第12章 高级数据结构及其实现

自顶向下伸展树

// 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 SplayTree<AnyType extends Comparable<? super AnyType>>
{
    /**
     * Construct the tree.
     */
    public SplayTree( )
    {
        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( )
    {
        if( isEmpty( ) )
            throw new UnderflowException( );

        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( )
    {
        if( isEmpty( ) )
            throw new UnderflowException( );

        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 )
    {
        SplayTree<Integer> t = new SplayTree<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 );
    }
}

红黑树

// RedBlackTree class
//
// CONSTRUCTION: with no parameters
//
// ******************PUBLIC OPERATIONS*********************
// void insert( x )       --> Insert x
// void remove( x )       --> Remove x (unimplemented)
// 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
// void printTree( )      --> Print all items
// ******************ERRORS********************************
// Throws UnderflowException as appropriate

/**
 * Implements a red-black tree.
 * Note that all "matching" is based on the compareTo method.
 * @author Mark Allen Weiss
 */
public class RedBlackTree<AnyType extends Comparable<? super AnyType>>
{
    /**
     * Construct the tree.
     */
    public RedBlackTree( )
    {
        nullNode = new RedBlackNode<>( null );
        nullNode.left = nullNode.right = nullNode;
        header      = new RedBlackNode<>( null );
        header.left = header.right = nullNode;
    }

    /**
     * Compare item and t.element, using compareTo, with
     * caveat that if t is header, then item is always larger.
     * This routine is called if is possible that t is header.
     * If it is not possible for t to be header, use compareTo directly.
     */
    private int compare( AnyType item, RedBlackNode<AnyType> t )
    {
        if( t == header )
            return 1;
        else
            return item.compareTo( t.element );    
    }
    
    /**
     * Insert into the tree.
     * @param item the item to insert.
     */
    public void insert( AnyType item )
    {
        current = parent = grand = header;
        nullNode.element = item;

        while( compare( item, current ) != 0 )
        {
            great = grand; grand = parent; parent = current;
            current = compare( item, current ) < 0 ?
                         current.left : current.right;

                // Check if two red children; fix if so
            if( current.left.color == RED && current.right.color == RED )
                 handleReorient( item );
        }

            // Insertion fails if already present
        if( current != nullNode )
            return;
        current = new RedBlackNode<>( item, nullNode, nullNode );

            // Attach to parent
        if( compare( item, parent ) < 0 )
            parent.left = current;
        else
            parent.right = current;
        handleReorient( item );
    }

    /**
     * Remove from the tree.
     * @param x the item to remove.
     * @throws UnsupportedOperationException if called.
     */
    public void remove( AnyType x )
    {
        throw new UnsupportedOperationException( );
    }

    /**
     * Find the smallest item  the tree.
     * @return the smallest item or throw UnderflowExcepton if empty.
     */
    public AnyType findMin( )
    {
        if( isEmpty( ) )
            throw new UnderflowException( );

        RedBlackNode<AnyType> itr = header.right;

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

        return itr.element;
    }

    /**
     * Find the largest item in the tree.
     * @return the largest item or throw UnderflowExcepton if empty.
     */
    public AnyType findMax( )
    {
        if( isEmpty( ) )
            throw new UnderflowException( );

        RedBlackNode<AnyType> itr = header.right;

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

        return itr.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 )
    {
        nullNode.element = x;
        current = header.right;

        for( ; ; )
        {
            if( x.compareTo( current.element ) < 0 )
                current = current.left;
            else if( x.compareTo( current.element ) > 0 ) 
                current = current.right;
            else if( current != nullNode )
                return true;
            else
                return false;
        }
    }

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

    /**
     * Print the tree contents in sorted order.
     */
    public void printTree( )
    {
        if( isEmpty( ) )
            System.out.println( "Empty tree" );
        else
            printTree( header.right );
    }
    
    /**
     * Internal method to print a subtree in sorted order.
     * @param t the node that roots the subtree.
     */
    private void printTree( RedBlackNode<AnyType> t )
    {
        if( t != nullNode )
        {
            printTree( t.left );
            System.out.println( t.element );
            printTree( t.right );
        }
    }
     
    /**
     * Test if the tree is logically empty.
     * @return true if empty, false otherwise.
     */
    public boolean isEmpty( )
    {
        return header.right == nullNode;
    }

    /**
     * Internal routine that is called during an insertion
     * if a node has two red children. Performs flip and rotations.
     * @param item the item being inserted.
     */
    private void handleReorient( AnyType item )
    {
            // Do the color flip
        current.color = RED;
        current.left.color = BLACK;
        current.right.color = BLACK;

        if( parent.color == RED )   // Have to rotate
        {
            grand.color = RED;
            if( ( compare( item, grand ) < 0 ) !=
                ( compare( item, parent ) < 0 ) )
                parent = rotate( item, grand );  // Start dbl rotate
            current = rotate( item, great );
            current.color = BLACK;
        }
        header.right.color = BLACK; // Make root black
    }

    /**
     * Internal routine that performs a single or double rotation.
     * Because the result is attached to the parent, there are four cases.
     * Called by handleReorient.
     * @param item the item in handleReorient.
     * @param parent the parent of the root of the rotated subtree.
     * @return the root of the rotated subtree.
     */
    private RedBlackNode<AnyType> rotate( AnyType item, RedBlackNode<AnyType> parent )
    {
        if( compare( item, parent ) < 0 )
            return parent.left = compare( item, parent.left ) < 0 ?
                rotateWithLeftChild( parent.left )  :  // LL
                rotateWithRightChild( parent.left ) ;  // LR
        else
            return parent.right = compare( item, parent.right ) < 0 ?
                rotateWithLeftChild( parent.right ) :  // RL
                rotateWithRightChild( parent.right );  // RR
    }

    /**
     * Rotate binary tree node with left child.
     */
    private RedBlackNode<AnyType> rotateWithLeftChild( RedBlackNode<AnyType> k2 )
    {
        RedBlackNode<AnyType> k1 = k2.left;
        k2.left = k1.right;
        k1.right = k2;
        return k1;
    }

    /**
     * Rotate binary tree node with right child.
     */
    private RedBlackNode<AnyType> rotateWithRightChild( RedBlackNode<AnyType> k1 )
    {
        RedBlackNode<AnyType> k2 = k1.right;
        k1.right = k2.left;
        k2.left = k1;
        return k2;
    }

    private static class RedBlackNode<AnyType>
    {
            // Constructors
        RedBlackNode( AnyType theElement )
        {
            this( theElement, null, null );
        }

        RedBlackNode( AnyType theElement, RedBlackNode<AnyType> lt, RedBlackNode<AnyType> rt )
        {
            element  = theElement;
            left     = lt;
            right    = rt;
            color    = RedBlackTree.BLACK;
        }

        AnyType               element;    // The data in the node
        RedBlackNode<AnyType> left;       // Left child
        RedBlackNode<AnyType> right;      // Right child
        int                   color;      // Color
    }
    
    private RedBlackNode<AnyType> header;
    private RedBlackNode<AnyType> nullNode;

    private static final int BLACK = 1;    // BLACK must be 1
    private static final int RED   = 0;

        // Used in insert routine and its helpers
    private RedBlackNode<AnyType> current;
    private RedBlackNode<AnyType> parent;
    private RedBlackNode<AnyType> grand;
    private RedBlackNode<AnyType> great;


        // Test program
    public static void main( String [ ] args )
    {
        RedBlackTree<Integer> t = new RedBlackTree<>( );
        final int NUMS = 400000;
        final int GAP  =  35461;

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

        for( int i = GAP; i != 0; i = ( i + GAP ) % NUMS )
            t.insert( i );

        if( t.findMin( ) != 1 || t.findMax( ) != NUMS - 1 )
            System.out.println( "FindMin or FindMax error!" );

        for( int i = 1; i < NUMS; i++ )
             if( !t.contains( i ) )
                 System.out.println( "Find error1!" );
    }
}

Treap树

// Treap 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
// void printTree( )      --> Print tree in sorted order
// ******************ERRORS********************************
// Throws UnderflowException as appropriate

/**
 * Implements a treap.
 * Note that all "matching" is based on the compareTo method.
 * @author Mark Allen Weiss
 */
public class Treap<AnyType extends Comparable<? super AnyType>>
{
    /**
     * Construct the treap.
     */
    public Treap( )
    {
        nullNode = new TreapNode<>( null );
        nullNode.left = nullNode.right = nullNode;
        nullNode.priority = Integer.MAX_VALUE;
        root = nullNode;
    }

    /**
     * Insert into the tree. Does nothing if x is already present.
     * @param x the item to insert.
     */
    public void insert( AnyType x )
    {
        root = insert( x, root );
    }

    /**
     * Remove from the tree. Does nothing if x is not found.
     * @param x the item to remove.
     */
    public void remove( AnyType x )
    {
        root = remove( x, root );
    }

    /**
     * Find the smallest item in the tree.
     * @return the smallest item, or throw UnderflowException if empty.
     */
    public AnyType findMin( )
    {
        if( isEmpty( ) )
            throw new UnderflowException( );

        TreapNode<AnyType> ptr = root;

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

        return ptr.element;
    }

    /**
     * Find the largest item in the tree.
     * @return the largest item, or throw UnderflowException if empty.
     */
    public AnyType findMax( )
    {
        if( isEmpty( ) )
            throw new UnderflowException( );

        TreapNode<AnyType> ptr = root;

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

        return ptr.element;
    }

    /**
     * Find an item in the tree.
     * @param x the item to search for.
     * @return true if x is found.
     */
    public boolean contains( AnyType x )
    {
        TreapNode<AnyType> current = root;
        nullNode.element = x;

        for( ; ; )
        {
            int compareResult = x.compareTo( current.element );
            
            if( compareResult < 0 )
                current = current.left;
            else if( compareResult > 0 ) 
                current = current.right;
            else
                return current != nullNode;
        }
    }

    /**
     * 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;
    }

    /**
     * Print the tree contents in sorted order.
     */
    public void printTree( )
    {
        if( isEmpty( ) )
            System.out.println( "Empty tree" );
        else
            printTree( root );
    }

    /**
     * Internal method to insert into a subtree.
     * @param x the item to insert.
     * @param t the node that roots the subtree.
     * @return the new root of the subtree.
     */
    private TreapNode<AnyType> insert( AnyType x, TreapNode<AnyType> t )
    {
        if( t == nullNode )
            return new TreapNode<>( x, nullNode, nullNode );
            
        int compareResult = x.compareTo( t.element );
        
        if( compareResult < 0 )
        {
            t.left = insert( x, t.left );
            if( t.left.priority < t.priority )
                t = rotateWithLeftChild( t );
        }
        else if( compareResult > 0  )
        {
            t.right = insert( x, t.right );
            if( t.right.priority < t.priority )
                t = rotateWithRightChild( t );
        }
        // Otherwise, it's a duplicate; do nothing

        return t;
    }

    /**
     * Internal method to remove from a subtree.
     * @param x the item to remove.
     * @param t the node that roots the subtree.
     * @return the new root of the subtree.
     */
    private TreapNode<AnyType> remove( AnyType x, TreapNode<AnyType> t )
    {
        if( t != nullNode )
        {
            int compareResult = x.compareTo( t.element );
            
            if( compareResult < 0 )
                t.left = remove( x, t.left );
            else if( compareResult > 0 )
                t.right = remove( x, t.right );
            else
            {
                    // Match found
                if( t.left.priority < t.right.priority )
                    t = rotateWithLeftChild( t );
                else
                    t = rotateWithRightChild( t );

                if( t != nullNode )     // Continue on down
                    t = remove( x, t );
                else
                    t.left = nullNode;  // At a leaf
            }
        }
        return t;
    }

    /**
     * Internal method to print a subtree in sorted order.
     * @param t the node that roots the tree.
     */
    private void printTree( TreapNode<AnyType> t )
    {
        if( t != t.left )
        {
            printTree( t.left );
            System.out.println( t.element.toString( ) );
            printTree( t.right );
        }
    }

    /**
     * Rotate binary tree node with left child.
     */
    private TreapNode<AnyType> rotateWithLeftChild( TreapNode<AnyType> k2 )
    {
        TreapNode<AnyType> k1 = k2.left;
        k2.left = k1.right;
        k1.right = k2;
        return k1;
    }

    /**
     * Rotate binary tree node with right child.
     */
    private TreapNode<AnyType> rotateWithRightChild( TreapNode<AnyType> k1 )
    {
        TreapNode<AnyType> k2 = k1.right;
        k1.right = k2.left;
        k2.left = k1;
        return k2;
    }

    private static class TreapNode<AnyType>
    {
            // Constructors
        TreapNode( AnyType theElement )
        {
            this( theElement, null, null );
        }

        TreapNode( AnyType theElement, TreapNode<AnyType> lt, TreapNode<AnyType> rt )
        {
            element  = theElement;
            left     = lt;
            right    = rt;
            priority = randomObj.randomInt( );
        }

            // Friendly data; accessible by other package routines
        AnyType            element;      // The data in the node
        TreapNode<AnyType> left;         // Left child
        TreapNode<AnyType> right;        // Right child
        int                priority;     // Priority

        private static Random randomObj = new Random( );
    }
    
    private TreapNode<AnyType> root;
    private TreapNode<AnyType> nullNode;
 

        // Test program
    public static void main( String [ ] args )
    {
        Treap<Integer> t = new Treap<>( );
        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( NUMS < 40 )
            t.printTree( );
        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 );
    }
}

后缀树

后缀树的一小部分应用列举如下：

1. 找到T中最长的重复子串：遍历树，找到带最大字母深度的内部节点，这就表示了最大的LCP。运行时间是O(|T|)。好可以推广到重复了至少k遍的最长子串。

2. 找到两个字符串T1和T2的最长公共子串：合成一个字符串T1#T2，其中#是不在任一字符串的某个字符。然后为这个字符串建立后缀树，找到最深的那个内部节点，其至少存在一个后缀是在#之前起始的，一个是在#之后起始的。这个完成时间可以做到跟字符串的总长度成正比，并且推广为解决总长度为N的k个字符串问题的O(kN)算法。

3. 找到模式P出现的次数：假设后缀树增加了记录，简单地沿着内部节点向下的路径，使每个叶子保持跟踪其下面的后缀的个数；第一个是P的前缀的内部节点提供了答案；如果这样的节点不存在，答案是0和1，可以通过检查搜索终止处的后缀得到。这个花费的时间跟模式P的长度成正比，而与|T|的大小无关。

4. 找到指定长度L > 1的最常见子串：返回那些字母深度至少为L的节点中规模最大的内部节点。所花时间是O(|T|)。 

import java.util.Arrays;

class SuffixArray
{
    
    /*
     * Create the LCP array from the suffix array
     * @param s the input array populated from 0..N-1, with available pos N
     * @param sa the already-computed suffix array 0..N-1
     * @param LCP the resulting LCP array 0..N-1
     */
    public static void makeLCPArray( int [ ] s, int [ ] sa, int [ ] LCP )
    {
        int N = sa.length;
        int [ ] rank = new int[ N ];
        
        s[ N ] = -1;
        for( int i = 0; i < N; i++ )
            rank[ sa[ i ] ] = i;
        
        int h = 0;
        for( int i = 0; i < N; i++ )
            if( rank[ i ] > 0 )
            {
                int j = sa[ rank[ i ] - 1 ];
                
                while( s[ i + h ] == s[ j + h ] )
                    h++;
                
                LCP[ rank[ i ] ] = h;
                if( h > 0 )
                    h--;
            }
    }
    
    /*
     * Fill in the suffix array information for String str
     * @param str the input String
     * @param sa existing array to place the suffix array
     */
    public static void createSuffixArray( String str, int [ ] sa, int [ ] LCP )
    {        
        int N = str.length( );
        
        int [ ] s = new int[ N + 3 ];
        int [ ] SA = new int[ N + 3 ];
        
        for( int i = 0; i < N; i++ )
            s[ i ] = str.charAt( i );
        
        makeSuffixArray( s, SA, N, 256 );
        
        for( int i = 0; i < N; i++ )
            sa[ i ] = SA[ i ];
        
        makeLCPArray( s, sa, LCP );
    }
    
    
    // find the suffix array SA of s[0..n-1] in {1..K}^n
    // require s[n]=s[n+1]=s[n+2]=0, n>=2
    public static void makeSuffixArray( int [ ] s, int [ ] SA, int n, int K )
    {
        int n0 = ( n + 2 ) / 3;
        int n1 = ( n + 1 ) / 3;
        int n2 = n / 3;
        int t = n0 - n1;  // 1 iff n%3 == 1
        int n12 = n1 + n2 + t;

        int [ ] s12  = new int[ n12 + 3 ];
        int [ ] SA12 = new int[ n12 + 3 ];
        int [ ] s0   = new int[ n0 ];
        int [ ] SA0  = new int[ n0 ];
        
        // generate positions in s for items in s12
        // the "+t" adds a dummy mod 1 suffix if n%3 == 1
        // at that point, the size of s12 is n12
        for( int i = 0, j = 0; i < n + t; i++ )
            if( i % 3 != 0 )
                s12[ j++ ] = i;
        
        int K12 = assignNames( s, s12, SA12, n0, n12, K );
  
        computeS12( s12, SA12, n12, K12 );
        computeS0( s, s0, SA0, SA12, n0, n12, K );
        merge( s, s12, SA, SA0, SA12, n, n0, n12, t );
    }
    
    // Assigns the new supercharacter names.
    // At end of routine, SA will have indices into s, in sorted order
    // and s12 will have new character names
    // Returns the number of names assigned; note that if
    // this value is the same as n12, then SA is a suffix array for s12.
    private static int assignNames( int [ ] s, int [ ] s12, int [ ] SA12,
                                   int n0, int n12, int K )
    {
           // radix sort the new character trios
        radixPass( s12 , SA12, s, 2, n12, K );
        radixPass( SA12, s12 , s, 1, n12, K );  
        radixPass( s12 , SA12, s, 0, n12, K );

          // find lexicographic names of triples
        int name = 0;
        int c0 = -1, c1 = -1, c2 = -1;
      
        for( int i = 0; i < n12; i++ )
        {
            if( s[ SA12[ i ] ] != c0 || s[ SA12[ i ] + 1 ] != c1
                                     || s[ SA12[ i ] + 2 ] != c2 )
            { 
                name++;
                c0 = s[ SA12[ i ] ];
                c1 = s[ SA12[ i ] + 1 ];
                c2 = s[ SA12[ i ] + 2 ];
            }
          
            if( SA12[ i ] % 3 == 1 )
                s12[ SA12[ i ] / 3 ]      = name;
            else
                s12[ SA12[ i ] / 3 + n0 ] = name; 
       }
      
       return name;
    }
    
    
    // stably sort in[0..n-1] with indices into s that has keys in 0..K
    // into out[0..n-1]; sort is relative to offset into s
    // uses counting radix sort
    private static void radixPass( int [ ] in, int [ ] out, int [ ] s, int offset,
                                  int n, int K ) 
    { 
        int [ ] count = new int[ K + 2 ];            // counter array
        
        for( int i = 0; i < n; i++ )
            count[ s[ in[ i ] + offset ] + 1 ]++;    // count occurences
        
        for( int i = 1; i <= K + 1; i++ )            // compute exclusive sums
            count[ i ] += count[ i - 1 ];

        for( int i = 0; i < n; i++ )
            out[ count[ s[ in[ i ] + offset ] ]++ ] = in[ i ];      // sort
    } 
    
    // stably sort in[0..n-1] with indices into s that has keys in 0..K
    // into out[0..n-1]
    // uses counting radix sort
    private static void radixPass( int [ ] in, int [ ] out, int [ ] s, int n, int K ) 
    { 
        radixPass( in, out, s, 0, n, K );
    }
   

    // Compute the suffix array for s12, placing result into SA12
    private static void computeS12( int [ ] s12, int [ ] SA12, int n12, int K12 )
    {
        if( K12 == n12 ) // if unique names, don't need recursion
            for( int i = 0; i < n12; i++ )
                SA12[ s12[i] - 1 ] = i; 
        else
        {
            makeSuffixArray( s12, SA12, n12, K12 );
            // store unique names in s12 using the suffix array 
            for( int i = 0; i < n12; i++ )
                s12[ SA12[ i ] ] = i + 1;
        }
    }
    
    private static void computeS0( int [ ] s, int [ ] s0, int [ ] SA0, int [ ] SA12,
                               int n0, int n12, int K )
    {
        for( int i = 0, j = 0; i < n12; i++ )
            if( SA12[ i ] < n0 )
                s0[ j++ ] = 3 * SA12[ i ];
        
        radixPass( s0, SA0, s, n0, K );
    }
    
    
    // merge sorted SA0 suffixes and sorted SA12 suffixes
    private static void merge( int [ ] s, int [ ] s12,
                              int [ ] SA, int [ ] SA0, int [ ] SA12,
                              int n, int n0, int n12, int t )
    {      
        int p = 0, k = 0;
        
        while( t != n12 && p != n0 )
        {
            int i = getIndexIntoS( SA12, t, n0 ); // S12
            int j = SA0[ p ];                     // S0
            
            if( suffix12IsSmaller( s, s12, SA12, n0, i, j, t ) )
            { 
                SA[ k++ ] = i;
                t++;
            }
            else
            { 
                SA[ k++ ] = j;
                p++;
            }  
        } 
        
        while( p < n0 )
            SA[ k++ ] = SA0[ p++ ];
        while( t < n12 )
            SA[ k++ ] = getIndexIntoS( SA12, t++, n0 ); 
    }
    
    private static int getIndexIntoS( int [ ] SA12, int t, int n0 )
    {
        if( SA12[ t ] < n0 )
            return SA12[ t ] * 3 + 1;
        else
            return ( SA12[ t ] - n0 ) * 3 + 2;
    }
    
    private static boolean leq( int a1, int a2, int b1, int b2 )
      { return a1 < b1 || a1 == b1 && a2 <= b2; }
    
    private static boolean leq( int a1, int a2, int a3, int b1, int b2, int b3 )
      { return a1 < b1 || a1 == b1 && leq( a2, a3,b2, b3 ); }
    
    private static boolean suffix12IsSmaller( int [ ] s, int [ ] s12, int [ ] SA12,
                                             int n0, int i, int j, int t )
    {
        if( SA12[ t ] < n0 )  // s1 vs s0; can break tie after 1 character
            return leq( s[ i ], s12[ SA12[ t ] + n0 ],
                        s[ j ], s12[ j / 3 ] );
        else                  // s2 vs s0; can break tie after 2 characters
            return leq( s[ i ], s[ i + 1 ], s12[ SA12[ t ] - n0 + 1 ],
                        s[ j ], s[ j + 1 ], s12[ j / 3 + n0 ] );
    }

    public static void printV( int [ ]  a, String comment )
    {
        System.out.print( comment + ":" );
        for( int x : a ) 
            System.out.print( x + " " );

        System.out.println( );
    }

    public static boolean isPermutation( int [ ] SA, int n )
    {
        boolean [ ] seen = new boolean [ n ];
        
        for( int i = 0; i < n; i++ )
            seen[ i ] = false;
        
        for( int i = 0; i < n; i++ )
            seen[ SA[ i ] ] = true;
        
        for( int i = 0; i < n; i++ )
            if( !seen[ i ] )
                return false;
        
        return true;
    }

    public static boolean sleq( int  [ ] s1, int start1, int [ ] s2, int start2 )
    {
        for( int i = start1, j = start2; ; i++, j++ )
        {
            if( s1[ i ] < s2[ j ] )
                return true;
            
            if( s1[ i ] > s2[ j ] )
                return false;
        }
    } 

    // Check if SA is a sorted suffix array for s
    public static boolean isSorted( int [ ] SA, int [ ] s, int n )
    {
        for( int i = 0; i < n-1; i++ )
            if( !sleq( s, SA[ i ], s, SA[ i + 1 ] ) )
                return false;
      
        return true;  
    }



    public static void assert0( boolean cond )
    {
        if( !cond )
            throw new AssertionException( );
    }


    public static void test( String str )
    {
        int [ ] sufarr = new int[ str.length( ) ];
        int [ ] LCP = new int[ str.length( ) ];

        createSuffixArray( str, sufarr, LCP );
        
        System.out.println( str + ":" );
        for( int i = 0; i < str.length( ); i++ )
            System.out.println( i + " " + sufarr[ i ] + " " + LCP[ i ] );
        System.out.println( );
    }

    public static void main( String [ ] args )
    {
        test( "banana" );
        test( "aaaaaa" );
    } 

    /*
     * Returns the LCP for any two strings
     */
    public static int computeLCP( String s1, String s2 )
    {
        int i = 0;
        
        while( i < s1.length( ) && i < s2.length( ) && s1.charAt( i ) == s2.charAt( i ) )
            i++;
        
        return i;
    }

    /*
     * Fill in the suffix array and LCP information for String str
     * @param str the input String
     * @param SA existing array to place the suffix array
     * @param LCP existing array to place the LCP information
     * Note: Starting in Java 7, this will use quadratic space.
     */
    public static void createSuffixArraySlow( String str, int [ ] SA, int [ ] LCP )
    {
        if( SA.length != str.length( ) || LCP.length != str.length( ) )
            throw new IllegalArgumentException( );
        
        int N = str.length( );
        
        String [ ] suffixes = new String[ N ];
        for( int i = 0; i < N; i++ )
            suffixes[ i ] = str.substring( i );
        
        Arrays.sort( suffixes );
        
        for( int i = 0; i < N; i++ )
            SA[ i ] = N - suffixes[ i ].length( );
        
        LCP[ 0 ] = 0;
        for( int i = 1; i < N; i++ )
            LCP[ i ] = computeLCP( suffixes[ i - 1 ], suffixes[ i ] );
    }
}


class AssertionException extends RuntimeException
{
}

k - d树

/**
 * Quick illustration of a two-dimensional tree.
 */
public class KdTree<AnyType extends Comparable<? super AnyType>>
{
    private static class KdNode<AnyType>
    {
        AnyType [ ]     data;
        KdNode<AnyType> left;
        KdNode<AnyType> right;

        KdNode( AnyType item[ ] )
        {
            data = (AnyType[]) new Comparable[ 2 ];
            data[ 0 ] = item[ 0 ];
            data[ 1 ] = item[ 1 ];
            left = right = null;
        }
    }

    private KdNode<AnyType> root;

    public KdTree( )
    {
        root = null;
    }

    public void insert( AnyType [ ] x )
    {
        root = insert( x, root, 0 );
    }

    private KdNode<AnyType> insert( AnyType [ ] x, KdNode<AnyType> t, int level )
    {
        if( t == null )
            t = new KdNode<>( x );
        else if( x[ level ].compareTo( t.data[ level ] ) < 0 )
            t.left = insert( x, t.left, 1 - level );
        else
            t.right = insert( x, t.right, 1 - level );
        return t;
    }

    /**
     * Print items satisfying
     * low[ 0 ] <= x[ 0 ] <= high[ 0 ] and
     * low[ 1 ] <= x[ 1 ] <= high[ 1 ].
     */
    public void printRange( AnyType [ ] low, AnyType [ ] high )
    {
        printRange( low, high, root, 0 );
    }

    private void printRange( AnyType [ ] low, AnyType [ ] high,
                             KdNode<AnyType> t, int level )
    {
        if( t != null )
        {
            if( low[ 0 ].compareTo( t.data[ 0 ] ) <= 0 &&
                        low[ 1 ].compareTo( t.data[ 1 ] ) <= 0 &&
                       high[ 0 ].compareTo( t.data[ 0 ] ) >= 0 &&
                       high[ 1 ].compareTo( t.data[ 1 ] ) >= 0 )
                System.out.println( "(" + t.data[ 0 ] + ","
                                        + t.data[ 1 ] + ")" );

            if( low[ level ].compareTo( t.data[ level ] ) <= 0 )
                printRange( low, high, t.left, 1 - level );
            if( high[ level ].compareTo( t.data[ level ] ) >= 0 )
                printRange( low, high, t.right, 1 - level );
        }
    }

    public static void main( String [ ] args )
    {
        KdTree<Integer> t = new KdTree<>( );
        
        System.out.println( "Starting program" );
        for( int i = 300; i < 370; i++ )
        {
            Integer [ ] it = new Integer[ 2 ];
            it[ 0 ] = i;
            it[ 1 ] = 2500 - i;
            t.insert( it );
        }

        Integer [ ] low = { 70, 2186 };
        Integer [ ] high = { 1200, 2200 };

        t.printRange( low, high );
    }
}

配对堆

import java.util.ArrayList;

// PairingHeap class
//
// CONSTRUCTION: with no initializer
//
// ******************PUBLIC OPERATIONS*********************
// Position insert( x )   --> Insert x, return position
// Comparable deleteMin( )--> Return and remove smallest item
// Comparable findMin( )  --> Return smallest item
// boolean isEmpty( )     --> Return true if empty; else false
// int size( )            --> Return size of priority queue
// void makeEmpty( )      --> Remove all items
// void decreaseKey( Position p, newVal )
//                        --> Decrease value in node p
// ******************ERRORS********************************
// Exceptions thrown for various operations

/**
 * Implements a pairing heap.
 * Supports a decreaseKey operation.
 * Note that all "matching" is based on the compareTo method.
 * @author Mark Allen Weiss
 * @see PriorityQueue.Position
 */
public class PairingHeap<AnyType extends Comparable<? super AnyType>>
{    
    /**
     * The Position interface represents a type that can
     * be used for the decreaseKey operation.
     */
    public interface Position<AnyType>
    {
        /**
         * Returns the value stored at this position.
         * @return the value stored at this position.
         */
        AnyType getValue( );
    }
    
    /**
     * Construct the pairing heap.
     */
    public PairingHeap( )
    {
        root = null;
        theSize = 0;
    }

    /**
     * Insert into the priority queue, and return a Position
     * that can be used by decreaseKey.
     * Duplicates are allowed.
     * @param x the item to insert.
     * @return the node containing the newly inserted item.
     */
    public Position<AnyType> insert( AnyType x )
    {
        PairNode<AnyType> newNode = new PairNode<>( x );

        if( root == null )
            root = newNode;
        else
            root = compareAndLink( root, newNode );
            
        theSize++;
        return newNode;
    }

    /**
     * Find the smallest item in the priority queue.
     * @return the smallest item.
     * @throws UnderflowException if pairing heap is empty.
     */
    public AnyType findMin( )
    {
        if( isEmpty( ) )
            throw new UnderflowException( );
        return root.element;
    }

    /**
     * Remove the smallest item from the priority queue.
     * @return the smallest item.
     * @throws UnderflowException if pairing heap is empty.
     */
    public AnyType deleteMin( )
    {
        if( isEmpty( ) )
            throw new UnderflowException( );

        AnyType x = findMin( );
        root.element = null; // null it out in case used in decreaseKey
        if( root.leftChild == null )
            root = null;
        else
            root = combineSiblings( root.leftChild );

        theSize--;
        return x;
    }

    /**
     * Change the value of the item stored in the pairing heap.
     * @param pos any Position returned by insert.
     * @param newVal the new value, which must be smaller
     *    than the currently stored value.
     * @throws IllegalArgumentException if pos is null.
     * @throws IllegalArgumentException if new value is larger than old.
     */
    public void decreaseKey( Position<AnyType> pos, AnyType newVal )
    {
        if( pos == null )
            throw new IllegalArgumentException( "null Position passed to decreaseKey" );

        PairNode<AnyType> p = (PairNode<AnyType>) pos;
        
        if( p.element == null )
            throw new IllegalArgumentException( "pos already deleted" );
        if( p.element.compareTo( newVal ) < 0 )
            throw new IllegalArgumentException( "newVal/oldval: " + newVal + " /" + p.element );
        p.element = newVal;
        if( p != root )
        {
            if( p.nextSibling != null )
                p.nextSibling.prev = p.prev;
            if( p.prev.leftChild == p )
                p.prev.leftChild = p.nextSibling;
            else
                p.prev.nextSibling = p.nextSibling;

            p.nextSibling = null;
            root = compareAndLink( root, p );
        }
    }

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

    /**
     * Returns number of items stored in the priority queue.
     * @return size of the priority queue.
     */
    public int size( )
    {
        return theSize;
    }

    /**
     * Make the priority queue logically empty.
     */
    public void makeEmpty( )
    {
        root = null;
        theSize = 0;
    }
    
    /**
     * Private static class for use with PairingHeap.
     */
    private static class PairNode<AnyType> implements Position<AnyType> 
    {
        /**
         * Construct the PairNode.
         * @param theElement the value stored in the node.
         */
        public PairNode( AnyType theElement )
        {
            element     = theElement;
            leftChild   = null;
            nextSibling = null;
            prev        = null;
        }

        /**
         * Returns the value stored at this position.
         * @return the value stored at this position.
         */
        public AnyType getValue( )
        {
            return element;
        }
        
            // Friendly data; accessible by other package routines
        public AnyType    element;
        public PairNode<AnyType>   leftChild;
        public PairNode<AnyType>   nextSibling;
        public PairNode<AnyType>   prev;
    }

    private PairNode<AnyType> root;
    private int theSize;

    /**
     * Internal method that is the basic operation to maintain order.
     * Links first and second together to satisfy heap order.
     * @param first root of tree 1, which may not be null.
     *    first.nextSibling MUST be null on entry.
     * @param second root of tree 2, which may be null.
     * @return result of the tree merge.
     */
    private PairNode<AnyType> compareAndLink( PairNode<AnyType> first, PairNode<AnyType> second )
    {
        if( second == null )
            return first;

        if( second.element.compareTo( first.element ) < 0 )
        {
            // Attach first as leftmost child of second
            second.prev = first.prev;
            first.prev = second;
            first.nextSibling = second.leftChild;
            if( first.nextSibling != null )
                first.nextSibling.prev = first;
            second.leftChild = first;
            return second;
        }
        else
        {
            // Attach second as leftmost child of first
            second.prev = first;
            first.nextSibling = second.nextSibling;
            if( first.nextSibling != null )
                first.nextSibling.prev = first;
            second.nextSibling = first.leftChild;
            if( second.nextSibling != null )
                second.nextSibling.prev = second;
            first.leftChild = second;
            return first;
        }
    }

    private PairNode<AnyType> [ ] doubleIfFull( PairNode<AnyType> [ ] array, int index )
    {
        if( index == array.length )
        {
            PairNode<AnyType> [ ] oldArray = array;

            array = new PairNode[ index * 2 ];
            for( int i = 0; i < index; i++ )
                array[ i ] = oldArray[ i ];
        }
        return array;
    }
   
        // The tree array for combineSiblings
    private PairNode<AnyType> [ ] treeArray = new PairNode[ 5 ];

    /**
     * Internal method that implements two-pass merging.
     * @param firstSibling the root of the conglomerate;
     *     assumed not null.
     */
    private PairNode<AnyType> combineSiblings( PairNode<AnyType> firstSibling )
    {
        if( firstSibling.nextSibling == null )
            return firstSibling;

            // Store the subtrees in an array
        int numSiblings = 0;
        for( ; firstSibling != null; numSiblings++ )
        {
            treeArray = doubleIfFull( treeArray, numSiblings );
            treeArray[ numSiblings ] = firstSibling;
            firstSibling.prev.nextSibling = null;  // break links
            firstSibling = firstSibling.nextSibling;
        }
        treeArray = doubleIfFull( treeArray, numSiblings );
        treeArray[ numSiblings ] = null;

            // Combine subtrees two at a time, going left to right
        int i = 0;
        for( ; i + 1 < numSiblings; i += 2 )
            treeArray[ i ] = compareAndLink( treeArray[ i ], treeArray[ i + 1 ] );

            // j has the result of last compareAndLink.
            // If an odd number of trees, get the last one.
        int j = i - 2;
        if( j == numSiblings - 3 )
            treeArray[ j ] = compareAndLink( treeArray[ j ], treeArray[ j + 2 ] );

            // Now go right to left, merging last tree with
            // next to last. The result becomes the new last.
        for( ; j >= 2; j -= 2 )
            treeArray[ j - 2 ] = compareAndLink( treeArray[ j - 2 ], treeArray[ j ] );

        return (PairNode<AnyType>) treeArray[ 0 ];
    }

        // Test program
    public static void main( String [ ] args )
    {
        PairingHeap<Integer> h = new PairingHeap<>( );
        int numItems = 10000;
        int i = 37;
        int j;

        System.out.println( "Checking; no bad output is good" );
        for( i = 37; i != 0; i = ( i + 37 ) % numItems )
           h.insert( i );
        for( i = 1; i < numItems; i++ )
            if( h.deleteMin( ) != i )
                System.out.println( "Oops! " + i );

        ArrayList<PairingHeap.Position<Integer>> p = new ArrayList<>( );
        for( i = 0; i < numItems; i++ )
                p.add( null );
        
        for( i = 0, j = numItems / 2; i < numItems; i++, j =(j+71)%numItems )
            p.set( j, h.insert( j + numItems ) );
        for( i = 0, j = numItems / 2; i < numItems; i++, j =(j+53)%numItems )
            h.decreaseKey( p.get( j ), p.get( j ).getValue( ) - numItems );
        i = -1;
        while( !h.isEmpty( ) )
            if( h.deleteMin( ) != ++i )
                System.out.println( "Oops! " + i + " " );
        System.out.println( "Check completed" );
    }
}