跳跃表的实现
------------------

Redis 的跳跃表由 ``redis.h/zskiplistNode`` 和 ``redis.h/zskiplist`` 两个结构定义，
其中 ``zskiplistNode`` 结构用于表示跳跃表节点，
而 ``zskiplist`` 结构则用于保存跳跃表节点的相关信息，
比如节点的数量，
以及指向表头节点和表尾节点的指针，
等等。

.. graphviz::

    digraph {

        rankdir = LR;

        node [shape = record, width = "0.5"];

        //

        l [label = " <header> header | <tail> tail | level \n 5 | length \n 3 "];

        subgraph cluster_nodes {

            style = invisible;

            header [label = " <l32> L32 | ... | <l5> L5 | <l4> L4 | <l3> L3 | <l2> L2 | <l1> L1 "];

            bw_null [label = "NULL", shape = plaintext];

            level_null [label = "NULL", shape = plaintext];

            A [label = " <l4> L4 | <l3> L3 | <l2> L2 | <l1> L1 | <backward> BW | 1.0 | o1 "];

            B [label = " <l2> L2 | <l1> L1 | <backward> BW | 2.0 | o2 "];

            C [label = " <l5> L5 | <l4> L4 | <l3> L3 | <l2> L2 | <l1> L1 | <backward> BW | 3.0 | o3 "];

        }

        subgraph cluster_nulls {

            style = invisible;

            n1 [label = "NULL", shape = plaintext];
            n2 [label = "NULL", shape = plaintext];
            n3 [label = "NULL", shape = plaintext];
            n4 [label = "NULL", shape = plaintext];
            n5 [label = "NULL", shape = plaintext];

        }

        //

        l:header -> header;
        l:tail -> C;

        header:l32 -> level_null [label = "0"];
        header:l5 -> C:l5 [label = "3"];
        header:l4 -> A:l4 [label = "1"];
        header:l3 -> A:l3 [label = "1"];
        header:l2 -> A:l2 [label = "1"];
        header:l1 -> A:l1 [label = "1"];

        A:l4 -> C:l4 [label = "2"];
        A:l3 -> C:l3 [label = "2"];
        A:l2 -> B:l2 [label = "1"];
        A:l1 -> B:l1 [label = "1"];

        B:l2 -> C:l2 [label = "1"];
        B:l1 -> C:l1 [label = "1"];

        C:l5 -> n5 [label = "0"];
        C:l4 -> n4 [label = "0"];
        C:l3 -> n3 [label = "0"];
        C:l2 -> n2 [label = "0"];
        C:l1 -> n1 [label = "0"];

        bw_null -> A:backward -> B:backward -> C:backward [dir = back];

        label = "\n 图 5-1    一个跳跃表";
    }

图 5-1 展示了一个跳跃表示例，
位于图片最左边的是 ``zskiplist`` 结构，
该结构包含以下属性：

- ``header`` ：指向跳跃表的表头节点。

- ``tail`` ：指向跳跃表的表尾节点。

- ``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）。

- ``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）。

位于 ``zskiplist`` 结构右方的是四个 ``zskiplistNode`` 结构，
该结构包含以下属性：

- 层（level）：节点中用 ``L1`` 、 ``L2`` 、 ``L3`` 等字样标记节点的各个层， ``L1`` 代表第一层， ``L2`` 代表第二层，以此类推。每个层都带有两个属性：前进指针和跨度。前进指针用于访问位于表尾方向的其他节点，而跨度则记录了前进指针所指向节点和当前节点的距离。在上面的图片中，连线上带有数字的箭头就代表前进指针，而那个数字就是跨度。当程序从表头向表尾进行遍历时，访问会沿着层的前进指针进行。

- 后退（backward）指针：节点中用 ``BW`` 字样标记节点的后退指针，它指向位于当前节点的前一个节点。后退指针在程序从表尾向表头遍历时使用。

- 分值（score）：各个节点中的 ``1.0`` 、 ``2.0`` 和 ``3.0`` 是节点所保存的分值。在跳跃表中，节点按各自所保存的分值从小到大排列。

- 成员对象（obj）：各个节点中的 ``o1`` 、 ``o2`` 和 ``o3`` 是节点所保存的成员对象。

注意表头节点和其他节点的构造是一样的：
表头节点也有后退指针、分值和成员对象，
不过表头节点的这些属性都不会被用到，
所以图中省略了这些部分，
只显示了表头节点的各个层。

本节接下来的内容将对 ``zskiplistNode`` 和 ``zskiplist`` 两个结构进行更详细的介绍。


跳跃表节点
^^^^^^^^^^^^^^^^

跳跃表节点的实现由 ``redis.h/zskiplistNode`` 结构定义：

::

    typedef struct zskiplistNode {
        
        // 后退指针
        struct zskiplistNode *backward;

        // 分值
        double score;

        // 成员对象
        robj *obj;

        // 层
        struct zskiplistLevel {

            // 前进指针
            struct zskiplistNode *forward;

            // 跨度
            unsigned int span;

        } level[];

    } zskiplistNode;

层
""""""

跳跃表节点的 ``level`` 数组可以包含多个元素，
每个元素都包含一个指向其他节点的指针，
程序可以通过这些层来加快访问其他节点的速度，
一般来说，
层的数量越多，
访问其他节点的速度就越快。

每次创建一个新跳跃表节点的时候，
程序都根据幂次定律
（\ `power law <http://en.wikipedia.org/wiki/Power_law>`_\ ，越大的数出现的概率越小）
随机生成一个介于 ``1`` 和 ``32`` 之间的值作为 ``level`` 数组的大小，
这个大小就是层的“高度”。

图 5-2 分别展示了三个高度为 ``1`` 层、 ``3`` 层和 ``5`` 层的节点，
因为 C 语言的数组索引总是从 ``0`` 开始的，
所以节点的第一层是 ``level[0]`` ，
而第二层是 ``level[1]`` ，
以此类推。

.. graphviz:: 

    digraph {

        label = "\n 图 5-2    带有不同层高的节点";

        rankdir = LR;

        //

        node [shape = record];

        n1 [label = " zskiplistNode | level[0] | backward | score | obj "];
        n2 [label = " zskiplistNode | level[2] | level[1] | level[0] | backward | score | obj "];
        n3 [label = " zskiplistNode | level[4] | level[3] | level[2] | level[1] | level[0] | backward | score | obj "];

        //

        edge [style = invis];

        n1 -> n2 -> n3;
    }


前进指针
"""""""""""""""""""

每个层都有一个指向表尾方向的前进指针（\ ``level[i].forward`` 属性），
用于从表头向表尾方向访问节点。

图 5-3 用虚线表示出了程序从表头向表尾方向，
遍历跳跃表中所有节点的路径：

1. 迭代程序首先访问跳跃表的第一个节点（表头），
   然后从第四层的前进指针移动到表中的第二个节点。

2. 在第二个节点时，
   程序沿着第二层的前进指针移动到表中的第三个节点。

3. 在第三个节点时，
   程序同样沿着第二层的前进指针移动到表中的第四个节点。

4. 当程序再次沿着第四个节点的前进指针移动时，
   它碰到一个 ``NULL`` ，
   程序知道这时已经到达了跳跃表的表尾，
   于是结束这次遍历。

.. graphviz::

    digraph {


        rankdir = LR;

        node [shape = record, width = "0.5"];

        //

        l [label = " <header> header | <tail> tail | level \n 5 | length \n 3 "];

        subgraph cluster_nodes {

            style = invisible;

            header [label = " <l32> L32 | ... | <l5> L5 | <l4> L4 | <l3> L3 | <l2> L2 | <l1> L1 "];

            bw_null [label = "NULL", shape = plaintext];

            level_null [label = "NULL", shape = plaintext];

            A [label = " <l4> L4 | <l3> L3 | <l2> L2 | <l1> L1 | <backward> BW | 1.0 | o1 "];

            B [label = " <l2> L2 | <l1> L1 | <backward> BW | 2.0 | o2 "];

            C [label = " <l5> L5 | <l4> L4 | <l3> L3 | <l2> L2 | <l1> L1 | <backward> BW | 3.0 | o3 "];

        }

        subgraph cluster_nulls {

            style = invisible;

            n1 [label = "NULL", shape = plaintext];
            n2 [label = "NULL", shape = plaintext];
            n3 [label = "NULL", shape = plaintext];
            n4 [label = "NULL", shape = plaintext];
            n5 [label = "NULL", shape = plaintext];

        }

        //

        l:header -> header [style = dashed];
        l:tail -> C;

        header:l32 -> level_null [label = "0"];
        header:l5 -> C:l5 [label = "3"];
        header:l4 -> A:l4 [label = "1", style = dashed];
        header:l3 -> A:l3 [label = "1"];
        header:l2 -> A:l2 [label = "1"];
        header:l1 -> A:l1 [label = "1"];

        A:l4 -> C:l4 [label = "2"];
        A:l3 -> C:l3 [label = "2"];
        A:l2 -> B:l2 [label = "1", style = dashed];
        A:l1 -> B:l1 [label = "1"];

        B:l2 -> C:l2 [label = "1", style = dashed];
        B:l1 -> C:l1 [label = "1"];

        C:l5 -> n5 [label = "0"];
        C:l4 -> n4 [label = "0"];
        C:l3 -> n3 [label = "0"];
        C:l2 -> n2 [label = "0", style = dashed];
        C:l1 -> n1 [label = "0"];

        bw_null -> A:backward -> B:backward -> C:backward [dir = back];


        label = "\n 图 5-3    遍历整个跳跃表";
    }

跨度
"""""""""""""""""""""

层的跨度（\ ``level[i].span`` 属性）用于记录两个节点之间的距离：

- 两个节点之间的跨度越大，
  它们相距得就越远。

- 指向 ``NULL`` 的所有前进指针的跨度都为 ``0`` ，
  因为它们没有连向任何节点。

初看上去，
很容易以为跨度和遍历操作有关，
但实际上并不是这样 ——
遍历操作只使用前进指针就可以完成了，
跨度实际上是用来计算排位（rank）的：
在查找某个节点的过程中，
将沿途访问过的所有层的跨度累计起来，
得到的结果就是目标节点在跳跃表中的排位。

举个例子，
图 5-4 用虚线标记了在跳跃表中查找分值为 ``3.0`` 、
成员对象为 ``o3`` 的节点时，
沿途经历的层：
查找的过程只经过了一个层，
并且层的跨度为 ``3`` ，
所以目标节点在跳跃表中的排位为 ``3`` 。

.. graphviz::

    digraph {

        rankdir = LR;

        node [shape = record, width = "0.5"];

        //

        l [label = " <header> header | <tail> tail | level \n 5 | length \n 3 "];

        subgraph cluster_nodes {

            style = invisible;

            header [label = " <l32> L32 | ... | <l5> L5 | <l4> L4 | <l3> L3 | <l2> L2 | <l1> L1 "];

            bw_null [label = "NULL", shape = plaintext];

            level_null [label = "NULL", shape = plaintext];

            A [label = " <l4> L4 | <l3> L3 | <l2> L2 | <l1> L1 | <backward> BW | 1.0 | o1 "];

            B [label = " <l2> L2 | <l1> L1 | <backward> BW | 2.0 | o2 "];

            C [label = " <l5> L5 | <l4> L4 | <l3> L3 | <l2> L2 | <l1> L1 | <backward> BW | 3.0 | o3 "];

        }

        subgraph cluster_nulls {

            style = invisible;

            n1 [label = "NULL", shape = plaintext];
            n2 [label = "NULL", shape = plaintext];
            n3 [label = "NULL", shape = plaintext];
            n4 [label = "NULL", shape = plaintext];
            n5 [label = "NULL", shape = plaintext];

        }

        //

        l:header -> header [style = dashed];
        l:tail -> C;

        header:l32 -> level_null [label = "0"];
        header:l5 -> C:l5 [label = "3", style = dashed];
        header:l4 -> A:l4 [label = "1"];
        header:l3 -> A:l3 [label = "1"];
        header:l2 -> A:l2 [label = "1"];
        header:l1 -> A:l1 [label = "1"];

        A:l4 -> C:l4 [label = "2"];
        A:l3 -> C:l3 [label = "2"];
        A:l2 -> B:l2 [label = "1"];
        A:l1 -> B:l1 [label = "1"];

        B:l2 -> C:l2 [label = "1"];
        B:l1 -> C:l1 [label = "1"];

        C:l5 -> n5 [label = "0"];
        C:l4 -> n4 [label = "0"];
        C:l3 -> n3 [label = "0"];
        C:l2 -> n2 [label = "0"];
        C:l1 -> n1 [label = "0"];

        bw_null -> A:backward -> B:backward -> C:backward [dir = back];

        label = "\n 图 5-4    计算节点的排位";

    }

再举个例子，
图 5-5 用虚线标记了在跳跃表中查找分值为 ``2.0`` 、
成员对象为 ``o2`` 的节点时，
沿途经历的层：
在查找节点的过程中，
程序经过了两个跨度为 ``1`` 的节点，
因此可以计算出，
目标节点在跳跃表中的排位为 2 。

.. graphviz::

    digraph {

        rankdir = LR;

        node [shape = record, width = "0.5"];

        //

        l [label = " <header> header | <tail> tail | level \n 5 | length \n 3 "];

        subgraph cluster_nodes {

            style = invisible;

            header [label = " <l32> L32 | ... | <l5> L5 | <l4> L4 | <l3> L3 | <l2> L2 | <l1> L1 "];

            bw_null [label = "NULL", shape = plaintext];

            level_null [label = "NULL", shape = plaintext];

            A [label = " <l4> L4 | <l3> L3 | <l2> L2 | <l1> L1 | <backward> BW | 1.0 | o1 "];

            B [label = " <l2> L2 | <l1> L1 | <backward> BW | 2.0 | o2 "];

            C [label = " <l5> L5 | <l4> L4 | <l3> L3 | <l2> L2 | <l1> L1 | <backward> BW | 3.0 | o3 "];

        }

        subgraph cluster_nulls {

            style = invisible;

            n1 [label = "NULL", shape = plaintext];
            n2 [label = "NULL", shape = plaintext];
            n3 [label = "NULL", shape = plaintext];
            n4 [label = "NULL", shape = plaintext];
            n5 [label = "NULL", shape = plaintext];

        }

        //

        l:header -> header [style = dashed];
        l:tail -> C;

        header:l32 -> level_null [label = "0"];
        header:l5 -> C:l5 [label = "3"];
        header:l4 -> A:l4 [label = "1", style = dashed];
        header:l3 -> A:l3 [label = "1"];
        header:l2 -> A:l2 [label = "1"];
        header:l1 -> A:l1 [label = "1"];

        A:l4 -> C:l4 [label = "2"];
        A:l3 -> C:l3 [label = "2"];
        A:l2 -> B:l2 [label = "1", style = dashed];
        A:l1 -> B:l1 [label = "1"];

        B:l2 -> C:l2 [label = "1"];
        B:l1 -> C:l1 [label = "1"];

        C:l5 -> n5 [label = "0"];
        C:l4 -> n4 [label = "0"];
        C:l3 -> n3 [label = "0"];
        C:l2 -> n2 [label = "0"];
        C:l1 -> n1 [label = "0"];

        bw_null -> A:backward -> B:backward -> C:backward [dir = back];

        label = "\n 图 5-5    另一个计算节点排位的例子";
    }


后退指针
"""""""""""

节点的后退指针（\ ``backward`` 属性）用于从表尾向表头方向访问节点：
跟可以一次跳过多个节点的前进指针不同，
因为每个节点只有一个后退指针，
所以每次只能后退至前一个节点。

图 5-6 用虚线展示了如果从表尾向表头遍历跳跃表中的所有节点：
程序首先通过跳跃表的 ``tail`` 指针访问表尾节点，
然后通过后退指针访问倒数第二个节点，
之后再沿着后退指针访问倒数第三个节点，
再之后遇到指向 ``NULL`` 的后退指针，
于是访问结束。

.. graphviz::

    digraph {

        rankdir = LR;

        node [shape = record, width = "0.5"];

        //

        l [label = " <header> header | <tail> tail | level \n 5 | length \n 3 "];

        subgraph cluster_nodes {

            style = invisible;

            header [label = " <l32> L32 | ... | <l5> L5 | <l4> L4 | <l3> L3 | <l2> L2 | <l1> L1 "];

            bw_null [label = "NULL", shape = plaintext];

            level_null [label = "NULL", shape = plaintext];

            A [label = " <l4> L4 | <l3> L3 | <l2> L2 | <l1> L1 | <backward> BW | 1.0 | o1 "];

            B [label = " <l2> L2 | <l1> L1 | <backward> BW | 2.0 | o2 "];

            C [label = " <l5> L5 | <l4> L4 | <l3> L3 | <l2> L2 | <l1> L1 | <backward> BW | 3.0 | o3 "];

        }

        subgraph cluster_nulls {

            style = invisible;

            n1 [label = "NULL", shape = plaintext];
            n2 [label = "NULL", shape = plaintext];
            n3 [label = "NULL", shape = plaintext];
            n4 [label = "NULL", shape = plaintext];
            n5 [label = "NULL", shape = plaintext];

        }

        //

        l:header -> header;
        l:tail -> C [style = dashed];

        header:l32 -> level_null [label = "0"];
        header:l5 -> C:l5 [label = "3"];
        header:l4 -> A:l4 [label = "1"];
        header:l3 -> A:l3 [label = "1"];
        header:l2 -> A:l2 [label = "1"];
        header:l1 -> A:l1 [label = "1"];

        A:l4 -> C:l4 [label = "2"];
        A:l3 -> C:l3 [label = "2"];
        A:l2 -> B:l2 [label = "1"];
        A:l1 -> B:l1 [label = "1"];

        B:l2 -> C:l2 [label = "1"];
        B:l1 -> C:l1 [label = "1"];

        C:l5 -> n5 [label = "0"];
        C:l4 -> n4 [label = "0"];
        C:l3 -> n3 [label = "0"];
        C:l2 -> n2 [label = "0"];
        C:l1 -> n1 [label = "0"];

        bw_null -> A:backward -> B:backward -> C:backward [dir = back, style = dashed];

        label = "\n 图 5-6    从表尾向表头方向遍历跳跃表";
    }


分值和成员
""""""""""""""

节点的分值（\ ``score`` 属性）是一个 ``double`` 类型的浮点数，
跳跃表中的所有节点都按分值从小到大来排序。

节点的成员对象（\ ``obj`` 属性）是一个指针，
它指向一个字符串对象，
而字符串对象则保存着一个 SDS 值。

在同一个跳跃表中，
各个节点保存的成员对象必须是唯一的，
但是多个节点保存的分值却可以是相同的：
分值相同的节点将按照成员对象在字典序中的大小来进行排序，
成员对象较小的节点会排在前面（靠近表头的方向），
而成员对象较大的节点则会排在后面（靠近表尾的方向）。

举个例子，
在图 5-7 所示的跳跃表中，
三个跳跃表节点都保存了相同的分值 ``10086.0`` ，
但保存成员对象 ``o1`` 的节点却排在保存成员对象 ``o2`` 和 ``o3`` 的节点之前，
而保存成员对象 ``o2`` 的节点又排在保存成员对象 ``o3`` 的节点之前，
由此可见，
``o1`` 、 ``o2`` 、 ``o3`` 三个成员对象在字典中的排序为 ``o1 <= o2 <= o3`` 。

.. graphviz::

    digraph {

        rankdir = LR;

        node [shape = record, width = "0.5"];

        //

        l [label = " <header> header | <tail> tail | level \n 5 | length \n 3 "];

        subgraph cluster_nodes {

            style = invisible;

            header [label = " <l32> L32 | ... | <l5> L5 | <l4> L4 | <l3> L3 | <l2> L2 | <l1> L1 "];

            bw_null [label = "NULL", shape = plaintext];

            level_null [label = "NULL", shape = plaintext];

            A [label = " <l4> L4 | <l3> L3 | <l2> L2 | <l1> L1 | <backward> BW | 10086.0 | o1 "];

            B [label = " <l2> L2 | <l1> L1 | <backward> BW | 10086.0 | o2 "];

            C [label = " <l5> L5 | <l4> L4 | <l3> L3 | <l2> L2 | <l1> L1 | <backward> BW | 10086.0 | o3 "];

        }

        subgraph cluster_nulls {

            style = invisible;

            n1 [label = "NULL", shape = plaintext];
            n2 [label = "NULL", shape = plaintext];
            n3 [label = "NULL", shape = plaintext];
            n4 [label = "NULL", shape = plaintext];
            n5 [label = "NULL", shape = plaintext];

        }

        //

        l:header -> header;
        l:tail -> C;

        header:l32 -> level_null [label = "0"];
        header:l5 -> C:l5 [label = "3"];
        header:l4 -> A:l4 [label = "1"];
        header:l3 -> A:l3 [label = "1"];
        header:l2 -> A:l2 [label = "1"];
        header:l1 -> A:l1 [label = "1"];

        A:l4 -> C:l4 [label = "2"];
        A:l3 -> C:l3 [label = "2"];
        A:l2 -> B:l2 [label = "1"];
        A:l1 -> B:l1 [label = "1"];

        B:l2 -> C:l2 [label = "1"];
        B:l1 -> C:l1 [label = "1"];

        C:l5 -> n5 [label = "0"];
        C:l4 -> n4 [label = "0"];
        C:l3 -> n3 [label = "0"];
        C:l2 -> n2 [label = "0"];
        C:l1 -> n1 [label = "0"];

        bw_null -> A:backward -> B:backward -> C:backward [dir = back];

        label = "\n 图 5-7    三个带有相同分值的跳跃表节点";
    }


跳跃表
^^^^^^^^^^^

虽然仅靠多个跳跃表节点就可以组成一个跳跃表，
如图 5-8 所示。

.. graphviz::

    digraph {

        rankdir = LR;

        node [shape = record, width = "0.5"];

        //

        //l [label = " <header> header | <tail> tail | level \n 5 | length \n 3 "];

        subgraph cluster_nodes {

            style = invisible;

            header [label = " <l32> L32 | ... | <l5> L5 | <l4> L4 | <l3> L3 | <l2> L2 | <l1> L1 "];

            bw_null [label = "NULL", shape = plaintext];

            level_null [label = "NULL", shape = plaintext];

            A [label = " <l4> L4 | <l3> L3 | <l2> L2 | <l1> L1 | <backward> BW | 1.0 | o1 "];

            B [label = " <l2> L2 | <l1> L1 | <backward> BW | 2.0 | o2 "];

            C [label = " <l5> L5 | <l4> L4 | <l3> L3 | <l2> L2 | <l1> L1 | <backward> BW | 3.0 | o3 "];

        }

        subgraph cluster_nulls {

            style = invisible;

            n1 [label = "NULL", shape = plaintext];
            n2 [label = "NULL", shape = plaintext];
            n3 [label = "NULL", shape = plaintext];
            n4 [label = "NULL", shape = plaintext];
            n5 [label = "NULL", shape = plaintext];

        }

        //

        //l:header -> header;
        //l:tail -> C;

        header:l32 -> level_null [label = "0"];
        header:l5 -> C:l5 [label = "3"];
        header:l4 -> A:l4 [label = "1"];
        header:l3 -> A:l3 [label = "1"];
        header:l2 -> A:l2 [label = "1"];
        header:l1 -> A:l1 [label = "1"];

        A:l4 -> C:l4 [label = "2"];
        A:l3 -> C:l3 [label = "2"];
        A:l2 -> B:l2 [label = "1"];
        A:l1 -> B:l1 [label = "1"];

        B:l2 -> C:l2 [label = "1"];
        B:l1 -> C:l1 [label = "1"];

        C:l5 -> n5 [label = "0"];
        C:l4 -> n4 [label = "0"];
        C:l3 -> n3 [label = "0"];
        C:l2 -> n2 [label = "0"];
        C:l1 -> n1 [label = "0"];

        bw_null -> A:backward -> B:backward -> C:backward [dir = back];

        label = "\n 图 5-8    由多个跳跃表节点组成的跳跃表";
    }

但通过使用一个 ``zskiplist`` 结构来持有这些节点，
程序可以更方便地对整个跳跃表进行处理，
比如快速访问跳跃表的表头节点和表尾节点，
又或者快速地获取跳跃表节点的数量（也即是跳跃表的长度）等信息，
如图 5-9 所示。

.. graphviz::

    digraph {

        rankdir = LR;

        node [shape = record, width = "0.5"];

        //

        l [label = " <header> header | <tail> tail | level \n 5 | length \n 3 "];

        subgraph cluster_nodes {

            style = invisible;

            header [label = " <l32> L32 | ... | <l5> L5 | <l4> L4 | <l3> L3 | <l2> L2 | <l1> L1 "];

            bw_null [label = "NULL", shape = plaintext];

            level_null [label = "NULL", shape = plaintext];

            A [label = " <l4> L4 | <l3> L3 | <l2> L2 | <l1> L1 | <backward> BW | 1.0 | o1 "];

            B [label = " <l2> L2 | <l1> L1 | <backward> BW | 2.0 | o2 "];

            C [label = " <l5> L5 | <l4> L4 | <l3> L3 | <l2> L2 | <l1> L1 | <backward> BW | 3.0 | o3 "];

        }

        subgraph cluster_nulls {

            style = invisible;

            n1 [label = "NULL", shape = plaintext];
            n2 [label = "NULL", shape = plaintext];
            n3 [label = "NULL", shape = plaintext];
            n4 [label = "NULL", shape = plaintext];
            n5 [label = "NULL", shape = plaintext];

        }

        //

        l:header -> header;
        l:tail -> C;

        header:l32 -> level_null [label = "0"];
        header:l5 -> C:l5 [label = "3"];
        header:l4 -> A:l4 [label = "1"];
        header:l3 -> A:l3 [label = "1"];
        header:l2 -> A:l2 [label = "1"];
        header:l1 -> A:l1 [label = "1"];

        A:l4 -> C:l4 [label = "2"];
        A:l3 -> C:l3 [label = "2"];
        A:l2 -> B:l2 [label = "1"];
        A:l1 -> B:l1 [label = "1"];

        B:l2 -> C:l2 [label = "1"];
        B:l1 -> C:l1 [label = "1"];

        C:l5 -> n5 [label = "0"];
        C:l4 -> n4 [label = "0"];
        C:l3 -> n3 [label = "0"];
        C:l2 -> n2 [label = "0"];
        C:l1 -> n1 [label = "0"];

        bw_null -> A:backward -> B:backward -> C:backward [dir = back];

        label = "\n 图 5-9    带有 zskiplist 结构的跳跃表";
    }

``zskiplist`` 结构的定义如下：

::

    typedef struct zskiplist {

        // 表头节点和表尾节点
        struct zskiplistNode *header, *tail;

        // 表中节点的数量
        unsigned long length;

        // 表中层数最大的节点的层数
        int level;

    } zskiplist;

``header`` 和 ``tail`` 指针分别指向跳跃表的表头和表尾节点，
通过这两个指针，
程序定位表头节点和表尾节点的复杂度为 :math:`O(1)` 。

通过使用 ``length`` 属性来记录节点的数量，
程序可以在 :math:`O(1)` 复杂度内返回跳跃表的长度。

``level`` 属性则用于在 :math:`O(1)` 复杂度内获取跳跃表中层高最大的那个节点的层数量，
注意表头节点的层高并不计算在内。