MySQL：强制查询在WHERE子句中使用带有局部变量的索引

上下文

我有一个应用程序从表中选择加权随机条目,其中前缀总和(权重)是关键部分.简化的表定义如下所示：

CREATE TABLE entries (
    id INT NOT NULL PRIMARY KEY AUTO_INCREMENT,weight DECIMAL(9,3),fenwick DECIMAL(9,3)
) ENGINE=MEMORY;

其中`fenwick`将值存储在`weights`的Fenwick树表示中.

让每个条目的“范围”跨越其前缀和与其前缀相加的权重.应用程序必须在0和SUM(权重)之间生成一个随机数@r,并找到其范围包含@r的条目,如下所示：

Fenwick树结合MEMORY引擎和二进制搜索,应该允许我在O(lg ^ 2(n))时间内找到适当的条目,而不是使用朴素查询的O(n)时间：

SELECT a.id-1 FROM (SELECT *,(@x:=@x+weight) AS counter FROM entries 
    CROSS JOIN (SELECT @x:=0) a
    HAVING counter>@r LIMIT 1) a;

研究

由于多个查询的开销,我一直在尝试将前缀sum操作压缩成一个查询(而不是脚本语言中的几个数组访问).在这个过程中,我意识到传统的求和方法,即涉及按降序键顺序访问元素,只会求和第一个元素.我怀疑当WHERE子句中存在变量时,MysqL会线性地运行表.这是查询：

SELECT
SUM(1) INTO @garbage
FROM entries 
CROSS JOIN (
    SELECT @sum:=0,@n:=@entryid
) a
WHERE id=@n AND @n>0 AND (@n:=@n-(@n&(-@n))) AND (@sum:=@sum+entries.fenwick);
/*SELECT @sum*/

其中@entryid是我们正在计算的前缀和的条目的ID.我确实创建了一个有效的查询(以及返回整数最左边位的函数lft)：

SET @n:=lft(@entryid);
SET @sum:=0;
SELECT
    SUM(1) INTO @garbage
    FROM entries
    WHERE id=@n 
      AND @n<=@entryid 
      AND (@n:=@n+lft(@entryid^@n)) 
      AND (@sum:=@sum+entries.fenwick);
/*SELECT @sum*/

但它只证实了我对线性搜索的怀疑. EXPLAIN查询也是如此：

+------+-------------+---------+------+---------------+------+---------+------+--------+-------------+
| id   | select_type | table   | type | possible_keys | key  | key_len | ref  | rows   | Extra       |
+------+-------------+---------+------+---------------+------+---------+------+--------+-------------+
|    1 | SIMPLE      | entries | ALL  | NULL          | NULL | NULL    | NULL | 752544 | Using where |
+------+-------------+---------+------+---------------+------+---------+------+--------+-------------+
1 row in set (0.00 sec)

指数：

SHOW INDEXES FROM entries;
+---------+------------+----------+--------------+-------------+-----------+-------------+----------+--------+------+------------+---------+---------------+
| Table   | Non_unique | Key_name | Seq_in_index | Column_name | Collation | Cardinality | Sub_part | Packed | Null | Index_type | Comment | Index_comment |
+---------+------------+----------+--------------+-------------+-----------+-------------+----------+--------+------+------------+---------+---------------+
| entries |          0 | PRIMARY  |            1 | id          | NULL      |       752544 |     NULL | NULL   |      | HASH       |         |               |
+---------+------------+----------+--------------+-------------+-----------+-------------+----------+--------+------+------------+---------+---------------+
1 row in set (0.00 sec)

现在,我已经看到很多问题,询问如何消除WHERE子句中的变量,以便优化器可以处理查询.但是,如果没有id = @n,我无法想到这个查询的方式.我已经考虑将我想要求的条目的关键值放入一个表并使用连接,但我相信我会得到不良影响：要么过多的表,要么通过评估@entryid来进行线性搜索.

题

有没有办法强制MysqL使用此查询的索引？如果他们提供此功能,我甚至会尝试不同的DBMS.

最佳答案

放弃

芬威克树对我来说是新的,我发现这篇文章时才发现它们.
这里给出的结果是基于我的理解和一些研究,
但我绝不是一个芬威克树专家,我可能错过了一些东西.

参考资料

说明fenwick树是如何工作的

https://stackoverflow.com/a/15444954/1157540转载自
https://cs.stackexchange.com/a/10541/38148

https://cs.stackexchange.com/a/42816/38148

使用芬威克树

https://en.wikipedia.org/wiki/Fenwick_tree

https://en.wikipedia.org/wiki/Prefix_sum

问题1,找到给定条目的权重

鉴于下表

CREATE TABLE `entries` (
  `id` int(11) NOT NULL AUTO_INCREMENT,`weight` decimal(9,3) DEFAULT NULL,`fenwick` decimal(9,3) NOT NULL DEFAULT '0.000',PRIMARY KEY (`id`)
) ENGINE=INNODB;

并且已经填充了数据(参见concat提供的http://sqlfiddle.com/#!9/be1f2/1),
如何计算给定条目@entryid的权重？

这里要理解的关键概念是,fenwick索引的结构基于id值本身的数学和按位运算.

查询通常应仅使用主键查找(WHERE ID = value).

使用排序(ORDER BY)或范围(WHERE(value1< ID)AND(ID< value2))的任何查询都会错过该点,并且不会按预期的顺序遍历树. 例如,使用密钥60：

SET @entryid := 60;

让我们用二进制分解值60

MysqL> SELECT (@entryid & 0x0080) as b8,->        (@entryid & 0x0040) as b7,->        (@entryid & 0x0020) as b6,->        (@entryid & 0x0010) as b5,->        (@entryid & 0x0008) as b4,->        (@entryid & 0x0004) as b3,->        (@entryid & 0x0002) as b2,->        (@entryid & 0x0001) as b1;
+------+------+------+------+------+------+------+------+
| b8   | b7   | b6   | b5   | b4   | b3   | b2   | b1   |
+------+------+------+------+------+------+------+------+
|    0 |    0 |   32 |   16 |    8 |    4 |    0 |    0 |
+------+------+------+------+------+------+------+------+
1 row in set (0.00 sec)

换句话说,只保留位设置,我们有

32 + 16 + 8 + 4 = 60

现在,逐个删除设置的最低位以导航树：

32 + 16 + 8 + 4 = 60
32 + 16 + 8 = 56
32 + 16 = 48
32

这给出了访问元件60的路径(32,48,56,60).

注意,将60转换为(32,60)仅需要对ID值本身进行位计算：不需要访问表或数据库,并且可以在发出查询的客户端中完成此计算.

然后是元素60的分数权重

MysqL> select sum(fenwick) from entries where id in (32,60);
+--------------+
| sum(fenwick) |
+--------------+
|       32.434 |
+--------------+
1 row in set (0.00 sec)

验证

MysqL> select sum(weight) from entries where id <= @entryid;
+-------------+
| sum(weight) |
+-------------+
|      32.434 |
+-------------+
1 row in set (0.00 sec)

现在,让我们比较这些查询的效率.

MysqL> explain select sum(fenwick) from entries where id in (32,60);
+----+-------------+---------+------------+-------+---------------+---------+---------+------+------+----------+-------------+
| id | select_type | table   | partitions | type  | possible_keys | key     | key_len | ref  | rows | filtered | Extra       |
+----+-------------+---------+------------+-------+---------------+---------+---------+------+------+----------+-------------+
|  1 | SIMPLE      | entries | NULL       | range | PRIMARY       | PRIMARY | 4       | NULL |    4 |   100.00 | Using where |
+----+-------------+---------+------------+-------+---------------+---------+---------+------+------+----------+-------------+

或者,有所不同

explain format=json select sum(fenwick) from entries where id in (32,60);
{
  "query_block": {
    "select_id": 1,"cost_info": {
      "query_cost": "5.61"
    },"table": {
      "table_name": "entries","access_type": "range","possible_keys": [
        "PRIMARY"
      ],"key": "PRIMARY","used_key_parts": [
        "id"
      ],"key_length": "4","rows_examined_per_scan": 4,"rows_produced_per_join": 4,"filtered": "100.00","cost_info": {
        "read_cost": "4.81","eval_cost": "0.80","prefix_cost": "5.61","data_read_per_join": "64"
      },"used_columns": [
        "id","fenwick"
      ],"attached_condition": "(`test`.`entries`.`id` in (32,60))"
    }
  }

因此,优化器通过主键获取4行(IN子句中有4个值).

当不使用fenwick索引时,我们有

MysqL> explain select sum(weight) from entries where id <= @entryid;
+----+-------------+---------+------------+-------+---------------+---------+---------+------+------+----------+-------------+
| id | select_type | table   | partitions | type  | possible_keys | key     | key_len | ref  | rows | filtered | Extra       |
+----+-------------+---------+------------+-------+---------------+---------+---------+------+------+----------+-------------+
|  1 | SIMPLE      | entries | NULL       | range | PRIMARY       | PRIMARY | 4       | NULL |   60 |   100.00 | Using where |
+----+-------------+---------+------------+-------+---------------+---------+---------+------+------+----------+-------------+

或者,表达方式不同

explain format=json select sum(weight) from entries where id <= @entryid;
{
  "query_block": {
    "select_id": 1,"cost_info": {
      "query_cost": "25.07"
    },"rows_examined_per_scan": 60,"rows_produced_per_join": 60,"cost_info": {
        "read_cost": "13.07","eval_cost": "12.00","prefix_cost": "25.07","data_read_per_join": "960"
      },"weight"
      ],"attached_condition": "(`test`.`entries`.`id` <= (@`entryid`))"
    }
  }

优化器在此执行索引扫描,读取60行.

ID = 60时,fenwick的好处是4次,而60次.

现在,考虑一下如何扩展,例如值高达64K.

对于fenwick,16位值将设置最多16位,因此要查找的元素数量最多为16.

如果没有fenwick,扫描最多可以读取64K条目(平均读数为32K).

问题2,找到给定重量的条目

OP问题是找到给定重量的条目.

例如

SET @search_weight := 35.123;

为了说明算法,这篇文章详细说明了如何完成查找(对不起,如果这太冗长了)

SET @found_id := 0;

首先,找出有多少条目.

SET @max_id := (select id from entries order by id desc limit 1);

在测试数据中,max_id为156.

因为128< = max_id< 256,开始搜索的最高位是128.

MysqL> set @search_id := @found_id + 128;
MysqL> select id,fenwick,@search_weight,->    if (fenwick <= @search_weight,"keep","discard") as action
    ->    from entries where id = @search_id;
+-----+---------+----------------+---------+
| id  | fenwick | @search_weight | action  |
+-----+---------+----------------+---------+
| 128 |  66.540 |         35.123 | discard |
+-----+---------+----------------+---------+

重量66.540大于我们的搜索,因此丢弃128,继续下一位.

MysqL> set @search_id := @found_id + 64;
MysqL> select id,"discard") as action
    ->    from entries where id = @search_id;
+----+---------+----------------+--------+
| id | fenwick | @search_weight | action |
+----+---------+----------------+--------+
| 64 |  33.950 |         35.123 | keep   |
+----+---------+----------------+--------+

在这里我们需要保持这个位(64),并计算找到的重量：

set @found_id := @search_id,@search_weight := @search_weight - 33.950;

然后继续下一个位：

MysqL> set @search_id := @found_id + 32;
MysqL> select id,"discard") as action
    ->    from entries where id = @search_id;
+----+---------+----------------+---------+
| id | fenwick | @search_weight | action  |
+----+---------+----------------+---------+
| 96 |  16.260 |          1.173 | discard |
+----+---------+----------------+---------+

MysqL> set @search_id := @found_id + 16;
MysqL> select id,"discard") as action
    ->    from entries where id = @search_id;
+----+---------+----------------+---------+
| id | fenwick | @search_weight | action  |
+----+---------+----------------+---------+
| 80 |   7.394 |          1.173 | discard |
+----+---------+----------------+---------+

MysqL> set @search_id := @found_id + 8;
MysqL> select id,"discard") as action
    ->    from entries where id = @search_id;
+----+---------+----------------+---------+
| id | fenwick | @search_weight | action  |
+----+---------+----------------+---------+
| 72 |   3.995 |          1.173 | discard |
+----+---------+----------------+---------+

MysqL> set @search_id := @found_id + 4;
MysqL> select id,"discard") as action
    ->    from entries where id = @search_id;
+----+---------+----------------+---------+
| id | fenwick | @search_weight | action  |
+----+---------+----------------+---------+
| 68 |   1.915 |          1.173 | discard |
+----+---------+----------------+---------+

MysqL> set @search_id := @found_id + 2;
MysqL> select id,"discard") as action
    ->    from entries where id = @search_id;
+----+---------+----------------+--------+
| id | fenwick | @search_weight | action |
+----+---------+----------------+--------+
| 66 |   1.146 |          1.173 | keep   |
+----+---------+----------------+--------+

我们在这里找到了另一个

set @found_id := @search_id,@search_weight := @search_weight - 1.146;

MysqL> set @search_id := @found_id + 1;
MysqL> select id,"discard") as action
    ->    from entries where id = @search_id;
+----+---------+----------------+--------+
| id | fenwick | @search_weight | action |
+----+---------+----------------+--------+
| 67 |   0.010 |          0.027 | keep   |
+----+---------+----------------+--------+

还有一个

set @found_id := @search_id,@search_weight := @search_weight - 0.010;

最终的搜索结果是：

MysqL> select @found_id,@search_weight;
+-----------+----------------+
| @found_id | @search_weight |
+-----------+----------------+
|        67 |          0.017 |
+-----------+----------------+

验证

MysqL> select sum(weight) from entries where id <= 67;        
+-------------+                                               
| sum(weight) |                                               
+-------------+                                               
|      35.106 |                                               
+-------------+                                               

MysqL> select sum(weight) from entries where id <= 68;
+-------------+
| sum(weight) |
+-------------+
|      35.865 |
+-------------+

事实上,

35.106 (fenwick[67]) <= 35.123 (search) <= 35.865 (fenwick[68])

搜索查找值一次解析1位,每个查找结果决定要搜索的下一个ID的值.

此处给出的查询仅供参考.在实际应用程序中,代码应该只是一个包含以下内容的循环：

SELECT fenwick from entries where id = ?;

使用应用程序代码(或存储过程)实现与@ found_id,@ search_id和@search_weight相关的逻辑.

普通的留言

> Fenwick树用于前缀计算.如果要解决的问题首先涉及前缀,那么使用这些树才有意义.维基百科有一些应用程序的指针.不幸的是,OP没有描述fenwick树的用途.
> Fenwick树是查找复杂性和更新复杂性之间的权衡,因此它们只对非静态数据有意义.对于静态数据,计算一次完整前缀将更有效.
>执行的测试使用了一个INNODB表,主键索引被排序,因此计算max_id是一个简单的O(1)操作.如果max_id已在其他地方可用,我认为即使使用带有HASH索引ID的MEMORY表也可以,因为只使用主键查找.

附：

sqlfiddle今天已经关闭了,所以发布使用的原始数据(最初由concat提供),以便有兴趣的人可以重新运行测试.

INSERT INTO `entries` VALUES (1,0.480,0.480),(2,0.542,1.022),(3,0.269,0.269),(4,0.721,2.012),(5,0.798,0.798),(6,0.825,1.623),(7,0.731,0.731),(8,0.181,4.547),(9,0.711,0.711),(10,0.013,0.724),(11,0.930,0.930),(12,0.613,2.267),(13,0.276,0.276),(14,0.539,0.815),(15,0.867,0.867),(16,0.718,9.214),(17,0.991,0.991),(18,0.801,1.792),(19,0.033,0.033),(20,0.759,2.584),(21,0.698,0.698),(22,0.212,0.910),(23,0.965,0.965),(24,0.189,4.648),(25,0.049,0.049),(26,0.678,0.727),(27,0.245,0.245),(28,0.190,1.162),(29,0.214,0.214),(30,0.502,0.716),(31,0.868,0.868),(32,0.834,17.442),(33,0.566,0.566),(34,0.327,0.893),(35,0.939,0.939),(36,0.713,2.545),(37,0.747,0.747),(38,0.595,1.342),(39,0.733,0.733),(40,0.884,5.504),(41,0.218,0.218),(42,0.437,0.655),(43,0.532,0.532),(44,0.350,1.537),(45,0.154,0.154),(46,0.875),(47,0.140,0.140),(48,0.538,8.594),(49,0.271,0.271),(50,0.739,1.010),(51,0.884),(52,0.203,2.097),(53,0.361,0.361),(54,0.197,0.558),(55,0.903,0.903),(56,0.923,4.481),(57,0.906,0.906),(58,0.761,1.667),(59,0.089,0.089),(60,0.161,1.917),(61,0.537,0.537),(62,0.201,0.738),(63,0.397,0.397),(64,0.381,33.950),(65,0.715,0.715),(66,0.431,1.146),(67,0.010,0.010),(68,1.915),(69,0.763,0.763),(70,1.300),(71,0.399,0.399),(72,3.995),(73,0.709,0.709),(74,0.401,1.110),(75,0.880,0.880),(76,0.198,2.188),(77,0.348,0.348),(78,0.148,0.496),(79,0.693,0.693),(80,0.022,7.394),(81,0.031,0.031),(82,0.120),(83,0.353,0.353),(84,0.498,0.971),(85,0.428,0.428),(86,0.650,1.078),(87,0.963,0.963),(88,0.866,3.878),(89,0.442,0.442),(90,0.610,1.052),(91,0.725,0.725),(92,0.797,2.574),(93,0.808,0.808),(94,0.648,1.456),(95,0.817,0.817),(96,0.141,16.260),(97,0.256,0.256),(98,0.855,1.111),(99,0.508,0.508),(100,0.976,2.595),(101,(102,0.840,1.193),(103,0.139,0.139),(104,0.178,4.105),(105,0.469,0.469),(106,0.814,1.283),(107,0.664,0.664),(108,0.876,2.823),(109,0.390,0.390),(110,0.323,0.713),(111,(112,0.241,8.324),(113,0.881,0.881),(114,0.681,1.562),(115,0.760,0.760),(116,3.082),(117,0.518,0.518),(118,0.313,0.831),(119,0.008,0.008),(120,0.103,4.024),(121,0.488,0.488),(122,0.135,0.623),(123,0.207,0.207),(124,0.633,1.463),(125,0.542),(126,0.812,1.354),(127,0.433,0.433),(128,0.732,66.540),(129,0.358,0.358),(130,0.594,0.952),(131,0.897,0.897),(132,0.701,2.550),(133,0.815,(134,0.973,1.788),(135,0.419,0.419),(136,0.175,4.932),(137,0.620,0.620),(138,0.573,(139,0.004,0.004),(140,0.304,1.501),(141,(142,0.629,1.137),(143,0.618,0.618),(144,0.206,8.394),(145,0.175),(146,0.255,0.430),(147,0.750,0.750),(148,0.987,2.167),(149,0.683,0.683),(150,0.453,1.136),(151,0.219,0.219),(152,0.734,4.256),(153,0.016,0.016),(154,0.874,0.891),(155,0.325,0.325),(156,0.002,1.217);

附： 2

现在有一个完整的sqlfiddle：

http://sqlfiddle.com/#!9/d2c82/1

MySQL：强制查询在WHERE子句中使用带有局部变量的索引

猜你在找的MySQL相关文章