自己動手實現java資料結構（九） 跳錶

1. 跳錶介紹
   　　在之前關於資料結構的部落格中已經介紹過兩種最基礎的資料結構：基於連續記憶體空間的向量(線性表)和基於鏈式節點結構的連結串列。

有序的向量可以通過二分查詢以logn對數複雜度完成隨機查詢，但由於插入/刪除元素時可能導致內部陣列內整體資料的平移複製，導致隨機插入/刪除的效率較低。而普通的一維連結串列結構雖然可以做到高效的插入/刪除元素(只是關聯的節點拓撲結構改變)，但是在隨機查詢時卻效率較低，因為其只能從頭/尾節點順序的進行遍歷才能找到對應節點。

電腦科學家發明了能夠兼具向量與連結串列優點的平衡二叉搜尋樹(Balance Binary Search Tree BBST)，這其中紅黑樹是平均效能最高，也最複雜的一種BBST。

正是因為高效能的平衡二叉樹過於複雜，使得電腦科學家另闢蹊徑，發明了被稱為跳錶(Skip List)的資料結構。跳錶通過建立具有層次結構的索引節點，解決了普通連結串列無法進行二分查詢的缺陷。跳錶是基於連結串列的，因此其插入和刪除效率和連結串列一樣優秀；而由於索引節點的引入，也使得跳錶可以以類似二分查詢的形式進行特定元素的搜尋，其查詢效能也達到了O(logn)的對數複雜度，和有序向量以及平衡二叉樹查詢漸進時間複雜度一致。

總的來說，跳錶是一個平均效能很優秀，結構相對簡單的資料結構，在redis以及LevelDB、RocksDB等KV鍵值對資料庫中被廣泛使用。

2. 跳錶工作原理
   跳錶查詢：
   　　跳錶是一個擁有多層索引節點的連結串列，最低層是一個連結串列，儲存著全部的原始資料節點。而索引節點是建立在最底層連結串列節點之上的，且從下到上索引節點的數量逐漸稀疏。

在查詢時，從最高層開始以類似二分查詢的方式跳躍著的逐步向下層逼近查詢最終的目標節點。跳錶插入：
　　瞭解了跳錶的結構，以及其能夠高效隨機查詢的原理之後。很自然的會想到一個問題，跳錶的索引節點是如何維護的？換句話說，當插入/刪除節點時跳錶的索引結構是如何變化的？

要想保證跳錶高效的查詢效率，需要令跳錶相鄰的上下層節點的數量之比大致為1:2，且同一層索引節點的分佈儘量均勻(二分查詢)。

一種自然的想法是每次插入新節點時，詳細的檢查每一層的索引節點，精心維護相鄰水平層索引節點1:2的數量，並控制節點排布的稀疏程度(必要時甚至可以重建整個索引)。但這樣使得跳錶的插入效能大大降低，所以實際上跳錶並沒有選擇這種容易想到但低效方式維護索引。

在跳錶中，通過類似丟硬幣的方式，以概率來決定索引節點是否需要被建立。具體的說，每當插入一個新節點時，根據某種概率演算法計算是否需要為其建立上一層的索引節點。如果判斷需要建立，那麼再接著進行一次基於概率的判斷，如果為真則在更高一層也建立索引節點，並迴圈往復。

假設概率演算法為真的數學期望為1/2，則插入新節點時有50%(1/2)的概率建立第一層的索引節點，25%(1/2^{2)的概率建立第兩層的索引節點，12.5%(1/2}3)的概率建立第三層的索引節點，以此類推。這種基於概率的索引節點建立方式，從巨集觀的數學期望上也能保證相鄰上下層d的索引節點個數之比為1:2。同時由於插入節點數值的大小和插入順序都是完全隨機的，因此從期望上來說，同一水平層索引節點的分佈也是大致均勻的。

總的來說，插入新節點時基於概率的索引建立演算法插入效率相對來說非常高，雖然在極端情況下會導致索引節點上下、水平的分佈不均，但依然是非常優秀的實現。同時，可以通過控制概率演算法的數學期望，靈活的調整跳錶的空間佔用與查詢效率的取捨(概率演算法為真的數學期望從1/2降低到1/4時，建立上級索引的概率降低，索引的密度下降，因此其隨機查詢效率降低，但其索引節點將會大大減少以節約空間，跳錶的這一特性在對空間佔用敏感的記憶體資料庫應用中是很有價值的)。跳錶刪除：
　　在理解了跳錶插入的原理後，跳錶的刪除就很好理解了。當最底層的資料節點被刪除時，只需要將其之上的所有索引節點一併刪除即可。3. 跳錶實現細節
　　下面介紹跳錶的實現細節。本篇部落格的跳錶SkipListMap是用java實現的，為了令程式碼更容易理解，在一些地方選擇了效率相對較低，但更容易理解的實現。

跳錶節點實現
　　為了令整個跳錶的實現更加簡單，區別於jdk的ConcurrentSkipListMap。當前版本跳錶的定義的節點結構既用於最底層的資料節點，也用於上層的索引節點；且節點持有上、下、左、右關聯的四個節點的引用。在每一水平層引入了左右兩個哨兵節點，通過節點中的NodeType列舉區分哨兵節點與普通的索引/資料節點。

為了能夠在後續介紹的插入/刪除等操作中，更加簡單的定位到臨近的節點，簡化程式碼的理解難度。相比jdk、redis等工程化的高效能跳錶實現，當前版本實現的跳錶節點冗餘了一些不必要的欄位屬性以及額外的哨兵節點，額外的浪費了一些空間，但跳錶實現的核心思路是一致的。跳錶的基礎結構
　　跳錶是一個能夠支援高效增刪改查、平均效能很高的資料結構，對標的是紅黑樹為首的平衡二叉搜尋樹。因此在我們參考jdk的實現，跳錶和之前系列部落格中的TreeMap一樣也實現了Map介面。

跳錶的每一個水平層是從左到右，有小到大進行排序的，具體的比較邏輯由compare函式來完成。跳錶查詢實現
　　跳錶實現的一個關鍵就是如何進行快速的隨機查詢。

對於指定key的查詢，首先從最上層的跳錶head節點開始，從左到右的進行比對，當找到一個節點比key小，而且其相鄰的右節點比key大時，則沿著找到的節點進入下一層繼續查詢。(每一個水平層的左哨兵節點視為無窮小，而右哨兵節點視為無窮大)
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https://www.apple.com.cn/cn/search/%E5%B1%B1%E8%A5%BF%E5%8C%BB%E7%A7%91%E5%A4%A7%E5%AD%A67969.1638%28%E8%96%87%29%E6%AF%95%E4%B8%9A%E8%AF%81%E6%A0%B7%E6%9C%AC
https://www.apple.com.cn/cn/search/%E5%B1%B1%E8%A5%BF%E8%B4%A2%E7%BB%8F%E5%A4%A7%E5%AD%A67969.1638%28%E8%96%87%29%E6%AF%95%E4%B8%9A%E8%AF%81%E6%A0%B7%E6%9C%AC
https://www.apple.com.cn/cn/search/%E5%B1%B1%E8%A5%BF%E5%B7%A5%E5%95%86%E5%AD%A6%E9%99%A27969.1638%28%E8%96%87%29%E6%AF%95%E4%B8%9A%E8%AF%81%E6%A0%B7%E6%9C%AC
　　由於跳錶的相鄰上下兩層的節點稀疏程度不同，進入下一水平層更有可能逼近指定key對應的資料節點。通過在水平層大跨步的跳躍，並在對應的節點處進入下一層，迴圈往復的如此操作直到最底層。跳躍式的進行連結串列節點的查詢方式，也是跳錶名稱SkipList的來源。

從程式碼實現中可以看到，跳錶通過建立在其上的索引節點進行查詢，比起原始的一維連結串列，能夠更快的定位到所要查詢的節點。且如果按照概率演算法構建的索引節點分佈比較平均的話，跳錶的查詢效率將能夠媲美有序向量、平衡二叉樹的查詢效率。