Splay Trees. Splay Trees 1

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1 Spla Trees v Spla Trees 1

2 Spla Trees are Binar Search Trees BST Rules: items stored onl at internal nodes kes stored at nodes in the left subtree of v are less than or equal to the ke stored at v kes stored at nodes in the right subtree of v are greater than or equal to the ke stored at v An inorder traversal will return the kes in order note that two kes of equal value ma be wellseparated (1,C) (1,Q) (2,R) (5,H) (8,N) (5,G) (7,P) all the kes in the blue region are 20 all the kes in the ellow region are 20 Spla Trees 2

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4 Eample Searching in a BST, continued search for ke 8, ends at an internal node. (1,Q) (8,N) (1,C) (5,H) (7,P) (2,R) (5,G) Spla Trees 4

5 Spla Trees do Rotations after Ever Operation (Even Search) right rotation new operation: spla splaing moves a node to the root using rotations makes the left child of a node into s parent; becomes the right child of left rotation makes the right child of a node into s parent; becomes the left child of a right rotation about a left rotation about (structure of tree above is not modified) (structure of tree above is not modified) Spla Trees 5

6 start with node Splaing: is a left-left grandchild means is a left child of its parent, which is itself a left child of its parent p is s parent; g is p s parent is the root? no is a child of the root? es is the left child of the root? es ig right-rotate about the root es no no ig stop left-rotate about the root is a left-left grandchild? es is a right-right grandchild? es is a right-left grandchild? es is a left-right grandchild? es ig-ig right-rotate about g, right-rotate about p ig-ig left-rotate about g, left-rotate about p ig-ag left-rotate about p, right-rotate about g ig-ag right-rotate about p, left-rotate about g Spla Trees 6

7 Visualiing the Splaing Cases ig-ag ig-ig ig w w Spla Trees 7

8 Splaing Eample let = (8,N) is the right child of its parent, which is the left child of the grandparent left-rotate around p, then rightrotate around g p g (1,Q) (8,N) (1,C) (5,H) (7,P) 1. (before rotating) (2,R) (5,G) g p (8,N) p (8,N) g (1,Q) (7,P) (1,C) (5,H) (2,R) (5,G) 2. (1,C) (5,H) (after first rotation) 3. (1,Q) (2,R) (7,P) (5,G) (after second rotation) Spla Trees 8 is not et the root, so we spla again

9 Splaing Eample, Continued now is the left child of the root right-rotate around root (8,N) (1,Q) (7,P) (1,C) (5,H) (2,R) (5,G) 1. (before appling rotation) (1,Q) (1,C) (5,H) (7,P) (8,N) 2. (after rotation) (2,R) (5,G) is the root, so stop Spla Trees 9

10 Eample Result of Splaing tree might not be more balanced e.g. spla before (1,Q) (8,N) before, the depth of the shallowest leaf is 3 and the deepest is 7 after, the depth of shallowest leaf is 1 and deepest is 8 (1,C) (2,R) (5,H) (7,P) (5,G) (1,Q) (8,N) (1,Q) (8,N) (1,C) (5,H) (7,P) (1,C) (5,H) (7,P) (2,R) (5,G) after first spla (2,R) (5,G) after second spla Spla Trees 10

11 Spla Tree Definition a spla tree is a binar search tree where a node is splaed after it is accessed (for a search or update) deepest internal node accessed is splaed splaing costs O(h), where h is height of the tree which is still O(n) worst-case O(h) rotations, each of which is O(1) Spla Trees 11

12 Spla Trees & Ordered Dictionaries which nodes are splaed after each operation? method findelement spla node if ke found, use that node if ke not found, use parent of ending eternal node insertelement use the new node containing the item inserted removeelement use the parent of the internal node that was actuall removed from the tree (the parent of the node that the removed item was swapped with) Spla Trees 12

13 Amortied Analsis of Spla Trees Running time of each operation is proportional to time for splaing. Define rank(v) as the logarithm (base 2) of the number of nodes in subtree rooted at v. Costs: ig = $1, ig-ig = $2, ig-ag = $2. Thus, cost for plaing a node at depth d = $d. Imagine that we store rank(v) cber-dollars at each node v of the spla tree (just for the sake of analsis). Spla Trees 13

14 Cost per ig ig w w Doing a ig at costs at most rank () - rank(): cost = rank () + rank () - rank() - rank() < rank () - rank(). Spla Trees 14

15 Cost per ig-ig and ig-ag ig-ig Doing a ig-ig or ig-ag at costs at most 3(rank () - rank()) - 2. Proof: See Theorem 3.9, Page 192. ig-ag Spla Trees 15

16 Cost of Splaing Cost of splaing a node at depth d of a tree rooted at r: at most 3(rank(r) - rank()) - d + 2: Proof: Splaing takes d/2 splaing substeps: cost d / 2 i= 1 d / 2 i= 1 cost (3(rank = 3(rank( r) i i ( ) rank rank 0 ( )) i 1 3(rank( r) rank( )) d ( )) 2( d 2. 2) Spla Trees / d ) + 2 2

17 Performance of Spla Trees Recall: rank of a node is logarithm of its sie. Thus, amortied cost of an spla operation is O(log n). In fact, the analsis goes through for an reasonable definition of rank(). This implies that spla trees can actuall adapt to perform searches on frequentlrequested items much faster than O(log n) in some cases. (See Theorems 3.10 and 3.11.) Spla Trees 17