROUP BY d.uid2
ORDER BY d.id DESC )
LIMIT 0, 20
mysql> show index from users;
+-------+------------+-------------+--------------+--------------+-----------+-------------+----------+--------+------+------------+---------+---------------+
| Table | Non_unique | Key_name | Seq_in_index | Column_name | Collation | Cardinality | Sub_part | Packed | Null | Index_type | Comment | Index_comment |
+-------+------------+-------------+--------------+--------------+-----------+-------------+----------+--------+------+------------+---------+---------------+
| users | 0 | PRIMARY | 1 | id | A | 606396 | NULL | NULL | | BTREE | | |
| users | 1 | number | 1 | number | A | 606396 | NULL | NULL | | BTREE | | |
| users | 1 | name | 1 | name | A | 121279 | NULL | NULL | | BTREE | | |
| users | 1 | show_phone | 1 | show_phone | A | 10 | NULL | NULL | | BTREE | | |
| users | 1 | show_mail | 1 | show_mail | A | 14 | NULL | NULL | | BTREE | | |
| users | 1 | show_on_map | 1 | show_on_map | A | 18 | NULL | NULL | | BTREE | | |
| users | 1 | show_on_map | 2 | map_activity | A | 606396 | NULL | NULL | | BTREE | | |
| users | 1 | udid | 1 | udid | A | 606396 | NULL | NULL | YES | BTREE | | |
| users | 1 | blocked | 1 | blocked | A | 10 | NULL | NULL | | BTREE | | |
+-------+------------+-------------+--------------+--------------+-----------+-------------+----------+--------+------+------------+---------+---------------+
mysql> show index from dialog;
+--------+------------+-----------+--------------+-------------+-----------+-------------+----------+--------+------+------------+---------+---------------+
| Table | Non_unique | Key_name | Seq_in_index | Column_name | Collation | Cardinality | Sub_part | Packed | Null | Index_type | Comment | Index_comment |
+--------+------------+-----------+--------------+-------------+-----------+-------------+----------+--------+------+------------+---------+---------------+
| dialog | 0 | PRIMARY | 1 | id | A | 2964484 | NULL | NULL | | BTREE | | |
| dialog | 1 | uid1 | 1 | uid1 | A | 211748 | NULL | NULL | | BTREE | | |
| dialog | 1 | uid1 | 2 | uid2 | A | 2964484 | NULL | NULL | | BTREE | | |
| dialog | 1 | uid2 | 1 | uid2 | A | 1482242 | NULL | NULL | | BTREE | | |
| dialog | 1 | anonym_id | 1 | anonym_id | A | 18 | NULL | NULL | | BTREE | | |
| dialog | 1 | idx_mid | 1 | mid | A | 200 | NULL | NULL | | BTREE | | |
+--------+------------+-----------+--------------+-------------+-----------+-------------+----------+--------+------+------------+---------+---------------+
mysql> show index from messages;
+----------+------------+-----------+--------------+-------------+-----------+-------------+----------+--------+------+------------+---------+---------------+
| Table | Non_unique | Key_name | Seq_in_index | Column_name | Collation | Cardinality | Sub_part | Packed | Null | Index_type | Comment | Index_comment |
+----------+------------+-----------+--------------+-------------+-----------+-------------+----------+--------+------+------------+---------+---------------+
| messages | 0 | PRIMARY | 1 | id | A | 23227116 | NULL | NULL | | BTREE | | |
| messages | 1 | user_id_2 | 1 | user_id | A | 188 | NULL | NULL | | BTREE | | |
| messages | 1 | user_id_2 | 2 | read_state | A | 188 | NULL | NULL | | BTREE | | |
| messages | 1 | user_id_2 | 3 | removed | A | 188 | NULL | NULL | | BTREE | | |
| messages | 1 | from_id | 1 | from_id | A | 188 | NULL | NULL | | BTREE | | |
| messages | 1 | from_id | 2 | to_number | A | 188 | NULL | NULL | | BTREE | | |
+----------+------------+-----------+--------------+-------------+-----------+-------------+----------+--------+------+------------+---------+---------------+
EXPLAIN EXTENDED
1 PRIMARY d index_merge uid1,uid2,idx_mid uid1,uid2 4,4 26 100.00 Using sort_union(uid1,uid2); Using where; Using temporary; Using filesort
1 PRIMARY u ALL PRIMARY 625866 100.00 Using where; Using join buffer
1 PRIMARY m eq_ref PRIMARY PRIMARY 4 numbers.d.mid 1 100.00
2 UNION d ref anonym_id anonym_id 4 const 1 100.00
2 UNION u eq_ref PRIMARY PRIMARY 4 numbers.d.uid2 1 100.00
2 UNION m eq_ref PRIMARY PRIMARY 4 numbers.d.mid 1 100.00
UNION RESULT <union1,2> ALL