预排序遍历树算法(非递归无限极分类算法)

本文是我学习MySQL官方教程Managing Hierarchical Data in MySQL的笔记

多层数据结构估计所有的web开发者估计都不会陌生，各种软件的分类都是基于多层结构来设计的。

下面是一个典型的多层数据结构示意图：



相关创建数据语句：

CREATE TABLE category(
category_id INT AUTO_INCREMENT PRIMARY KEY,
name VARCHAR(20) NOT NULL,
parent INT DEFAULT NULL);


INSERT INTO category
VALUES(1,'ELECTRONICS',NULL),(2,'TELEVISIONS',1),(3,'TUBE',2),
(4,'LCD',2),(5,'PLASMA',2),(6,'PORTABLE ELECTRONICS',1),
(7,'MP3 PLAYERS',6),(8,'FLASH',7),
(9,'CD PLAYERS',6),(10,'2 WAY RADIOS',6);

SELECT * FROM category ORDER BY category_id;

在这种数据结构中，各层之间通过字段 parent 来形成邻接表，我们查询某些层级的关系的时候一般都是通过递归的方式，遍历某个层级关系的SQL的查询次数会顺着层级的增加，想想在层级有20的时候，根据某个底层节点取它到顶层节点的查询次数吧。

为了解决这个问题，人们想出了嵌套集模型（The Nested Set Model），请看下图：

上图依然是表现的与图一相同的层级关系，但是却更换了一种表现形式 下面是新的关系表和数据（关系和数据与之前相同，但是表结构不一样）：

CREATE TABLE nested_category (
category_id INT AUTO_INCREMENT PRIMARY KEY,
name VARCHAR(20) NOT NULL,
lft INT NOT NULL,
rgt INT NOT NULL
);


INSERT INTO nested_category
VALUES(1,'ELECTRONICS',1,20),(2,'TELEVISIONS',2,9),(3,'TUBE',3,4),
(4,'LCD',5,6),(5,'PLASMA',7,8),(6,'PORTABLE ELECTRONICS',10,19),
(7,'MP3 PLAYERS',11,14),(8,'FLASH',12,13),
(9,'CD PLAYERS',15,16),(10,'2 WAY RADIOS',17,18);


SELECT * FROM nested_category ORDER BY category_id;

这里将 left,right 修改为 lft,rgt因为这两个词在MYSQL中属于关键字 下面我们将插入的数据标识在图上： 

同样，我们将数据标识在原来的结构上：




怎么样，是不是很明确了

下面使我自己标定一种形式，方便理解

[1
      [2
           [3 4] 
           [5 6] 
           [7 8]
      9] 
      [10
           [11
                 [12 13]
           14]
           [15 16]
           [17 18]
      19]
20]

遍历整个树，查询子集 条件：左边 > 父级L， 右边 < 父级R

SELECT node.name
FROM nested_category AS node,
nested_category AS parent
WHERE node.lft BETWEEN parent.lft AND parent.rgt
AND parent.name = 'ELECTRONICS'
ORDER BY node.lft;

+----------------------+
| name                 |
+----------------------+
| ELECTRONICS          |
| TELEVISIONS          |
| TUBE                 |
| LCD                  |
| PLASMA               |
| PORTABLE ELECTRONICS |
| MP3 PLAYERS          |
| FLASH                |
| CD PLAYERS           |
| 2 WAY RADIOS         |
+----------------------+

- 查询所有无分支的节点 条件：右边 = 左边L + 1

SELECT name
FROM nested_category
WHERE rgt = lft + 1;

- 查询某个字节点到根节点的路径

SELECT parent.name
FROM nested_category AS node,
nested_category AS parent
WHERE node.lft BETWEEN parent.lft AND parent.rgt
AND node.name = 'FLASH'
ORDER BY parent.lft;


SELECT node.name, (COUNT(parent.name) - 1) AS depth
FROM nested_category AS node,
nested_category AS parent
WHERE node.lft BETWEEN parent.lft AND parent.rgt
GROUP BY node.name
ORDER BY node.lft;

- 查询子节点的深度

SELECT node.name, (COUNT(parent.name) - (sub_tree.depth + 1)) AS depth
FROM nested_category AS node,
    nested_category AS parent,
    nested_category AS sub_parent,
    (
        SELECT node.name, (COUNT(parent.name) - 1) AS depth
        FROM nested_category AS node,
        nested_category AS parent
        WHERE node.lft BETWEEN parent.lft AND parent.rgt
        AND node.name = 'PORTABLE ELECTRONICS'
        GROUP BY node.name
        ORDER BY node.lft
    )AS sub_tree
WHERE node.lft BETWEEN parent.lft AND parent.rgt
    AND node.lft BETWEEN sub_parent.lft AND sub_parent.rgt
    AND sub_parent.name = sub_tree.name
GROUP BY node.name
ORDER BY node.lft;

- 插入新节点
算法详解： 
1.所有分类 左边和右边的值 > 插入节点的左边节点记录的右值 的全部 + 2
2.插入节点 左值 = 插入位置左边节点记录的右值 + 1， 右值 = 插入位置左边节点记录的右值 + 2
例子:
在 R = 9（L8, R9）与 L = 10（L10，R11） 节点之间插入一个新节点
那么所有 左值 和 右值 > 9 的节点的左值和右值需要 + 2
例如新节点右边的节点（L10，R11）左值右值都需要 + 2 那么插入后的新值为 L12 R13
新节点的左值为 9 + 1 = 10 右值为 9 + 2 = 11
SQL语句实现

LOCK TABLE nested_category WRITE;
SELECT @myRight := rgt FROM nested_category
WHERE name = 'TELEVISIONS';
UPDATE nested_category SET rgt = rgt + 2 WHERE rgt > @myRight;
UPDATE nested_category SET lft = lft + 2 WHERE lft > @myRight;
INSERT INTO nested_category(name, lft, rgt) VALUES('GAME CONSOLES', @myRight + 1,@myRight + 2);
UNLOCK TABLES;

- 删除新节点
删除节点的算法与添加一个节点的算法相反

删除一个没有子节点的节点

LOCK TABLE nested_category WRITE;
SELECT @myLeft := lft, @myRight := rgt, @myWidth := rgt - lft + 1
FROM nested_category
WHERE name = 'GAME CONSOLES';
DELETE FROM nested_category WHERE lft BETWEEN @myLeft AND @myRight;
UPDATE nested_category SET rgt = rgt - @myWidth WHERE rgt > @myRight;
UPDATE nested_category SET lft = lft - @myWidth WHERE lft > @myRight;
UNLOCK TABLES;

删除一个分支节点和它所有的子节点

LOCK TABLE nested_category WRITE;
SELECT @myLeft := lft, @myRight := rgt, @myWidth := rgt - lft + 1
FROM nested_category
WHERE name = 'MP3 PLAYERS';
DELETE FROM nested_category WHERE lft BETWEEN @myLeft AND @myRight;
UPDATE nested_category SET rgt = rgt - @myWidth WHERE rgt > @myRight;
UPDATE nested_category SET lft = lft - @myWidth WHERE lft > @myRight;
UNLOCK TABLES;

删除一个节点后移动其字节点到

LOCK TABLE nested_category WRITE;
SELECT @myLeft := lft, @myRight := rgt, @myWidth := rgt - lft + 1
FROM nested_category
WHERE name = 'PORTABLE ELECTRONICS';
DELETE FROM nested_category WHERE lft = @myLeft;
UPDATE nested_category SET rgt = rgt - 1, lft = lft - 1 WHERE lft BETWEEN @myLeft AND@myRight;
UPDATE nested_category SET rgt = rgt - 2 WHERE rgt > @myRight;
UPDATE nested_category SET lft = lft - 2 WHERE lft > @myRight;
UNLOCK TABLES;

总结：

预排序遍历树算法的核心就是牺牲了写的性能来换取读取的性能

在你的开发的应用遇到此类问题的时（读压力 > 写压力），尝试下使用预排序遍历树算法来提高你的程序的性能吧。