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AVL Tree • Definition: – binary search tree – For any node, the heights of left subtree and right subtree differ at most 1. – (if differ at most k, called BH(k) tree). • Theorem (Adel'son-Vel'skii and Landis 1962): – The height of an AVL tree with N internal nodes always lies between log (N +1 ) and 1..4404 log(N + 2) - 0.328 AVL Tree Examples Worst case AVL trees Figures in these slides are copied from: http://www.eli.sdsu.edu/courses/fall96/cs660/notes/avl/avl.html#RTFToC2 AVL Tree • Insertion: – As normal binary search tree insertion – Reconstruct tree by rotation if some nodes become unbalanced • Deletion: – as normal binary search tree deletion – The node deleted will be either a leaf or have just one subtree (this is true for all binary search tree deletion) – if the deleted node has one subtree, then that subtree contains only one node – Traverse up the tree from the deleted node checking the balance of each node, balance the node by rotation if unbalanced • Theorem: – An insertion into an AVL tree requires at most one rotation to rebalance a tree. A deletion may require log(N) rotations to rebalance the tree. AVL tree insertion examples Insert into subtree 3 Insert into subtree 2 or 3 Perform single rotation Perform double rotation AVL tree deletion examples Deletion from subtree 1 Perform single rotation Deletion from subtree 1 Perform double rotation B-Tree • Similar to Red-Black tree or others, balanced search tree • Different from other balanced search tree, nodes may have many children. So mainly used for Disk I/O. • A node x with n[x] keys – – – – has n[x]+1 children, the keys in x are stored in non-decreasing order, the keys in children are delimited by the keys all paths from the root to a leaf are of equal length, i.e. height-balanced An example of B Tree Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. B-tree of degree t • Definition: – Root must have 1 key – Internal node has at least t-1 keys but at most 2t-1 keys, i.e. has at least t but at most 2t children. • Theorem: h≤logt(n+1)/2 • Insertion and deletions: – More complicate but still log (n) – Split and merge operation. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 2-3-4 Tree • A B-tree of degree t=2. – So every node has 2,3,4 children. • Recursive definition – – – – – • Satisfying search tree property nil is a 2-3-4 tree, a leaf is a 2-3-4 tree, an internal node has either 2, 3 or 4 children, all paths from the root to a leaf are of equal length i.e. a 2-3-4-tree is height-balanced Height log(n) in balance search tree of n nodes – Binary tree or non-binary, such as B-tree – Whether all n nodes are placed as leaves or also as internal nodes. • B+-Tree: – real application data nodes are leaves, internal nodes are keys through which the search can be traced to real data. Treap • Consider as Tree + Heap -organize the tree as binary search tree by keys – Assign random chosen priorities to nodes, adjust the tree to obey minheap order property – i.e. Assume that all keys are distinct, so do priorities, for any node u, • if v is a left child of u, key[v]<key[u] • if v is a right child of u, key[v]>key[u] • If v is a child of u, priority[v]>priority[u] • As a result, the expected height is log(n), so are insertion and deletion. Splay tree (See the end of amortized analysis) • • • • A binary search tree (not balanced) Height may be larger than log n, even n-1. However a sequence of n operations takes O(nlog n). Assumptions: data values are distinct and form a totally order set • Operations: – – – – – – Member(i,S) Insert(i,S) Delete(i,S) Merge(S,S’) Split(i,S) All based on • splay(i,S), reorganize tree so that i to be root if iS, otherwise, the new root is either max{k S |k<i} or min{k S |k>i} Splay tree (cont.) • For examples, – merge(S,S’) • Call Splay(, S) and then make S’ the right child – Delete(i,S), call Splay(i,S), remove I, then merge(left(i), right(i)). – Similar for others. – Constant number of splays called. Splay tree (cont.) • Splay operation is based on basic rotate(x) operation (either left or right). • Three cases: – y is the parent of x and x has no grandparent • rotate(x) – x is the left (or right) child of y and y is the left (or right) child of z, • rotate(y) and then rotate(x)\ – x is the left (or right) child of y and y is the right (or left) child of z, • rotate(x) and then rotate(x) Splay tree (cont.) • Credit invariant: Node x always has at least log (x) credits on deposit. – Where (S)=log (|S|) and (x)=(S(x)) • Lemma: – Each operation splay(x,S) requires no more than 3((S)-(x))+1 credits to perform the operation and maintain the credit invariant. • Theorem: – A sequence of m operations involving n inserts takes time O(mlog(n)).