1.内部结构

zset是由ziplist和skiplist组成的。

当数据比较少时，有序集合使用的是 ziplist 存储的，有序集合使用 ziplist 格式存储必须满足以下两个条件：

有序集合保存的元素个数要小于 128 个；
有序集合保存的所有元素成员的长度都必须小于 64 字节。
1.1 压缩列表(ziplist)
之前讲过，自取。
1.2 跳跃表（skiplist）

1.2.1 什么是跳跃表？
1.2.1.1 zset
```
typedef struct zset {
  dict *dict;
  zskiplist *zsl;
} zset;
```

1.2.1.2 zskiplist

typedef struct zskiplist {
    struct zskiplistNode *header, *tail;
    unsigned long length;
    int level;
} zskiplist;

header

指向跳跃表的表头节点，通过这个指针程序定位表头节点的时间复杂度就为O(1)
tail

指向跳跃表的表尾节点,通过这个指针程序定位表尾节点的时间复杂度就为O(1)

level

记录目前跳跃表内,层数最大的那个节点的层数(表头节点的层数不计算在内)，通过这个属性可以再O(1)的时间复杂度内获取层高最好的节点的层数。

length

记录跳跃表的长度,也即是,跳跃表目前包含节点的数量(表头节点不计算在内)，通过这个属性，程序可以再O(1)的时间复杂度内返回跳跃表的长度。

1.2.1.3 zskiplistNode

1.5 zset - 图3

typedef struct zskiplistNode {
    sds ele;
    double score;
    struct zskiplistNode *backward; //后退指针
    struct zskiplistLevel {
        struct zskiplistNode *forward; //前进指针
        unsigned long span;//跨度
    } level[]; //层数
} zskiplistNode;

level 层

节点中用L1、L2、L3等字样标记节点的各个层,L1代表第一层,L代表第二层,以此类推。

2.查找与插入过程

2.1 查找过程

我们先来看一个有序链表，如下图（最左侧的灰色节点表示一个空的头结点）：

在这样一个链表中，如果我们要查找某个数据，那么需要从头开始逐个进行比较，直到找到包含数据的那个节点，或者找到第一个比给定数据大的节点为止（没找到）。也就是说，时间复杂度为O(n)。同样，当我们要插入新数据的时候，也要经历同样的查找过程，从而确定插入位置。
假如我们每相邻两个节点增加一个指针，让指针指向下下个节点，如下图：
1.5 zset - 图6
这样所有新增加的指针连成了一个新的链表，但它包含的节点个数只有原来的一半（上图中是7, 19, 26）。现在当我们想查找数据的时候，可以先沿着这个新链表进行查找。当碰到比待查数据大的节点时，再回到原来的链表中进行查找。比如，我们想查找23，查找的路径是沿着下图中标红的指针所指向的方向进行的：
1.5 zset - 图7

23首先和7比较，再和19比较，比它们都大，继续向后比较。
但23和26比较的时候，比26要小，因此回到下面的链表（原链表），与22比较。
23比22要大，沿下面的指针继续向后和26比较。23比26小，说明待查数据23在原链表中不存在，而且它的插入位置应该在22和26之间。

在这个查找过程中，由于新增加的指针，我们不再需要与链表中每个节点逐个进行比较了。需要比较的节点数大概只有原来的一半。
利用同样的方式，我们可以在上层新产生的链表上，继续为每相邻的两个节点增加一个指针，从而产生第三层链表。如下图：

在这个新的三层链表结构上，如果我们还是查找23，那么沿着最上层链表首先要比较的是19，发现23比19大，接下来我们就知道只需要到19的后面去继续查找，从而一下子跳过了19前面的所有节点。可以想象，当链表足够长的时候，这种多层链表的查找方式能让我们跳过很多下层节点，大大加快查找的速度。

1.5 zset - 图9
会从最底部的节点开始搜索；

2.2 插入过程

1.5 zset - 图10
从上面skiplist的创建和插入过程可以看出，每一个节点的层数（level）是随机出来的，而且新插入一个节点不会影响其它节点的层数。因此，插入操作只需要修改插入节点前后的指针，而不需要对很多节点都进行调整。这就降低了插入操作的复杂度。实际上，这是skiplist的一个很重要的特性，这让它在插入性能上明显优于平衡树的方案。

疑问：

节点插入时随机出一个层数，仅仅依靠这样一个简单的随机数操作而构建出来的多层链表结构，能保证它有一个良好的查找性能吗？
为了回答这个疑问，我们需要分析skiplist的统计性能。

在分析之前，我们还需要着重指出的是，执行插入操作时计算随机数的过程，是一个很关键的过程，它对skiplist的统计特性有着很重要的影响。这并不是一个普通的服从均匀分布的随机数，它的计算过程如下：

首先，每个节点肯定都有第1层指针（每个节点都在第1层链表里）。
如果一个节点有第i层(i>=1)指针（即节点已经在第1层到第i层链表中），那么它有第(i+1)层指针的概率为p。
节点最大的层数不允许超过一个最大值，记为MaxLevel。

这个计算随机层数的伪码如下所示：

    double p = 1.0 / 4;
    int MAX_LEVEL = 32;
    double random() {
        return (double) (rand() / (double) RAND_MAX);
    }
    int randomLevel() {
        int level = 1;
        while (random() < p && level < MAX_LEVEL) {
            level += 1;
        }
        return level;
    }

randomLevel()的伪码中包含两个参数，一个是p，一个是MaxLevel。在Redis的skiplist实现中，这两个参数的取值为：

p = 1/4
MaxLevel = 32

2.3 算法性能分析

2.3.1 空间复杂度

根据前面randomLevel()的伪码，我们很容易看出，产生越高的节点层数，概率越低。定量的分析如下：

节点层数至少为1。而大于1的节点层数，满足一个概率分布。
节点层数恰好等于1的概率为1-p。
节点层数大于等于2的概率为p，而节点层数恰好等于2的概率为p(1-p)。
节点层数大于等于3的概率为p2，而节点层数恰好等于3的概率为p2(1-p)。
节点层数大于等于4的概率为p3，而节点层数恰好等于4的概率为p3(1-p)。

因此，一个节点的平均层数（也即包含的平均指针数目,数学期望），计算如下：
1.5 zset - 图11
1.5 zset - 图12 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因为 1.5 zset - 图13 , 会级数收敛， 1.5 zset - 图14