Java進階：17.3 伸展樹

1、伸展樹簡介

伸展樹(Splay Tree)是特殊的二叉查找樹(BST)。
它的特殊是指，它除了本身是棵二叉查找樹之外，它還具備一個特點: 當某個節點被訪問時，伸展樹會通過旋轉使該節點成爲樹根。這樣做的好處是，下次要訪問該節點時，能夠迅速的訪問到該節點。

通過之前的學習，知道連續m次查找，對於AVL樹來說，共需要O(mlogn)時間。根據局部性原理，我們可以改進AVL樹！！-----> 引入伸展樹

局部性：剛被訪問過的數據，極有可能很快地再次被訪問。下一將要訪問的節點，極有可能就在剛被訪問過的節點附近。

1.2、特性

1.和普通的二叉查找樹相比，具有任何情況下、任何操作的平攤O(log2n)的複雜度，時間性能上更好
2.和一般的平衡二叉樹比如 紅黑樹、AVL樹相比，維護更少的節點額外信息，空間性能更優，同時編程複雜度更低
3.在很多情況下，對於查找操作，後面的查詢和之前的查詢有很大的相關性。這樣每次查詢操作將被查到的節點旋轉到樹的根節點位置，這樣下次查詢操作可以很快的完成
4.可以完成對區間的查詢、修改、刪除等操作，可以實現線段樹和樹狀數組的所有功能

2、 性能評價

無需記錄節點高度或者平衡因子，編程簡單——優於AVL樹
分攤複雜度O(logN) —— 與AVL樹相當
局部性強、緩存命中率極高時(即k<<n<<n)
效率甚至可以更高——自適應的O(logK)
任何連續的m次查找，都可在O(mlogk + nlogn)時間內完成

但是
仍不能保證單詞最壞情況的出現
不適用於對效率敏感的場所

3、逐層伸展（ X )

伸展方式有兩種：1 逐層伸展 2雙層伸展
逐層伸展：一步一步往上，自下而上，逐層單旋。直到V最終被推送至根。
但效率很低，最壞情況：對於下圖依次訪問1~7的話，每一週期總時間O(n^2)，分攤時間O(n)。遠小於AVL樹的O(logN)

Zig-zig / zag-zag 的逐層旋轉：

4、雙層伸展 ( √ )

雙層伸展：向上追溯兩層，而非一層！ （Tarjan提出）

實現O(log2n)量級的平攤複雜度依靠每次對伸展樹進行查詢、修改、刪除操作之後，都進行旋轉操作 Splay(x, root)，該操作將節點x旋轉到樹的根部。

優點：① 摺疊效果：一旦訪問壞節點，對應路徑的長度將隨機減半
② 最好情況不會持續發生，單趟伸展操作，分攤O(logN)時間

伸展樹的旋轉有六種類型，如果去掉鏡像的重複，則爲三種：
zig(zag)、zig-zig(zag-zag)、zig-zag(zag-zig)。

4.1 zig- zig 旋轉

如下圖所示。在逐層伸展方式中，先VP旋轉，再VG旋轉。在雙層旋轉中，先對祖父節點G進行PG的越級旋轉，再進行VP的旋轉！！！（和第3部分的圖對比一下區別）

優點：在一棵退化成單鏈的伸展樹中訪問其最深的節點，經過伸展後樹高大約爲原先的1/2

4.2 zig- zag 旋轉

如下圖所示，VPG三者在之字型鏈上。此時，先對P節點進行PV的zig旋轉，再對G節點進行GV的zag旋轉，最後變爲右圖所示，V成爲P和G的祖先節點。這個方法和AVL樹雙旋完全等效，與逐層伸展別無二致。

4.3 zig / zag 旋轉

如果V只有父親，沒有祖父。就會出現下圖所示。Parent(v) = root(T) 在每輪調整中，這種情況至多（在最後）出現一次。

5、節點

public class mySplayTree<AnyType extends Comparable<? super AnyType>>
{
    /**
     * Construct the tree.
     */
    public mySplayTree( )
    {
        nullNode = new BinaryNode<AnyType>( null );
        nullNode.left = nullNode.right = nullNode;
        root = nullNode;
    }

    private BinaryNode<AnyType> newNode = null;  // Used between different inserts
    private BinaryNode<AnyType> header = new BinaryNode<AnyType>( null );
    private BinaryNode<AnyType> root;
    private BinaryNode<AnyType> nullNode;
}

6、查找

不論成功與否，總會把最後被訪問的節點伸展到根。 —> 動態操作（回想一下之前學的二叉樹、BVL樹的查找都是靜態操作噢）

    private BinaryNode<AnyType> splay( AnyType x, BinaryNode<AnyType> t )
    {
        BinaryNode<AnyType> leftTreeMax, rightTreeMin;

        header.left = header.right = nullNode;
        leftTreeMax = rightTreeMin = header;

        nullNode.element = x;   // Guarantee a match

        for( ; ; )
        {
            int compareResult = x.compareTo( t.element );

            if( compareResult < 0 )
            {
                if( x.compareTo( t.left.element ) < 0 )
                    t = rotateWithLeftChild( t );
                if( t.left == nullNode )
                    break;
                // Link Right
                rightTreeMin.left = t;
                rightTreeMin = t;
                t = t.left;
            }
            else if( compareResult > 0 )
            {
                if( x.compareTo( t.right.element ) > 0 )
                    t = rotateWithRightChild( t );
                if( t.right == nullNode )
                    break;
                // Link Left
                leftTreeMax.right = t;
                leftTreeMax = t;
                t = t.right;
            }
            else
                break;
        }

        leftTreeMax.right = t.left;
        rightTreeMin.left = t.right;
        t.left = header.right;
        t.right = header.left;
        return t;
    }

    public AnyType findMin( )
    {
        BinaryNode<AnyType> ptr = root;

        while( ptr.left != nullNode )
            ptr = ptr.left;

        root = splay( ptr.element, root );
        return ptr.element;
    }

    public AnyType findMax( )
    {

        BinaryNode<AnyType> ptr = root;

        while( ptr.right != nullNode )
            ptr = ptr.right;

        root = splay( ptr.element, root );
        return ptr.element;
    }

7、插入

最直觀的思路：調用BST標準的插入算法，再將新節點伸展到根。但是，其中要調用BST.search()方法，而Splay.search()集成了splay()操作，查找失敗後，小於key的最後一個節點會被伸展到根，所以只用在此處插入即可。
如果要插入的節點V，首先調用Splay.search() 它將查找失敗、並把t伸展到根節點；只需把v插入成t的父節點。

    public void insert( AnyType x )
    {
        if( newNode == null )
            newNode = new BinaryNode<AnyType>( null );
        newNode.element = x;

        if( root == nullNode )
        {
            newNode.left = newNode.right = nullNode;
            root = newNode;
        }
        else
        {
            root = splay( x, root );

            int compareResult = x.compareTo( root.element );

            if( compareResult < 0 )
            {
                newNode.left = root.left;
                newNode.right = root;
                root.left = nullNode;
                root = newNode;
            }
            else
            if( compareResult > 0 )
            {
                newNode.right = root.right;
                newNode.left = root;
                root.right = nullNode;
                root = newNode;
            }
            else
                return;   // No duplicates
        }
        newNode = null;   // So next insert will call new
    }

8、刪除

    public void remove( AnyType x )
    {
        if( !contains( x ) )
            return;

        BinaryNode<AnyType> newTree;

        // If x is found, it will be splayed to the root by contains
        if( root.left == nullNode )
            newTree = root.right;
        else
        {
            // Find the maximum in the left subtree
            // Splay it to the root; and then attach right child
            newTree = root.left;
            newTree = splay( x, newTree );
            newTree.right = root.right;
        }
        root = newTree;
    }

9、代碼實現

// SplayTree class
//
// CONSTRUCTION: with no initializer
//
// ******************PUBLIC OPERATIONS*********************
// void insert( x )       --> Insert x
// void remove( x )       --> Remove x
// boolean contains( x )  --> Return true if x is found
// Comparable findMin( )  --> Return smallest item
// Comparable findMax( )  --> Return largest item
// boolean isEmpty( )     --> Return true if empty; else false
// void makeEmpty( )      --> Remove all items
// ******************ERRORS********************************
// Throws UnderflowException as appropriate

/**
 * Implements a top-down splay tree.
 * Note that all "matching" is based on the compareTo method.
 * @author Mark Allen Weiss
 */

public class mySplayTree<AnyType extends Comparable<? super AnyType>>
{
    /**
     * Construct the tree.
     */
    public mySplayTree( )
    {
        nullNode = new BinaryNode<AnyType>( null );
        nullNode.left = nullNode.right = nullNode;
        root = nullNode;
    }

    private BinaryNode<AnyType> newNode = null;  // Used between different inserts

    /**
     * Insert into the tree.
     * @param x the item to insert.
     */
    public void insert( AnyType x )
    {
        if( newNode == null )
            newNode = new BinaryNode<AnyType>( null );
        newNode.element = x;

        if( root == nullNode )
        {
            newNode.left = newNode.right = nullNode;
            root = newNode;
        }
        else
        {
            root = splay( x, root );

            int compareResult = x.compareTo( root.element );

            if( compareResult < 0 )
            {
                newNode.left = root.left;
                newNode.right = root;
                root.left = nullNode;
                root = newNode;
            }
            else
            if( compareResult > 0 )
            {
                newNode.right = root.right;
                newNode.left = root;
                root.right = nullNode;
                root = newNode;
            }
            else
                return;   // No duplicates
        }
        newNode = null;   // So next insert will call new
    }

    /**
     * Remove from the tree.
     * @param x the item to remove.
     */
    public void remove( AnyType x )
    {
        if( !contains( x ) )
            return;

        BinaryNode<AnyType> newTree;

        // If x is found, it will be splayed to the root by contains
        if( root.left == nullNode )
            newTree = root.right;
        else
        {
            // Find the maximum in the left subtree
            // Splay it to the root; and then attach right child
            newTree = root.left;
            newTree = splay( x, newTree );
            newTree.right = root.right;
        }
        root = newTree;
    }

    /**
     * Find the smallest item in the tree.
     * Not the most efficient implementation (uses two passes), but has correct
     *     amortized behavior.
     * A good alternative is to first call find with parameter
     *     smaller than any item in the tree, then call findMin.
     * @return the smallest item or throw UnderflowException if empty.
     */
    public AnyType findMin( )
    {
        BinaryNode<AnyType> ptr = root;

        while( ptr.left != nullNode )
            ptr = ptr.left;

        root = splay( ptr.element, root );
        return ptr.element;
    }

    /**
     * Find the largest item in the tree.
     * Not the most efficient implementation (uses two passes), but has correct
     *     amortized behavior.
     * A good alternative is to first call find with parameter
     *     larger than any item in the tree, then call findMax.
     * @return the largest item or throw UnderflowException if empty.
     */
    public AnyType findMax( )
    {

        BinaryNode<AnyType> ptr = root;

        while( ptr.right != nullNode )
            ptr = ptr.right;

        root = splay( ptr.element, root );
        return ptr.element;
    }

    /**
     * Find an item in the tree.
     * @param x the item to search for.
     * @return true if x is found; otherwise false.
     */
    public boolean contains( AnyType x )
    {
        if( isEmpty( ) )
            return false;

        root = splay( x, root );

        return root.element.compareTo( x ) == 0;
    }

    /**
     * Make the tree logically empty.
     */
    public void makeEmpty( )
    {
        root = nullNode;
    }

    /**
     * Test if the tree is logically empty.
     * @return true if empty, false otherwise.
     */
    public boolean isEmpty( )
    {
        return root == nullNode;
    }

    private BinaryNode<AnyType> header = new BinaryNode<AnyType>( null ); // For splay

    /**
     * Internal method to perform a top-down splay.
     * The last accessed node becomes the new root.
     * @param x the target item to splay around.
     * @param t the root of the subtree to splay.
     * @return the subtree after the splay.
     */
    private BinaryNode<AnyType> splay( AnyType x, BinaryNode<AnyType> t )
    {
        BinaryNode<AnyType> leftTreeMax, rightTreeMin;

        header.left = header.right = nullNode;
        leftTreeMax = rightTreeMin = header;

        nullNode.element = x;   // Guarantee a match

        for( ; ; )
        {
            int compareResult = x.compareTo( t.element );

            if( compareResult < 0 )
            {
                if( x.compareTo( t.left.element ) < 0 )
                    t = rotateWithLeftChild( t );
                if( t.left == nullNode )
                    break;
                // Link Right
                rightTreeMin.left = t;
                rightTreeMin = t;
                t = t.left;
            }
            else if( compareResult > 0 )
            {
                if( x.compareTo( t.right.element ) > 0 )
                    t = rotateWithRightChild( t );
                if( t.right == nullNode )
                    break;
                // Link Left
                leftTreeMax.right = t;
                leftTreeMax = t;
                t = t.right;
            }
            else
                break;
        }

        leftTreeMax.right = t.left;
        rightTreeMin.left = t.right;
        t.left = header.right;
        t.right = header.left;
        return t;
    }

    /**
     * Rotate binary tree node with left child.
     * For AVL trees, this is a single rotation for case 1.
     */
    private static <AnyType> BinaryNode<AnyType> rotateWithLeftChild( BinaryNode<AnyType> k2 )
    {
        BinaryNode<AnyType> k1 = k2.left;
        k2.left = k1.right;
        k1.right = k2;
        return k1;
    }

    /**
     * Rotate binary tree node with right child.
     * For AVL trees, this is a single rotation for case 4.
     */
    private static <AnyType> BinaryNode<AnyType> rotateWithRightChild( BinaryNode<AnyType> k1 )
    {
        BinaryNode<AnyType> k2 = k1.right;
        k1.right = k2.left;
        k2.left = k1;
        return k2;
    }

    // Basic node stored in unbalanced binary search trees
    private static class BinaryNode<AnyType>
    {
        // Constructors
        BinaryNode( AnyType theElement )
        {
            this( theElement, null, null );
        }

        BinaryNode( AnyType theElement, BinaryNode<AnyType> lt, BinaryNode<AnyType> rt )
        {
            element  = theElement;
            left     = lt;
            right    = rt;
        }

        AnyType element;            // The data in the node
        BinaryNode<AnyType> left;   // Left child
        BinaryNode<AnyType> right;  // Right child
    }

    private BinaryNode<AnyType> root;
    private BinaryNode<AnyType> nullNode;


    // Test program; should print min and max and nothing else
    public static void main( String [ ] args )
    {
        mySplayTree<Integer> t = new mySplayTree<Integer>( );
        final int NUMS = 40000;
        final int GAP  =   307;

        System.out.println( "Checking... (no bad output means success)" );

        for( int i = GAP; i != 0; i = ( i + GAP ) % NUMS )
            t.insert( i );
        System.out.println( "Inserts complete" );

        for( int i = 1; i < NUMS; i += 2 )
            t.remove( i );
        System.out.println( "Removes complete" );

        if( t.findMin() != 2 || t.findMax() != NUMS - 2 )
            System.out.println( "FindMin or FindMax error!" );

        for( int i = 2; i < NUMS; i += 2 )
            if( !t.contains( i ) )
                System.out.println( "Error: find fails for " + i );

        for( int i = 1; i < NUMS; i += 2 )
            if( t.contains( i ) )
                System.out.println( "Error: Found deleted item " + i );
    }
}