MySQL:强制查询在WHERE子句中使用带有局部变量的索引

语境

我有一个应用程序，该应用程序从一个表中选择一个加权随机条目，对于这些条目，前缀总和(权重)是至关重要的部分。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树表示形式内。

让每个条目的“范围”跨越其前缀和及其前缀和权重之间。应用程序必须在@r和0之间生成一个随机数SUM(weight)，并找到范围包括@r的条目，如下所示:

Fenwick树，结合MEMORY引擎和二进制搜索，应该可以让我在O(lg^2(n))时间中找到合适的条目，而不是朴素的查询中的O(n) time:

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，我一直试图将前缀和运算压缩为一个查询(与脚本语言中看到的多个数组访问相反)。在此过程中，我已经意识到，传统的求和方法(涉及以降序键访问元素)只会求和第一个元素。我怀疑WHERE子句中存在变量时，MySQL会线性地线性运行表。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是我们正在计算其前缀总和的条目的ID。我确实创建了一个行之有效的查询(连同返回整数最左位的lft函数):

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*/

但这只是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)

指标:

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，我看到了很多问题，询问如何消除WHERE子句中的变量，以便优化程序可以在查询中使用。但是，我想不出没有id=@n的查询方法。我已经考虑过将要汇总的条目的键值放入表中并使用联接，但是我相信我会得到不良影响:要么表过多，要么通过根据@entryid进行评估来进行线性搜索。

问题

有什么方法可以强制MySQL在此查询中使用索引？如果他们提供此功能，我什至会尝试使用其他DBMS。

最佳答案

免责声明

芬威克树对我来说是新的，我只是在找到这篇文章的时候才发现它们的。
这里给出的结果是基于我的理解和一些研究，
但是我绝不是芬威克树专家，我可能错过了一些事情。

引用资料

fenwick树如何工作的解释

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

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

芬威克树的用法

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

给出下表

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;

并给出已经填充的数据(请参见concat提供的http://sqlfiddle.com/#!9/be1f2/1)，
如何计算给定条目@entryid的重量？

这里要理解的关键概念是，fenwick索引的结构基于对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)。

任何使用排序(ORDER BY)或范围(WHERE (value1 < ID) AND (ID < value2)))的查询都会遗漏要点，并且不会按预期顺序遍历树。

例如，使用键60:

SET @entryid := 60;

让我们将值60分解为二进制

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，删除一一设置的最低位以导航树:

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

这给出了进入元件60的路径(32、48、56、60)。

请注意，将60转换为(32, 48, 56, 60)仅需要对ID值本身进行位数学运算:无需访问表或数据库，并且可以在发出查询的客户端中进行此计算。

则元素60的fenwick weigth为

mysql> select sum(fenwick) from entries where id in (32, 48, 56, 60);
+--------------+
| sum(fenwick) |
+--------------+
|       32.434 |
+--------------+
1 row in set (0.00 sec)

确认

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

现在，让我们比较一下这些查询的效率。

mysql> explain select sum(fenwick) from entries where id in (32, 48, 56, 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，以不同的方式呈现

explain format=json select sum(fenwick) from entries where id in (32, 48, 56, 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 (32,48,56,60))"
    }
  }

因此，优化器通过主键获取了4行(IN子句中有4个值)。

当不使用fenwick指数时，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，以不同的方式呈现

explain format=json select sum(weight) from entries where id <= @entryid;
{
  "query_block": {
    "select_id": 1,
    "cost_info": {
      "query_cost": "25.07"
    },
    "table": {
      "table_name": "entries",
      "access_type": "range",
      "possible_keys": [
        "PRIMARY"
      ],
      "key": "PRIMARY",
      "used_key_parts": [
        "id"
      ],
      "key_length": "4",
      "rows_examined_per_scan": 60,
      "rows_produced_per_join": 60,
      "filtered": "100.00",
      "cost_info": {
        "read_cost": "13.07",
        "eval_cost": "12.00",
        "prefix_cost": "25.07",
        "data_read_per_join": "960"
      },
      "used_columns": [
        "id",
        "weight"
      ],
      "attached_condition": "(`test`.`entries`.`id` <= (@`entryid`))"
    }
  }

优化程序在这里执行索引扫描，读取60行。

在ID = 60的情况下，fenwick的 yield 为60，而4则为fetch。

现在，考虑如何扩展，例如，最大值为64K。

使用fenwick，一个16位的值将最多设置16个位，因此要查找的元素数最多为16个。

如果没有fenwick，则一次扫描最多可以读取64K条目(平均读取32K)。

问题2，找到给定权重的条目

OP的问题是找到给定重量的条目。

例如

SET @search_weight := 35.123;

为了说明该算法，本文详细介绍了查找的完成方式(抱歉，如果太冗长的话)

SET @found_id := 0;

首先，找到多少个条目。

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，所以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 |
+-----+---------+----------------+---------+

权重66.540大于我们的搜索，因此128被丢弃，移至下一位。

mysql> set @search_id := @found_id + 64;
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 |
+----+---------+----------------+--------+
| 64 |  33.950 |         35.123 | keep   |
+----+---------+----------------+--------+

在这里，我们需要保留此位(64)，并计算找到的权重:

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

然后继续进行以下操作:

mysql> set @search_id := @found_id + 32;
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  |
+----+---------+----------------+---------+
| 96 |  16.260 |          1.173 | discard |
+----+---------+----------------+---------+

mysql> set @search_id := @found_id + 16;
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  |
+----+---------+----------------+---------+
| 80 |   7.394 |          1.173 | discard |
+----+---------+----------------+---------+

mysql> set @search_id := @found_id + 8;
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  |
+----+---------+----------------+---------+
| 72 |   3.995 |          1.173 | discard |
+----+---------+----------------+---------+

mysql> set @search_id := @found_id + 4;
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  |
+----+---------+----------------+---------+
| 68 |   1.915 |          1.173 | discard |
+----+---------+----------------+---------+

mysql> set @search_id := @found_id + 2;
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 |
+----+---------+----------------+--------+
| 66 |   1.146 |          1.173 | keep   |
+----+---------+----------------+--------+

我们在这里找到了另外一点

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

mysql> set @search_id := @found_id + 1;
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 |
+----+---------+----------------+--------+
| 67 |   0.010 |          0.027 | keep   |
+----+---------+----------------+--------+

还有一个

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

最终搜索结果为:

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

确认

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 |
+-------------+

的确，

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

搜索一次查找值以解析1位，并且每个查找结果决定要搜索的下一个ID的值。

此处给出的查询仅用于说明。在实际的应用程序中，代码应该只是一个包含以下内容的循环:

SELECT fenwick from entries where id = ?;

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。仅当要解决的问题首先涉及前缀时，才使用这些树。维基百科为应用程序提供了一些指导。不幸的是，OP没有描述芬威克树的用途。

Fenwick树是查找复杂度和更新复杂度之间的折衷，因此它们仅对于非静态数据有意义。对于静态数据，一次计算完整前缀会更有效。

所执行的测试使用了INNODB表，对其进行了主键索引排序，因此计算max_id是简单的O(1)操作。如果max_id在其他地方已经可用，我认为即使使用ID为HASH索引的MEMORY表也可以使用，因为仅使用了主键查找。

P.S.

sqlfiddle今天停产了，因此发布使用的原始数据(最初由concat提供)，以便被困的人们可以重新运行测试。

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),(46,0.721,0.875),(47,0.140,0.140),(48,0.538,8.594),(49,0.271,0.271),(50,0.739,1.010),(51,0.884,0.884),(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),(68,0.759,1.915),(69,0.763,0.763),(70,0.537,1.300),(71,0.399,0.399),(72,0.381,3.995),(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),(82,0.089,0.120),(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),(101,0.353,0.353),(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),(111,0.442,0.442),(112,0.241,8.324),(113,0.881,0.881),(114,0.681,1.562),(115,0.760,0.760),(116,0.760,3.082),(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),(125,0.542,0.542),(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),(133,0.815,0.815),(134,0.973,1.788),(135,0.419,0.419),(136,0.175,4.932),(137,0.620,0.620),(138,0.573,1.193),(139,0.004,0.004),(140,0.304,1.501),(141,0.508,0.508),(142,0.629,1.137),(143,0.618,0.618),(144,0.206,8.394),(145,0.175,0.175),(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);

P.S. 2个

现在使用完整的sqlfiddle:

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

关于MySQL:强制查询在WHERE子句中使用带有局部变量的索引，我们在Stack Overflow上找到一个类似的问题： https://stackoverflow.com/questions/29475470/

MySQL:强制查询在WHERE子句中使用带有局部变量的索引

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