二分查詢和AVL樹查詢

什麼是跳錶

傳統意義的單鏈表是一個線性結構，向有序的連結串列中插入一個節點需要O(n)的時間，查詢操作需要O

跳躍表的簡單示例：

如果我們使用上圖所示的跳躍表，就可以減少查詢所需時間為O(n/2)，因為我們可以先通過每個節點的最上面的指標先進行查詢，這樣子就能跳過一半的節點。

比如我們想查詢19，首先和6比較，大於6之後，在和9進行比較，然後在和12進行比較......最後比較到21的時候，發現21大於19，說明查詢的點在17和21之間，從這個過程中，我們可以看出，查詢的時候跳過了3、7、12等點，因此查詢的複雜度為O(n/2)。

查詢的過程如下圖：

其實，上面基本上就是跳躍表的思想，每一個結點不單單隻包含指向下一個結點的指標，可能包含很多個指向後續結點的指標，這樣就可以跳過一些不必要的結點，從而加快查詢、刪除等操作。對於一個連結串列內每一個結點包含多少個指向後續元素的指標，後續節點個數是通過一個隨機函式生成器得到，這樣子就構成了一個跳躍表。

隨機生成的跳躍表可能如下圖所示：

跳躍表其實也是一種通過“空間來換取時間”的一個演算法，通過在每個節點中增加了向前的指標，從而提升查詢的效率。

“Skip lists are data structures  that use probabilistic  balancing rather  than  strictly  enforced balancing. As a result, the algorithms  for insertion  and deletion in skip lists  are much simpler and significantly  faster  than  equivalent  algorithms  for balanced trees.  ”
譯文：跳躍表使用概率均衡技術

跳錶是一種隨機化的資料結構，目前開源軟體 Redis 和 LevelDB 都有用到它。

   SkipList的操作

如果我們想查詢19是否存在？如何查詢呢？我們從頭結點開始，首先和9進行判斷，此時大於9，然後和21進行判斷，小於21，此時這個值肯定在9結點和21結點之間，此時，我們和17進行判斷，大於17，然後和21進行判斷，小於21，此時肯定在17結點和21結點之間，此時和19進行判斷，找到了。具體的示意圖如圖所示：

我們結合下圖進行講解，查詢路徑如下圖的灰色的線所示  申請新的結點如17結點所示， 調整指向新結點17的指標以及17結點指向後續結點的指標。這裡有一個小技巧，就是使用update陣列儲存大於17結點的位置，update陣列的內容如紅線所示，這些位置才是有可能更新指標的位置。

  Key-Value資料結構

目前常用的key-value資料結構有三種：Hash表、紅黑樹、SkipList，它們各自有著不同的優缺點（不考慮刪除操作）：
Hash表：插入、查詢最快，為O(1)；如使用連結串列實現則可實現無鎖；資料有序化需要顯式的排序操作。
紅黑樹：插入、查詢為O(logn)，但常數項較小；無鎖實現的複雜性很高，一般需要加鎖；資料天然有序。
SkipList：插入、查詢為O(logn)，但常數項比紅黑樹要大；底層結構為連結串列，可無鎖實現；資料天然有序。如果要實現一個key-value結構，需求的功能有插入、查詢、迭代、修改，那麼首先Hash表就不是很適合了，因為迭代的時間複雜度比較高；而紅黑樹的插入很可能會涉及多個結點的旋轉、變色操作，因此需要在外層加鎖，這無形中降低了它可能的併發度。而SkipList底層是用連結串列實現的，可以實現為lock free，同時它還有著不錯的效能（單執行緒下只比紅黑樹略慢），非常適合用來實現我們需求的那種key-value結構。LevelDB、Reddis的底層儲存結構就是用的SkipList。

  基於鎖的併發

常見的lock free程式設計一般是基於CAS(Compare And Swap)操作：CAS(void *ptr, Any oldValue, Any newValue);
即檢視記憶體地址ptr處的值，如果為oldValue則將其改為newValue，並返回true，否則返回false。X86平臺上的CAS操作一般是通過CPU的CMPXCHG指令來完成的。CPU在執行此指令時會首先鎖住CPU匯流排，禁止其它核心對記憶體的訪問，然後再檢視或修改*ptr的值。簡單的說CAS利用了CPU的硬體鎖來實現對共享資源的序列使用。
優點：
1、開銷較小：不需要進入核心，不需要切換執行緒；
2、沒有死鎖：匯流排鎖最長持續為一次read+write的時間；
3、只有寫操作需要使用CAS，讀操作與序列程式碼完全相同，可實現讀寫不互斥。
缺點：
1、程式設計非常複雜，兩行程式碼之間可能發生任何事，很多常識性的假設都不成立。
2、CAS模型覆蓋的情況非常少，無法用CAS實現原子的複數操作。


而在效能層面上，CAS與mutex/readwrite lock各有千秋，簡述如下：
1、單執行緒下CAS的開銷大約為10次加法操作，mutex的上鎖+解鎖大約為20次加法操作，而readwrite lock的開銷則更大一些。
2、CAS的效能為固定值，而mutex則可以通過改變臨界區的大小來調節效能；
3、如果臨界區中真正的修改操作只佔一小部分，那麼用CAS可以獲得更大的併發度。
4、多核CPU中執行緒排程成本較高，此時更適合用CAS。
跳錶和紅黑樹的效能相當，最主要的優勢就是當調整(插入或刪除)時，紅黑樹需要使用旋轉來維護平衡性，這個操作需要動多個節點，在併發時候很難控制。而跳錶插入或刪除時只需定位後插入，插入時只需新增插入的那個節點及其多個層的複製，以及定位和插入的原子性維護。所以它更加可以利用CAS操作來進行無鎖程式設計。

JDK為我們提供了很多Map介面的實現，使得我們可以方便地處理Key-Value的資料結構。

當我們希望快速存取<Key, Value>鍵值對時我們可以使用HashMap。
當我們希望在多執行緒併發存取<Key, Value>鍵值對時，我們會選擇ConcurrentHashMap。
TreeMap則會幫助我們保證資料是按照Key的自然順序或者compareTo方法指定的排序規則進行排序。
OK，那麼當我們需要多執行緒併發存取<Key, Value>資料並且希望保證資料有序時，我們需要怎麼做呢？
也許，我們可以選擇ConcurrentTreeMap。不好意思，JDK沒有提供這麼好的資料結構給我們。
當然，我們可以自己新增lock來實現ConcurrentTreeMap，但是隨著併發量的提升，lock帶來的效能開銷也隨之增大。
Don't cry......，JDK6裡面引入的ConcurrentSkipListMap也許可以滿足我們的需求。

    儲存結構

public class ConcurrentSkipListMap<K,V> extends AbstractMap<K,V> implements ConcurrentNavigableMap<K,V>,  
               Cloneable,java.io.Serializable {  
   /**  Special value used to identify base-level header*/  
   private static final Object BASE_HEADER = new Object();//該值用於標記資料節點的頭結點  
  
    /** The topmost head index of the skiplist.*/  
    private transient volatile HeadIndex<K,V> head;//最高級別索引的索引頭  
    ......  
    /** Nodes hold keys and values, and are singly linked in sorted order, possibly with some intervening marker nodes.  
     The list is headed by a dummy node accessible as head.node. The value field is declared only as Object because it  
     takes special non-V values for marker and header nodes. */  
    static final class Node<K,V> {//儲存鍵值對的資料節點，並且是有序的單鏈表。  
        final K key;  
        volatile Object value;  
        volatile Node<K,V> next;//後繼資料節點  
        ......  
    }  
  
    /** Index nodes represent the levels of the skip list.  
        Note that even though both Nodes and Indexes have forward-pointing fields, they have different types and are handled  
        in different ways, that can't nicely be captured by placing field in a shared abstract class. 
     */  
    static class Index<K,V> {//索引節點  
        final Node<K,V> node;//索引節點關聯的資料節點  
        final Index<K,V> down;//下一級別索引節點（關聯的資料節點相同）  
        volatile Index<K,V> right;//當前索引級別中，後繼索引節點  
        ......  
    }  
    /**  Nodes heading each level keep track of their level.*/  
    static final class HeadIndex<K,V> extends Index<K,V> {//索引頭  
        final int level;//索引級別  
        HeadIndex(Node<K,V> node, Index<K,V> down, Index<K,V> right, int level) {  
            super(node, down, right);  
            this.level = level;  
        }  
    }  
......  
}  

查詢

    //Returns the value to which the specified key is mapped, or null if this map contains no mapping for the key.
    public V get(Object key) {
        return doGet(key);
    }

    private V doGet(Object okey) {
        Comparable<? super K> key = comparable(okey);
        // Loop needed here and elsewhere in case value field goes null just as it is about to be returned, in which case we
        // lost a race with a deletion, so must retry.
        // 這裡採用迴圈的方式來查詢資料節點，是為了防止返回剛好被刪除的資料節點，一旦出現這樣的情況，需要重試。
        for (;;) {
            Node<K,V> n = findNode(key);//根據key查詢資料節點
            if (n == null)
                return null;
            Object v = n.value;
            if (v != null)
                return (V)v;
        }
    }

    /**Returns node holding key or null if no such, clearing out any deleted nodes seen along the way.  
       Repeatedly traverses at base-level looking for key starting at predecessor returned from findPredecessor, 
       processing base-level deletions as encountered. Some callers rely on this side-effect of clearing deleted nodes.
     
     * Restarts occur, at traversal step centered on node n, if:
     *
     *   (1) After reading n's next field, n is no longer assumed predecessor b's current successor, which means that
     *       we don't have a consistent 3-node snapshot and so cannot unlink any subsequent deleted nodes encountered.
     *
     *   (2) n's value field is null, indicating n is deleted, in which case we help out an ongoing structural deletion
     *       before retrying.  Even though there are cases where such  unlinking doesn't require restart, they aren't sorted out
     *       here because doing so would not usually outweigh cost of  restarting.
     *
     *   (3) n is a marker or n's predecessor's value field is null, indicating (among other possibilities) that
     *       findPredecessor returned a deleted node. We can't unlink the node because we don't know its predecessor, so rely
     *       on another call to findPredecessor to notice and return some earlier predecessor, which it will do. This check is
     *       only strictly needed at beginning of loop, (and the b.value check isn't strictly needed at all) but is done
     *       each iteration to help avoid contention with other threads by callers that will fail to be able to change
     *       links, and so will retry anyway.
     *
     * The traversal loops in doPut, doRemove, and findNear all include the same three kinds of checks. And specialized
     * versions appear in findFirst, and findLast and their variants. They can't easily share code because each uses the
     * reads of fields held in locals occurring in the orders they were performed.
     *
     * @param key the key
     * @return node holding key, or null if no such
     */
    private Node<K,V> findNode(Comparable<? super K> key) {
        for (;;) {
            Node<K,V> b = findPredecessor(key);//根據key查詢前驅資料節點
            Node<K,V> n = b.next;
            for (;;) {
                if (n == null)
                    return null;
                Node<K,V> f = n.next;
                //1、b的後繼節點兩次讀取不一致，重試
                if (n != b.next)                // inconsistent read 
                    break;
                Object v = n.value;

                //2、資料節點的值為null，表示該資料節點標記為已刪除，移除該資料節點並重試。
                if (v == null) {                // n is deleted
                    n.helpDelete(b, f);
                    break;
                }
                //3、b節點被標記為刪除，重試
                if (v == n || b.value == null)  // b is deleted
                    break;
                int c = key.compareTo(n.key);
                if (c == 0)//找到返回
                    return n;
                if (c < 0)//給定key小於當前可以，不存在
                    return null;
                b = n;//否則繼續查詢
                n = f;
            }
        }
    }

    /**Returns a base-level node with key strictly less than given key, or the base-level header if there is no such node.  
       Also unlinks indexes to deleted nodes found along the way.  Callers rely on this side-effect of clearing indices to deleted nodes.
     * @param key the key 
     * @return a predecessor of key     */
    //返回“小於且最接近給定key”的資料節點，如果不存在這樣的資料節點就返回最低級別的索引頭。
    private Node<K,V> findPredecessor(Comparable<? super K> key) {
        if (key == null)
            throw new NullPointerException(); // don't postpone errors
        for (;;) {
            Index<K,V> q = head;//從頂層索引開始查詢
            Index<K,V> r = q.right;
            for (;;) {
                if (r != null) {
                    Node<K,V> n = r.node;
                    K k = n.key;
                    if (n.value == null) {//資料節點的值為null,表示該資料節點標記為已刪除，斷開連線並重試
                        if (!q.unlink(r))
                            break;           // restart
                        r = q.right;         // reread r
                        continue;
                    }
                    if (key.compareTo(k) > 0) {//給定key大於當前key，繼續往右查詢
                        q = r;
                        r = r.right;
                        continue;
                    }
                }
                //執行到這裡有兩種情況：
                //1、當前級別的索引查詢結束
                //2、給定key小於等於當前key
                Index<K,V> d = q.down;//在下一級別索引中查詢
                if (d != null) {//如果還存在更低級別的索引，在更低級別的索引中繼續查詢
                    q = d;
                    r = d.right;
                } else
                    return q.node;//如果當前已經是最低級別的索引，當前索引節點關聯的資料節點即為所求
            }
        }
    }

插入

    /**
     * Associates the specified value with the specified key in this map.
     * If the map previously contained a mapping for the key, the old value is replaced.
     *
     * @param key key with which the specified value is to be associated
     * @param value value to be associated with the specified key
     * @return the previous value associated with the specified key, or
     *         <tt>null</tt> if there was no mapping for the key
     * @throws ClassCastException if the specified key cannot be compared
     *         with the keys currently in the map
     * @throws NullPointerException if the specified key or value is null
     */
    public V put(K key, V value) {
        if (value == null)
            throw new NullPointerException();
        return doPut(key, value, false);
    }

    /**
     * Main insertion method.  Adds element if not present, or replaces value if present and onlyIfAbsent is false.
     * @param kkey the key
     * @param value  the value that must be associated with key
     * @param onlyIfAbsent if should not insert if already present
     * @return the old value, or null if newly inserted
     */
    private V doPut(K kkey, V value, boolean onlyIfAbsent) {
        Comparable<? super K> key = comparable(kkey);
        for (;;) {
            Node<K,V> b = findPredecessor(key);//查詢前驅資料節點
            Node<K,V> n = b.next;
            for (;;) {
                if (n != null) {
                    Node<K,V> f = n.next;
                    //1、b的後繼兩次讀取不一致，重試
                    if (n != b.next)               // inconsistent read
                        break;
                    Object v = n.value;
                    //2、資料節點的值為null,表示該資料節點標記為已刪除，移除該資料節點並重試。
                    if (v == null) {               // n is deleted
                        n.helpDelete(b, f);
                        break;
                    }
                    //3、b節點被標記為已刪除，重試
                    if (v == n || b.value == null) // b is deleted
                        break;
                    int c = key.compareTo(n.key);
                    if (c > 0) {//給定key大於當前可以，繼續尋找合適的插入點
                        b = n;
                        n = f;
                        continue;
                    }
                    if (c == 0) {//找到
                        if (onlyIfAbsent || n.casValue(v, value))
                            return (V)v;
                        else
                            break; // restart if lost race to replace value
                    }
                    // else c < 0; fall through
                }
                //沒有找到，新建資料節點
                Node<K,V> z = new Node<K,V>(kkey, value, n);
                if (!b.casNext(n, z))
                    break;         // restart if lost race to append to b
                int level = randomLevel();//隨機的索引級別
                if (level > 0)
                    insertIndex(z, level);
                return null;
            }
        }
    }

    /**
     * Creates and adds index nodes for the given node.
     * @param z the node
     * @param level the level of the index
     */
    private void insertIndex(Node<K,V> z, int level) {
        HeadIndex<K,V> h = head;
        int max = h.level;

        if (level <= max) {//索引級別已經存在，在當前索引級別以及底層索引級別上都新增該節點的索引
            Index<K,V> idx = null;
            for (int i = 1; i <= level; ++i)//首先得到一個包含1~level個索引級別的down關係的連結串列，最後的inx為最高level索引 
                idx = new Index<K,V>(z, idx, null);
            addIndex(idx, h, level);//Adds given index nodes from given level down to 1.新增索引
        } else { // Add a new level 新增索引級別
            /* To reduce interference by other threads checking for empty levels in tryReduceLevel, new levels are added
             * with initialized right pointers. Which in turn requires keeping levels in an array to access them while
             * creating new head index nodes from the opposite direction. */
            level = max + 1;
            Index<K,V>[] idxs = (Index<K,V>[])new Index[level+1];
            Index<K,V> idx = null;
            for (int i = 1; i <= level; ++i)
                idxs[i] = idx = new Index<K,V>(z, idx, null);

            HeadIndex<K,V> oldh;
            int k;
            for (;;) {
                oldh = head;
                int oldLevel = oldh.level;//更新head
                if (level <= oldLevel) { // lost race to add level
                    k = level;
                    break;
                }
                HeadIndex<K,V> newh = oldh;
                Node<K,V> oldbase = oldh.node;
                for (int j = oldLevel+1; j <= level; ++j)
                    newh = new HeadIndex<K,V>(oldbase, newh, idxs[j], j);
                if (casHead(oldh, newh)) {
                    k = oldLevel;
                    break;
                }
            }
            addIndex(idxs[k], oldh, k);
        }
    }

public class ConcurrentSkipListMap<K,V> extends AbstractMap<K,V> implements ConcurrentNavigableMap<K,V>,
               Cloneable,java.io.Serializable {
   /**  Special value used to identify base-level header*/
   private static final Object BASE_HEADER = new Object();//該值用於標記資料節點的頭結點

    /** The topmost head index of the skiplist.*/
    private transient volatile HeadIndex<K,V> head;//最高級別索引的索引頭
    ......
    /** Nodes hold keys and values, and are singly linked in sorted order, possibly with some intervening marker nodes. 
     The list is headed by a dummy node accessible as head.node. The value field is declared only as Object because it 
     takes special non-V values for marker and header nodes. */
    static final class Node<K,V> {//儲存鍵值對的資料節點，並且是有序的單鏈表。
        final K key;
        volatile Object value;
        volatile Node<K,V> next;//後繼資料節點
        ......
    }

    /** Index nodes represent the levels of the skip list. 
        Note that even though both Nodes and Indexes have forward-pointing fields, they have different types and are handled 
        in different ways, that can't nicely be captured by placing field in a shared abstract class.
     */
    static class Index<K,V> {//索引節點
        final Node<K,V> node;//索引節點關聯的資料節點
        final Index<K,V> down;//下一級別索引節點（關聯的資料節點相同）
        volatile Index<K,V> right;//當前索引級別中，後繼索引節點
        ......
    }
    /**  Nodes heading each level keep track of their level.*/
    static final class HeadIndex<K,V> extends Index<K,V> {//索引頭
        final int level;//索引級別
        HeadIndex(Node<K,V> node, Index<K,V> down, Index<K,V> right, int level) {
            super(node, down, right);
            this.level = level;
        }
    }
......
}

參考：

JDK 1.7原始碼

http://blog.csdn.NET/ict2014/article/details/17394259

http://blog.sina.com.cn/s/blog_72995dcc01017w1t.html

https://yq.aliyun.com/articles/38381

http://www.2cto.com/kf/201212/175026.html

http://ifeve.com/cas-skiplist/