Optimising MySQL queries across hierarchical data

Question

I have a fairly stable directed graph of order ~100k vertices and size ~1k edges. It is two-dimensional insofar as its vertices can be identified by a pair of integers (x, y) (of cardinality ~100 x ~1000) and all edges are strictly increasing in x.

There is furthermore a dictionary of ~1k (key, val) pairs associated with each vertex.

I am currently storing the graph in a MySQL database across three (InnoDB) tables: a table of vertices (which I don't think is relevant to my question, so I have omitted to include both it and the foreign key constraints that refer to it in my extracts below); a table which holds the dictionaries; and a 'closure table' of connected vertices as described so eloquently by Bill Karwin.

The table of vertex dictionaries is defined as follows:

CREATE TABLE `VertexDictionary` (
  `x`   smallint(6) unsigned NOT NULL,
  `y`   smallint(6) unsigned NOT NULL,
  `key` varchar(50) NOT NULL DEFAULT '',
  `val` smallint(1) DEFAULT NULL,
  PRIMARY KEY (`x`, `y`  , `key`),
  KEY  `dict` (`x`, `key`, `val`)
);

and the closure table of connected vertices as:

CREATE TABLE `ConnectedVertices` (
  `tail_x` smallint(6) unsigned NOT NULL,
  `tail_y` smallint(6) unsigned NOT NULL,
  `head_x` smallint(6) unsigned NOT NULL,
  `head_y` smallint(6) unsigned NOT NULL,
  PRIMARY KEY   (`tail_x`, `tail_y`, `head_x`),
  KEY `reverse` (`head_x`, `head_y`, `tail_x`),
  KEY `fx` (`tail_x`, `head_x`),
  KEY `rx` (`head_x`, `tail_x`)
);

There is also a dictionary of (x, key) pairs such that for each such pair, all vertices identified with that x have within their dictionaries a value for that key. This dictionary is stored in a fourth table:

CREATE TABLE `SpecialKeys` (
  `x`   smallint(6) unsigned NOT NULL,
  `key` varchar(50) NOT NULL DEFAULT '',
  PRIMARY KEY (`x`),
  KEY `xkey`  (`x`, `key`)
);

I often wish to extract the set of keys used in the dictionaries of all vertices having a particular x=X, together with the associated value of any SpecialKeys connected to the left:

SELECT DISTINCT
  `v`.`key`,
  `u`.`val`
FROM
       `ConnectedVertices` AS `c`
  JOIN `VertexDictionary`  AS `u` ON (`u`.`x`, `u`.`y`  ) = (`c`.`tail_x`, `c`.`tail_y`)
  JOIN `VertexDictionary`  AS `v` ON (`v`.`x`, `v`.`y`  ) = (`c`.`head_x`, `c`.`head_y`)
  JOIN `SpecialKeys`       AS `k` ON (`k`.`x`, `k`.`key`) = (`u`.`x`, `u`.`key`)
WHERE
  `v`.`x` = X
;

for which the EXPLAIN output is:

id   select_type   table   type     possible_keys           key       key_len   ref                                rows   Extra
 1   SIMPLE        k       index    PRIMARY,xkey            xkey          154   NULL                                 40   Using index; Using temporary
 1   SIMPLE        c       ref      PRIMARY,reverse,fx,rx   PRIMARY         2   db.k.x                                1   Using where
 1   SIMPLE        v       ref      PRIMARY,dict            PRIMARY         4   const,db.c.head_y                   136   Using index
 1   SIMPLE        u       eq_ref   PRIMARY,dict            PRIMARY       156   db.c.tail_x,db.c.tail_y,db.k.key      1   Using where

But this query takes ~10s to complete. Been banging my head against a brick wall trying to improve matters, but to no avail.

Can the query be improved, or should I consider a different data structure? Extremely grateful for your thoughts!

---

UPDATE

I'm still getting nowhere with this, although I did rebuild the tables and found the EXPLAIN output to be slightly different (as now shown above, the number of rows fetched from v had increased from 1 to 136!); the query is still taking ~10s to execute.

I really don't understand what's going on here. Queries to obtain all (x, y, SpecialValue) and all (x, y, key) tuples are both very fast (~30ms and ~150ms respectively), yet essentially joining the two takes over fifty times longer than their combined time... how can I improve the time taken to perform that join?

Output of SHOW VARIABLES LIKE '%innodb%'; below:

Variable_name                    Value
------------------------------------------------------------
have_innodb                      YES
ignore_builtin_innodb            ON
innodb_adaptive_flushing         ON
innodb_adaptive_hash_index       ON
innodb_additional_mem_pool_size  2097152
innodb_autoextend_increment      8
innodb_autoinc_lock_mode         1
innodb_buffer_pool_size          1179648000
innodb_change_buffering          inserts
innodb_checksums                 ON
innodb_commit_concurrency        0
innodb_concurrency_tickets       500
innodb_data_file_path            ibdata1:10M:autoextend
innodb_data_home_dir             /rdsdbdata/db/innodb
innodb_doublewrite               ON
innodb_fast_shutdown             1
innodb_file_format               Antelope
innodb_file_format_check         Barracuda
innodb_file_per_table            ON
innodb_flush_log_at_trx_commit   1
innodb_flush_method              O_DIRECT
innodb_force_recovery            0
innodb_io_capacity               200
innodb_lock_wait_timeout         50
innodb_locks_unsafe_for_binlog   OFF
innodb_log_buffer_size           8388608
innodb_log_file_size             134217728
innodb_log_files_in_group        2
innodb_log_group_home_dir        /rdsdbdata/log/innodb
innodb_max_dirty_pages_pct       75
innodb_max_purge_lag             0
innodb_mirrored_log_groups       1
innodb_old_blocks_pct            37
innodb_old_blocks_time           0
innodb_open_files                300
innodb_read_ahead_threshold      56
innodb_read_io_threads           4
innodb_replication_delay         0
innodb_rollback_on_timeout       OFF
innodb_spin_wait_delay           6
innodb_stats_method              nulls_equal
innodb_stats_on_metadata         ON
innodb_stats_sample_pages        8
innodb_strict_mode               OFF
innodb_support_xa                ON
innodb_sync_spin_loops           30
innodb_table_locks               ON
innodb_thread_concurrency        0
innodb_thread_sleep_delay        10000
innodb_use_sys_malloc            ON
innodb_version                   1.0.16
innodb_write_io_threads          4

Answer 1

Without spending time testing it, you provided an incomplete example? you should definitely try reordering of joined tables. Explain output provides some info, let's say ordering by key_len should be heuristically fastest. First table to be filtered on should be listed as last in case the optimizer is not able to figure that out, I believe.

So, let's say 'c, v, k, u' order is the best.

SELECT DISTINCT
  `v`.`key`,
  `u`.`val`
FROM
  `VertexDictionary`  AS `u`
  JOIN `SpecialKeys`       AS `k` ON (`k`.`x`, `k`.`key`) = (`u`.`x`, `u`.`key`)
  JOIN `VertexDictionary`  AS `v`
  JOIN `ConnectedVertices` AS `c` ON (`u`.`x`, `u`.`y`  ) = (`c`.`tail_x`, `c`.`tail_y`)
           AND (`v`.`x`, `v`.`y`  ) = (`c`.`head_x`, `c`.`head_y`)
WHERE
  `v`.`x` = X
;

'rows' would suggest 'c/u, k, v' order, but that depends on data:

SELECT DISTINCT
  `v`.`key`,
  `u`.`val`
FROM
  `VertexDictionary`  AS `u`
  JOIN `VertexDictionary`  AS `v`
  JOIN `SpecialKeys`       AS `k` ON (`k`.`x`, `k`.`key`) = (`u`.`x`, `u`.`key`)
  JOIN `ConnectedVertices` AS `c` ON (`u`.`x`, `u`.`y`  ) = (`c`.`tail_x`, `c`.`tail_y`)
                                 AND (`v`.`x`, `v`.`y`  ) = (`c`.`head_x`, `c`.`head_y`)
 WHERE
  `v`.`x` = X
;

Hope this helps.

UPDATE (avoiding the varchar join):

SELECT DISTINCT
  `v`.`key`,
  `u`.`val`
FROM
       `ConnectedVertices` AS `c`
  JOIN `VertexDictionary`  AS `u` ON (`u`.`x`, `u`.`y`  ) = (`c`.`tail_x`, `c`.`tail_y`)
  JOIN `VertexDictionary`  AS `v` ON (`v`.`x`, `v`.`y`  ) = (`c`.`head_x`, `c`.`head_y`)
WHERE
  (`u`.`x`, `u`.`key`) IN (SELECT `k`.`x`, `k`.`key` FROM `SpecialKeys` AS `k`)
AND
  `v`.`x` = X
;