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§5 Splay Trees

Target : Any M consecutive tree operations starting from an empty tree take at most O( M log N ) time. Idea : After a node is accessed, it is pushed to the root by a series of AVL tree rotations. §5 Splay Trees. But if one node takes O( N ) time

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§5 Splay Trees

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  1. Target : Any M consecutive tree operations starting from an empty tree take at most O(M log N) time. Idea : After a node is accessed, it is pushed to the root by a series of AVL tree rotations. §5 Splay Trees But if one node takes O(N) time to access, we can keep accessing it for M times, can’t we? Does it mean that every operation takes O(log N) time? So a single operation might still take O(N) time? Then what’s the point? Sure we can – that only means that whenever a node is accessed, it must be moved. The bound is weaker. But the effect is the same: There are no bad input sequences. No. It means that the amortized time is O(log N). 1/12

  2. §5 Splay Trees k5 k5 k5 k4 F k4 k5 k4 F F k1 k3 E k3 k1 E F E k1 k2 k5 k1 D k2 k2 D k3 k2 k4 k2 B k4 F C k1 A B D B k3 E k3 B E C A B C A C A A C D D Does NOT work! 2/12

  3. §5 Splay Trees 2 3 N N 1 N 3 2 2 3 3 1 2 1 N 2 2 3 1 1 1 2 3 An even worse case: Insert: 1, 2, 3, … N Find: 1 1 Find: 2 …… Find: N T (N) = O ( N2 ) 3/12

  4. §5 Splay Trees Rotate X and P G X Single rotation Double rotation P D P G X A A B C D B C G X P D P A X C G B A B C D Try again -- For any nonroot node X , denote its parent by P and grandparent by G: Case 1:P is the root Case 2:P is not the root Zig-zag Zig-zig 4/12

  5. §5 Splay Trees k5 k5 k4 k4 F F k1 k1 k3 E E k2 k2 k2 k3 D k4 k1 B B D A k3 k5 C D F A A C E C B Splaying not only moves the accessed node to the root, but also roughly halves the depth of most nodes on the path. 5/12

  6. §5 Splay Trees 7 7 6 7 1 6 5 6 6 5 4 1 7 4 4 3 4 2 5 1 2 5 2 3 2 1 3 3 Insert: 1, 2, 3, 4, 5, 6, 7 Find: 1 Read the 32-node example given in Figures 4.52 – 4.60 6/12

  7. §5 Splay Trees Deletions: X will be at the root.  Step 1: Find X ; Home work: p.136 4.23 Access a splay tree There will be two subtrees TL and TR .  Step 2: Remove X ; The largest element will be the root of TL , and has no right child.  Step 3: FindMax ( TL ) ;  Step 4: Make TR the right child of the root of TL . Are splay trees really better than AVL trees? 7/12

  8.  Other Operations on Binary Search Trees  Sort: List the elements in increasing order Solution: inorder traversal.  Get Height: Compute the heights of the nodes Solution: postorder traversal.  Get Depth: Compute the depths of the nodes Solution: preorder traversal. 8/12

  9. 21 25 31 48 72 41 12 15 59 84 91 1,4,8,11 12,13 15,18,19 21,24 25,26 31,38 41,43,46 48,49,50 59,68 72,78 84,88 91,92,99 < < < < < < < < < < < §6 B-Trees 【Definition】A B-tree of order M is a tree with the following structural properties: (1) The root is either a leaf or has between 2 and M children. (2) All nonleaf nodes (except the root) have between M/2 and M children. (3) All leaves are at the same depth. Assume each nonroot leaf also has between M/2 and M children. Each interior node contains M pointers to the children. And M  1 smallest key values in the subtrees except the 1st one. All the actual data are stored at the leaves. A B-tree of order 4 (2-3-4 tree) 9/12

  10. §6 B-Trees 22: 22: 16: 11:16 41:58 41:58 1, 8 8,11,12 11,12 16,17 16,17,18 22,23,31 22,23,31 41,52 41,52 58,59,61 58,59,61 16:22 11: 18: 41:58 1, 8 11,12 16,17 18,19 22,23,31 41,52 58,59,61 A B-tree of order 3 (2-3 tree)   16,17,18  Find:52  Insert:18  Insert:1  Insert:19  Insert:28 10/12

  11. §6 B-Trees 22: 16: 41: 11: 18: 28: 58: 1, 8 11,12 16,17 18,19 22,23 28,31 41,52 58,59,61  Insert:70 First find a sibling with 2 keys and adjust. Keep more nodes full.  Deletion is similar to insertion except that the root is removed when it loses two children. 11/12

  12. §6 B-Trees T = O(M) For a general B-tree of order M Btree Insert ( ElementType X, Btree T ) { Search from root to leaf for X and find the proper leaf node; Insert X; while ( this node has M+1 keys ) { split it into 2 nodes with (M+1)/2 and (M+1)/2  keys, respectively; if (this node is the root) create a new root with two children; check its parent; } } Home work: p.138 4.36 Access a 2-3 tree T(M, N) = O( (M/log M) log N ) Depth(M, N) = O( logM/2 N  ) Note: The best choice of M is 3 or 4. TFind(M, N) = O( log N ) 12/12

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