Linux中间件之分析redis的跳表实现

zset对象的ziplist和skiplist转换

redis 存储转换如下：

执行zadd命令时会执行如下的函数（t_zset.c）：

/* Add a new element or update the score of an existing element in a sorted
 * set, regardless of its encoding.
 *
 * The set of flags change the command behavior. They are passed with an integer
 * pointer since the function will clear the flags and populate them with
 * other flags to indicate different conditions.
 *
 * The input flags are the following:
 *
 * ZADD_INCR: Increment the current element score by 'score' instead of updating
 *            the current element score. If the element does not exist, we
 *            assume 0 as previous score.
 * ZADD_NX:   Perform the operation only if the element does not exist.
 * ZADD_XX:   Perform the operation only if the element already exist.
 *
 * When ZADD_INCR is used, the new score of the element is stored in
 * '*newscore' if 'newscore' is not NULL.
 *
 * The returned flags are the following:
 *
 * ZADD_NAN:     The resulting score is not a number.
 * ZADD_ADDED:   The element was added (not present before the call).
 * ZADD_UPDATED: The element score was updated.
 * ZADD_NOP:     No operation was performed because of NX or XX.
 *
 * Return value:
 *
 * The function returns 1 on success, and sets the appropriate flags
 * ADDED or UPDATED to signal what happened during the operation (note that
 * none could be set if we re-added an element using the same score it used
 * to have, or in the case a zero increment is used).
 *
 * The function returns 0 on error, currently only when the increment
 * produces a NAN condition, or when the 'score' value is NAN since the
 * start.
 *
 * The command as a side effect of adding a new element may convert the sorted
 * set internal encoding from ziplist to hashtable+skiplist.
 *
 * Memory management of 'ele':
 *
 * The function does not take ownership of the 'ele' SDS string, but copies
 * it if needed. */
int zsetAdd(robj *zobj, double score, sds ele, int *flags, double *newscore) {
    /* Turn options into simple to check vars. */
    int incr = (*flags & ZADD_INCR) != 0;
    int nx = (*flags & ZADD_NX) != 0;
    int xx = (*flags & ZADD_XX) != 0;
    *flags = 0; /* We'll return our response flags. */
    double curscore;

    /* NaN as input is an error regardless of all the other parameters. */
    if (isnan(score)) {
        *flags = ZADD_NAN;
        return 0;
    }

    /* Update the sorted set according to its encoding. */
    if (zobj->encoding == OBJ_ENCODING_ZIPLIST) {
        unsigned char *eptr;

        if ((eptr = zzlFind(zobj->ptr,ele,&curscore)) != NULL) {
            /* NX? Return, same element already exists. */
            if (nx) {
                *flags |= ZADD_NOP;
                return 1;
            }

            /* Prepare the score for the increment if needed. */
            if (incr) {
                score += curscore;
                if (isnan(score)) {
                    *flags |= ZADD_NAN;
                    return 0;
                }
                if (newscore) *newscore = score;
            }

            /* Remove and re-insert when score changed. */
            if (score != curscore) {
                zobj->ptr = zzlDelete(zobj->ptr,eptr);
                zobj->ptr = zzlInsert(zobj->ptr,ele,score);
                *flags |= ZADD_UPDATED;
            }
            return 1;
        } else if (!xx) {
            /* check if the element is too large or the list
             * becomes too long *before* executing zzlInsert. */
            if (zzlLength(zobj->ptr)+1 > server.zset_max_ziplist_entries ||
                sdslen(ele) > server.zset_max_ziplist_value ||
                !ziplistSafeToAdd(zobj->ptr, sdslen(ele)))
            {
                zsetConvert(zobj,OBJ_ENCODING_SKIPLIST);
            } else {
                zobj->ptr = zzlInsert(zobj->ptr,ele,score);
                if (newscore) *newscore = score;
                *flags |= ZADD_ADDED;
                return 1;
            }
        } else {
            *flags |= ZADD_NOP;
            return 1;
        }
    }

    /* Note that the above block handling ziplist would have either returned or
     * converted the key to skiplist. */
    if (zobj->encoding == OBJ_ENCODING_SKIPLIST) {
        zset *zs = zobj->ptr;
        zskiplistNode *znode;
        dictEntry *de;

        de = dictFind(zs->dict,ele);
        if (de != NULL) {
            /* NX? Return, same element already exists. */
            if (nx) {
                *flags |= ZADD_NOP;
                return 1;
            }
            curscore = *(double*)dictGetVal(de);

            /* Prepare the score for the increment if needed. */
            if (incr) {
                score += curscore;
                if (isnan(score)) {
                    *flags |= ZADD_NAN;
                    return 0;
                }
                if (newscore) *newscore = score;
            }

            /* Remove and re-insert when score changes. */
            if (score != curscore) {
                znode = zslUpdateScore(zs->zsl,curscore,ele,score);
                /* Note that we did not removed the original element from
                 * the hash table representing the sorted set, so we just
                 * update the score. */
                dictGetVal(de) = &znode->score; /* Update score ptr. */
                *flags |= ZADD_UPDATED;
            }
            return 1;
        } else if (!xx) {
            ele = sdsdup(ele);
            znode = zslInsert(zs->zsl,score,ele);
            serverAssert(dictAdd(zs->dict,ele,&znode->score) == DICT_OK);
            *flags |= ZADD_ADDED;
            if (newscore) *newscore = score;
            return 1;
        } else {
            *flags |= ZADD_NOP;
            return 1;
        }
    } else {
        serverPanic("Unknown sorted set encoding");
    }
    return 0; /* Never reached. */
}

/*-----------------------------------------------------------------------------
 * Sorted set commands
 *----------------------------------------------------------------------------*/

/* This generic command implements both ZADD and ZINCRBY. */
void zaddGenericCommand(client *c, int flags) {
    static char *nanerr = "resulting score is not a number (NaN)";
    robj *key = c->argv[1];
    robj *zobj;
    sds ele;
    double score = 0, *scores = NULL;
    int j, elements;
    int scoreidx = 0;
    /* The following vars are used in order to track what the command actually
     * did during the execution, to reply to the client and to trigger the
     * notification of keyspace change. */
    int added = 0;      /* Number of new elements added. */
    int updated = 0;    /* Number of elements with updated score. */
    int processed = 0;  /* Number of elements processed, may remain zero with
                           options like XX. */

    /* Parse options. At the end 'scoreidx' is set to the argument position
     * of the score of the first score-element pair. */
    scoreidx = 2;
    while(scoreidx < c->argc) {
        char *opt = c->argv[scoreidx]->ptr;
        if (!strcasecmp(opt,"nx")) flags |= ZADD_NX;
        else if (!strcasecmp(opt,"xx")) flags |= ZADD_XX;
        else if (!strcasecmp(opt,"ch")) flags |= ZADD_CH;
        else if (!strcasecmp(opt,"incr")) flags |= ZADD_INCR;
        else break;
        scoreidx++;
    }

    /* Turn options into simple to check vars. */
    int incr = (flags & ZADD_INCR) != 0;
    int nx = (flags & ZADD_NX) != 0;
    int xx = (flags & ZADD_XX) != 0;
    int ch = (flags & ZADD_CH) != 0;

    /* After the options, we expect to have an even number of args, since
     * we expect any number of score-element pairs. */
    elements = c->argc-scoreidx;
    if (elements % 2 || !elements) {
        addReply(c,shared.syntaxerr);
        return;
    }
    elements /= 2; /* Now this holds the number of score-element pairs. */

    /* Check for incompatible options. */
    if (nx && xx) {
        addReplyError(c,
            "XX and NX options at the same time are not compatible");
        return;
    }

    if (incr && elements > 1) {
        addReplyError(c,
            "INCR option supports a single increment-element pair");
        return;
    }

    /* Start parsing all the scores, we need to emit any syntax error
     * before executing additions to the sorted set, as the command should
     * either execute fully or nothing at all. */
    scores = zmalloc(sizeof(double)*elements);
    for (j = 0; j < elements; j++) {
        if (getDoubleFromObjectOrReply(c,c->argv[scoreidx+j*2],&scores[j],NULL)
            != C_OK) goto cleanup;
    }

    /* Lookup the key and create the sorted set if does not exist. */
    zobj = lookupKeyWrite(c->db,key);
    if (zobj == NULL) {
        if (xx) goto reply_to_client; /* No key + XX option: nothing to do. */
        if (server.zset_max_ziplist_entries == 0 ||
            server.zset_max_ziplist_value < sdslen(c->argv[scoreidx+1]->ptr))
        {
            zobj = createZsetObject();
        } else {
            zobj = createZsetZiplistObject();
        }
        dbAdd(c->db,key,zobj);
    } else {
        if (zobj->type != OBJ_ZSET) {
            addReply(c,shared.wrongtypeerr);
            goto cleanup;
        }
    }

    for (j = 0; j < elements; j++) {
        double newscore;
        score = scores[j];
        int retflags = flags;

        ele = c->argv[scoreidx+1+j*2]->ptr;
        int retval = zsetAdd(zobj, score, ele, &retflags, &newscore);
        if (retval == 0) {
            addReplyError(c,nanerr);
            goto cleanup;
        }
        if (retflags & ZADD_ADDED) added++;
        if (retflags & ZADD_UPDATED) updated++;
        if (!(retflags & ZADD_NOP)) processed++;
        score = newscore;
    }
    server.dirty += (added+updated);

reply_to_client:
    if (incr) { /* ZINCRBY or INCR option. */
        if (processed)
            addReplyDouble(c,score);
        else
            addReplyNull(c);
    } else { /* ZADD. */
        addReplyLongLong(c,ch ? added+updated : added);
    }

cleanup:
    zfree(scores);
    if (added || updated) {
        signalModifiedKey(c,c->db,key);
        notifyKeyspaceEvent(NOTIFY_ZSET,
            incr ? "zincr" : "zadd", key, c->db->id);
    }
}

void zaddCommand(client *c) {
    zaddGenericCommand(c,ZADD_NONE);
}
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lookupKeyWrite(c->db,key)检查key是否存在。
createZsetObject()是创建跳表；createZsetZiplistObject()是创建ziplist。
zsetConvert(zobj,OBJ_ENCODING_SKIPLIST)转换存储类型。
创建调表的条件由zset_max_ziplist_entries和zset_max_ziplist_value两个参数决定。调试时可以在redis.conf文件中修改这两个参数。

# Similarly to hashes and lists, sorted sets are also specially encoded in
# order to save a lot of space. This encoding is only used when the length and
# elements of a sorted set are below the following limits:
zset-max-ziplist-entries 2
zset-max-ziplist-value 64
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跳表的实现

跳表（多层级有序链表）结构用来实现有序集合。
鉴于 redis 需要实现 zrange 以及 zrevrange 功能；需要节点间最好能直接相连并且增删改操作后结构依然有序。
B+ 树时间复杂度为$h\times O({\log }_{2}n)$，但 B+ 数有复杂的节点分裂操作；有序数组通过二分查找能获得 $O({\log }_{2}n)$ 时间复杂度；平衡二叉树也能获得$O({\log }_{2}n)$ 时间复杂度。

理想跳表

每隔一个节点生成一个层级节点；模拟二叉树结构，以此达到搜索时间复杂度为 $O({\log }_{2}n)$；通过空间换时间的结构。
但是如果对理想跳表结构进行删除增加操作，很有可能改变跳表结构；如果重构理想结构，将是巨大的运算。

考虑用概率的方法来进行优化：
从每一个节点出发，每增加一个节点都有 $\frac{1}{2}$ 的概率增加一个层级，$\frac{1}{4}$ 的概率增加两个层级， 的概率增加
3 个层级，以此类推。
经过证明，当数据量足够大（256）时，通过概率构造的跳表趋向于理想跳表，并且此时如果删除节点，无需重构跳表结构，此时依然趋向于理想跳表。此时时间复杂度为：
$$(1-\frac{1}{n^c})\times O({\log }_{2}n),c是常数$$

redis 跳表

从节约内存出发，redis 考虑牺牲一点时间复杂度让跳表结构更加变扁平，就像二叉堆改成四叉堆结构；并且 redis 还限制了跳
表的最高层级为 32。
节点数量大于 128 或者有一个字符串长度大于 64，则使用跳表（skiplist）。

数据结构

server.h

#define ZSKIPLIST_MAXLEVEL 32 /* Should be enough for 2^64 elements */
#define ZSKIPLIST_P 0.25      /* Skiplist P = 1/4 */

/* ZSETs use a specialized version of Skiplists */
typedef struct zskiplistNode {
    sds ele;
    double score;
    struct zskiplistNode *backward;
    struct zskiplistLevel {
        struct zskiplistNode *forward;
        unsigned long span;// 用于zrank
    } level[];
} zskiplistNode;

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

typedef struct zset {
    dict *dict;
    zskiplist *zsl;
} zset;

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1. 层级通过柔性数组来组织。

2. zset为了优化，加了dict，帮助快速索引到节点，因为dict的查询时间复杂度是$O(1)$。特别是删除和修改操作时，可以通过dict快速查找节点是否存在；如果存在，就可以找到该节点的左右两个节点以及有多少个层级，以达到以$O(1)$的时间复杂度进行删除或修改操作。

3. zset的添加节点操作的时间复杂度是$O({\log }_{2}n)$。

4. zset的修改操作是先删除节点再添加节点。

5. redis的跳表数据结构做了调整，它是一个循环双向链表的结构；这样的结构可以实现逆向查询。

结构图

要实现$O({\log }_{2}n)$的时间复杂度，有很多方式，redis使用调表的原因是调表中的所有元素都在第一个层级（即level 0），如果要进行范围查询，只需要找到两个边界，（因为调表是有序的）第一层往下遍历就行了，这是其他数据结构时间复杂度不能达到的高效范围查询。

总结

1. 有序数据查询数据的时间复杂度是$O({\log }_{2}n)$，有序链表查询数据的时间复杂度是$O(n)$。
2. 调表是多层级有序链表，可以二分查找数据，它的最底层包含所有的元素。
3. 调表的范围查询非常方便，通过$O({\log }_{2}n)$的时间复杂度快速找到边界，然后在最底层找到范围内所有元素。
4. 调表的增删改查时间复杂度都是$O({\log }_{2}n)$。