/ / Remove x from the tree / / Throws ItemNotFoundException if x is not in the tree

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template <class Comparable> void SplayTree<Comparable>::remove(const Comparable

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BinaryNode<Comparable> *newTree;

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/ / If x is found, it will be at the root splay ( x, root ) ; if( root->element ! = x ) throw ItemNotFoundException( ) ;

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if( root->left = = nullNode ) newTree = root->right; else i / / Find the maximum in the left subtree / / Splay it to the root; and then attach right child newTree = root->left; splay( x, newTree i ; newTree->right = root->right;

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delete root; root = newTree;

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Figure 2216 The top-down SplayTree class deletion routine

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If the new root contains a value larger than x, the new root and its right subtree become a right subtree of newNode, and the root's left subtree becomes a left subtree of newNode Similar logic applies if the new root contains a value smaller than x In either case, newNode is assigned to root to indicate that it is the new root Then we make newNode NULL at line 38 so that the next call to insert will call new Figure 2216 shows the deletion routine for splay trees A deletion procedure rarely is shorter than the corresponding insertion procedure Next, is the top-down splaying routine Our implementation, shown in Figure 2217 uses a header with left and right pointers to contain eventually the roots of the left and right trees These trees are initially empty, a header is used to correspond to the min or max node of the right or left tree, respectively, in this initial state In this way we can avoid checking for empty trees The first time the left tree becomes nonempty, the header's right pointer is initialized and does not change in the future Thus it contains the root of the left tree at the end of the top-down search Similarly, the header's left pointer eventually contains the root of the

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Implementation of Top-Down Splay

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46

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/ / Internal method to perform a top-down splay / / The last accessed node becomes the new root / / x is the target item to splay around

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/ / t is the root of the subtree to splay template <class Comparable> void SplayTree<Comparable>::splay( const Comparable & x , BinaryNodeiComparable> * & t ) const

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BinaryNode<Comparable> *leftTreeMax, *rightTreeMin; static BinaryNode<Comparable> header; headerleft = headerright = nullNode; 1eftTreeMax = rightTreeMin = &header; nullNode->element = x; / / Guarantee a match for( ; ; ) if( x < t->element

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if ( x < t->left->element ) rotateWithLeftChild( t if( t->left == nullNode ) break; / / Link Right rightTreeMin->left = t; rightTreeMin = t; t = t->left;

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1 else if( t->element < x ) i if( t->right->element < x rotateWithRightChild( t if( t->right == nullNode ) break; / / Link Left 1eftTreeMax->right = t; 1eftTreeMax = t; t = t->right;

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else break; 1eftTreeMax->right = t->left; rightTreeMin->left = t->right; t->left = headerright; t->right = headerleft;

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Figure 2217 A top-down splay algorithm

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Splay Trees

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1 / / Make the tree logically empty 2 template <class Comparable> 3 void SplayTree<Comparable>::makeEmpty()

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f indMax ( ) ; / / Splay max item to root while( ! isEmpty( ) ) remove( root->element i ;

Figure 2218 The makeEmpty routine, which runs in linear time without extra space

right tree The header variable is declared as static because we want to allocate it only once over the entire sequence of splays Before the reassembly at the end of the splay, headerleft and headerright point at R and L, respectively (this is not a typo-follow the links) Note that we are using the simplified top-down splay The destructor is implemented by calling the public makeEmpty routine We would expect that, as in 19, makeEmpty would then call a private recursive routine and perform a postorder deletion of all tree nodes However, that does not always work The problem is that splay trees can be very unbalanced even while giving good performance, and the recursion could run out of stack space Figure 2218 gives a simple alternative that is still O ( N )(though that is far from obvious) It is based on a theorem-which is very difficult to prove-that, if the items of a splay tree are accessed in sequential order, the total cost in linear Similar considerations are required for operator= and any other function that would normally have a recursive implementation However, only makeEmpty (which is called by the destructor) cannot be considered an optional method, and must be implemented