Optimizing MIN and MAX MySQL Functions

MySQL can optimize aggregate functions like MIN and MAX as long as the columns specified are indexed. This means that, in the case of MIN and MAX, the optimizer should be able to identify the highest and lowest values of an indexed column from the B-Tree index. Say I have a table like below:

CREATE TABLE `history` (
   `h_id` bigint(20) unsigned NOT NULL AUTO_INCREMENT,
   `u_id` int(10) unsigned NOT NULL,
   `cn_id` int(10) unsigned NOT NULL,
   `f_id` int(10) unsigned NOT NULL
   PRIMARY KEY (`h_id`)
 ) ENGINE=InnoDB

If I want to get the MAX value for cn_id, I’d to a query like this which will be a full table scan:

mysql (test) > EXPLAIN SELECT MAX(cn_id) FROM history \G
 *************************** 1. row ***************************
            id: 1
   select_type: SIMPLE
         table: history
          type: ALL
 possible_keys: NULL
           key: NULL
       key_len: NULL
           ref: NULL
          rows: 19908716
         Extra:
 1 row in set (0.00 sec)
If I add an index on cn_id, the query is optimized internally:
mysql (test) > EXPLAIN SELECT MAX(cn_id) FROM history \G
 *************************** 1. row ***************************
            id: 1
   select_type: SIMPLE
         table: NULL
          type: NULL
 possible_keys: NULL
           key: NULL
       key_len: NULL
           ref: NULL
          rows: NULL
         Extra: Select tables optimized away
 1 row in set (0.00 sec)
What if I’d want to get MAX(cn_id) for a particular u_id? Well, another full table scan!
mysql (test) > EXPLAIN SELECT MAX(cn_id) FROM history WHERE u_id = 106476 \G
 *************************** 1. row ***************************
            id: 1
   select_type: SIMPLE
         table: history
          type: ALL
 possible_keys: NULL
           key: NULL
       key_len: NULL
           ref: NULL
          rows: 19908716
         Extra: Using where
 1 row in set (0.00 sec)
Again the solution is simple, make sure that cn_id is part of an index including the u_id, in this case I created a composite key on (u_id, cn_id) so that once the optimizer found index records for u_id = 106476, within the same index record are the cn_id values.
mysql (test) > EXPLAIN SELECT MAX(cn_id) FROM history WHERE u_id = 106476 \G                    
 *************************** 1. row ***************************
            id: 1
   select_type: SIMPLE
         table: NULL
          type: NULL
 possible_keys: NULL
           key: NULL
       key_len: NULL
           ref: NULL
          rows: NULL
         Extra: Select tables optimized away
 1 row in set (0.00 sec)
Till next time!
UPDATE: Based on Daniel’s comment, to demonstrate the internal optimization for MIN and MAX with the proper keys in place, here is a sample query executed:
mysql (test) > EXPLAIN PARTITIONS SELECT MAX(h_date) FROM history WHERE t_id = 10947385 \G
*************************** 1. row ***************************
           id: 1
  select_type: SIMPLE
        table: NULL
   partitions: NULL
         type: NULL
possible_keys: NULL
          key: NULL
      key_len: NULL
          ref: NULL
         rows: NULL
        Extra: Select tables optimized away
1 row in set (0.00 sec)

mysql (test) > pager md5sum
PAGER set to 'md5sum'
mysql (test) > flush status;
Query OK, 0 rows affected (0.02 sec)

mysql (test) > SELECT MAX(h_date) FROM history WHERE t_id = 10947385 \G                   
54f1cd94f9bd1390899b13244443968f  -
1 row in set (0.00 sec)

mysql (test) > nopager
PAGER set to stdout
mysql (test) > SHOW STATUS LIKE 'handler%';     
+----------------------------+-------+
| Variable_name              | Value |
+----------------------------+-------+
| Handler_commit             | 1     |
| Handler_delete             | 0     |
| Handler_discover           | 0     |
| Handler_prepare            | 0     |
| Handler_read_first         | 0     |
| Handler_read_key           | 1     |
| Handler_read_last          | 0     |
| Handler_read_next          | 0     |
| Handler_read_prev          | 0     |
| Handler_read_rnd           | 0     |
| Handler_read_rnd_next      | 0     |
| Handler_rollback           | 0     |
| Handler_savepoint          | 0     |
| Handler_savepoint_rollback | 0     |
| Handler_update             | 0     |
| Handler_write              | 0     |
+----------------------------+-------+
16 rows in set (0.04 sec)