跳跃表的实现

Redis 的跳跃表由 redis.h/zskiplistNoderedis.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。

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 L1L2L3 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 1.02.03.0 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 o1o2o3 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 zskiplistNodezskiplist 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;

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,越大的数出现的概率越小) 随机生成一个介于 132 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。

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。

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

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 。

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。

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 o2o3 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, o1o2o3 三个成员对象在字典中的排序为 o1 <= o2 <= o3

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。

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。

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;

headertail 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 O(1)

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

level 属性则用于在 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。