AA-Tree
An AA tree in computer science is a red-black tree with one additional rule. Unlike red-black trees, RED nodes on an AA tree can only be added as a right subchild. In other words, no RED node can be a left subchild. This results in the simulation of a 2-3 tree instead of a 2-3-4 tree, which greatly simplifies the maintenance operations.
The following code shows how to implement a AA tree in Java:
// AATree class
//
// CONSTRUCTION: with no initializer
//
// ******************PUBLIC OPERATIONS*********************
// void insert( x ) --> Insert x
// void remove( x ) --> Remove x
// Comparable find( x ) --> Return item that matches x
// Comparable findMin( ) --> Return smallest item
// Comparable findMax( ) --> Return largest item
// boolean isEmpty( ) --> Return true if empty; else false
// void makeEmpty( ) --> Remove all items
// ******************ERRORS********************************
// Exceptions are thrown by insert and remove if warranted
/**
* Implements an AA-tree.
* Note that all "matching" is based on the compareTo method.
* @author Mark Allen Weiss
*/
public class AATree {
/**
* Construct the tree.
*/
public AATree( ) {
root = nullNode;
}
/**
* Insert into the tree.
* @param x the item to insert.
* @throws DuplicateItemException if x is already present.
*/
public void insert( Comparable x ) {
root = insert( x, root );
}
/**
* Remove from the tree.
* @param x the item to remove.
* @throws ItemNotFoundException if x is not found.
*/
public void remove( Comparable x ) {
deletedNode = nullNode;
root = remove( x, root );
}
/**
* Find the smallest item in the tree.
* @return the smallest item or null if empty.
*/
public Comparable findMin( ) {
if( isEmpty( ) )
return null;
AANode ptr = root;
while( ptr.left != nullNode )
ptr = ptr.left;
return ptr.element;
}
/**
* Find the largest item in the tree.
* @return the largest item or null if empty.
*/
public Comparable findMax( ) {
if( isEmpty( ) )
return null;
AANode 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 the matching item of null if not found.
*/
public Comparable find( Comparable x ) {
AANode current = root;
nullNode.element = x;
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 current.element;
else
return null;
}
}
/**
* 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;
}
/**
* Internal method to insert into a subtree.
* @param x the item to insert.
* @param t the node that roots the tree.
* @return the new root.
* @throws DuplicateItemException if x is already present.
*/
private AANode insert( Comparable x, AANode t ) {
if( t == nullNode )
t = new AANode( x );
else if( x.compareTo( t.element ) < 0 )
t.left = insert( x, t.left );
else if( x.compareTo( t.element ) > 0 )
t.right = insert( x, t.right );
else
throw new DuplicateItemException( x.toString( ) );
t = skew( t );
t = split( t );
return t;
}
/**
* Internal method to remove from a subtree.
* @param x the item to remove.
* @param t the node that roots the tree.
* @return the new root.
* @throws ItemNotFoundException if x is not found.
*/
private AANode remove( Comparable x, AANode t ) {
if( t != nullNode ) {
// Step 1: Search down the tree and set lastNode and deletedNode
lastNode = t;
if( x.compareTo( t.element ) < 0 )
t.left = remove( x, t.left );
else {
deletedNode = t;
t.right = remove( x, t.right );
}
// Step 2: If at the bottom of the tree and
// x is present, we remove it
if( t == lastNode ) {
if( deletedNode == nullNode || x.compareTo( deletedNode.element ) != 0 )
throw new ItemNotFoundException( x.toString( ) );
deletedNode.element = t.element;
t = t.right;
}
// Step 3: Otherwise, we are not at the bottom; rebalance
else
if( t.left.level < t.level - 1 || t.right.level < t.level - 1 ) {
if( t.right.level > --t.level )
t.right.level = t.level;
t = skew( t );
t.right = skew( t.right );
t.right.right = skew( t.right.right );
t = split( t );
t.right = split( t.right );
}
}
return t;
}
/**
* Skew primitive for AA-trees.
* @param t the node that roots the tree.
* @return the new root after the rotation.
*/
private static AANode skew( AANode t ) {
if( t.left.level == t.level )
t = rotateWithLeftChild( t );
return t;
}
/**
* Split primitive for AA-trees.
* @param t the node that roots the tree.
* @return the new root after the rotation.
*/
private static AANode split( AANode t ) {
if( t.right.right.level == t.level ) {
t = rotateWithRightChild( t );
t.level++;
}
return t;
}
/**
* Rotate binary tree node with left child.
*/
private static AANode rotateWithLeftChild( AANode k2 ) {
AANode k1 = k2.left;
k2.left = k1.right;
k1.right = k2;
return k1;
}
/**
* Rotate binary tree node with right child.
*/
private static AANode rotateWithRightChild( AANode k1 ) {
AANode k2 = k1.right;
k1.right = k2.left;
k2.left = k1;
return k2;
}
private static class AANode {
// Constructors
AANode( Comparable theElement ) {
element = theElement;
left = right = nullNode;
level = 1;
}
Comparable element; // The data in the node
AANode left; // Left child
AANode right; // Right child
int level; // Level
}
private AANode root;
private static AANode nullNode;
static // static initializer for nullNode
{
nullNode = new AANode( null );
nullNode.left = nullNode.right = nullNode;
nullNode.level = 0;
}
private static AANode deletedNode;
private static AANode lastNode;
// Test program; should print min and max and nothing else
public static void main( String [ ] args ) {
AATree t = new AATree( );
final int NUMS = 40000;
final int GAP = 307;
System.out.println( "Checking... (no bad output means success)" );
t.insert( new Integer( NUMS * 2 ) );
t.insert( new Integer( NUMS * 3 ) );
for( int i = GAP; i != 0; i = ( i + GAP ) % NUMS )
t.insert( new Integer( i ) );
System.out.println( "Inserts complete" );
t.remove( t.findMax( ) );
for( int i = 1; i < NUMS; i+= 2 )
t.remove( new Integer( i ) );
t.remove( t.findMax( ) );
System.out.println( "Removes complete" );
if( ((Integer)(t.findMin( ))).intValue( ) != 2 ||
((Integer)(t.findMax( ))).intValue( ) != NUMS - 2 )
System.out.println( "FindMin or FindMax error!" );
for( int i = 2; i < NUMS; i+=2 )
if( ((Integer)t.find( new Integer( i ) )).intValue( ) != i )
System.out.println( "Error: find fails for " + i );
for( int i = 1; i < NUMS; i+=2 )
if( t.find( new Integer( i ) ) != null )
System.out.println( "Error: Found deleted item " + i );
}
}
/**
* Exception class for duplicate item errors
* in search tree insertions.
* @author Mark Allen Weiss
*/
public class DuplicateItemException extends RuntimeException {
/**
* Construct this exception object.
*/
public DuplicateItemException( ) {
super( );
}
/**
* Construct this exception object.
* @param message the error message.
*/
public DuplicateItemException( String message ) {
super( message );
}
}
/**
* Exception class for failed finds/removes in search
* trees, hash tables, and list and tree iterators.
* @author Mark Allen Weiss
*/
public class ItemNotFoundException extends RuntimeException {
/**
* Construct this exception object.
*/
public ItemNotFoundException( ) {
super( );
}
/**
* Construct this exception object.
* @param message the error message.
*/
public ItemNotFoundException( String message ) {
super( message );
}
}