L09 - Balanced search trees.pdf

# L09 - Balanced search trees.pdf - Balanced search trees(2,3...

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Balanced search trees

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(2,3) Trees
Binary search trees (BST) } Recall the binary search tree data structure © University of Sydney 3

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© Goodrich and Tamassia 4 Binary Search Trees (review) } A binary search tree is a binary tree storing keys (or key-value entries) at its internal nodes and satisfying the following property: } Let u , v , and w be three nodes such that u is in the left subtree of v and w is in the right subtree of v . We have key ( u ) key ( v ) key ( w ) } External nodes do not store items } An inorder traversal of a binary search trees visits the keys in increasing order 6 9 2 4 1 8
© Goodrich and Tamassia 5 Search (review) } To search for a key k , we trace a downward path starting at the root } The next node visited depends on the comparison of k with the key of the current node } If we reach a leaf, the key is not found } Example: get (4): } Call TreeSearch(4,root) } The algorithms for floorEntry and ceilingEntry are similar Algorithm TreeSearch ( k , v ) if T.isExternal ( v ) return v if k < key ( v ) return TreeSearch ( k , T.left ( v )) else if k = key ( v ) return v else { k > key ( v ) } return TreeSearch ( k , T.right ( v )) 6 9 2 4 1 8 < > =

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© Goodrich and Tamassia 6 Insertion (review) } To perform operation put (k, o), we search for key k (using TreeSearch) } Assume k is not already in the tree, and let w be the leaf reached by the search } We insert k at node w and expand w into an internal node } Example: insert 5 6 9 2 4 1 8 6 9 2 4 1 8 5 < > > w w
© Goodrich and Tamassia 7 Deletion (review) } To perform operation remove ( k ), we search for key k } Assume key k is in the tree, and let let v be the node storing k } If node v has a leaf child w , we remove v and w from the tree with operation removeExternal ( w ), which removes w and its parent } Example: remove 4 6 9 2 4 1 8 5 v w 6 9 2 5 1 8 < >

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© Goodrich and Tamassia 8 Deletion (cont.) (review) } We consider the case where the key k to be removed is stored at a node v whose children are both internal } we find the internal node w that follows v in an inorder traversal } we copy key ( w ) into node v } we remove node w and its left child z (which must be a leaf) by means of operation removeExternal( z ) } Example: remove 3 3 1 8 6 9 5 v w z 2 5 1 8 6 9 v 2
© Goodrich and Tamassia 9 Performance (review) } Consider an ordered map with n items implemented by means of a binary search tree of height h } the space used is O ( n ) } methods get, floorEntry, ceilingEntry, put and remove take O ( h ) time } The height h is O ( n ) in the worst case and O (log n ) in the best case

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Inorder successors in BSTs (review) } Delete operation on an internal node with two children: } replace with inorder successor } Or, replace with inorder predecessor } Finding inorder successors: } The inorder successor of a node v is the “left-most” node of the right subtree } The inorder successor is either a leaf or an internal node that only a left child } Therefore inorder successors are “easy” to delete from a binary search tree © Goodrich and Tamassia 10
Inorder successor (review) InorderNext(v): (if v has no right subtree, then there is no successor) temp = right child of v while (temp has a left child) temp = left child of temp return temp © Goodrich and Tamassia

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