Splay Trees Goodrich, Tamassia, Dickerson Splay Trees 1

Transcription

1 Spla Trees v Goodrich, Tamassia, Dickerson Spla Trees 1

2 Spla Trees are Binar Search Trees BST Rules: entries 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) (5,H) (5,I) (7,P) (6,Y) all the kes in the blue region are 20 all the kes in the ellow region are Goodrich, Tamassia, Dickerson Spla Trees 2

3 Searching in a Spla Tree: Starts the Same as in a BST Search proceeds down the tree to found item or an eternal node. Eample: Search for time with ke 11. (1,Q) (1,C) (5,H) (7,P) (5,I) (6,Y) 2004 Goodrich, Tamassia, Dickerson Spla Trees 3

4 Eample Searching in a BST, continued search for ke 8, ends at an internal node. (1,Q) (1,C) (5,H) (7,P) (5,I) (6,Y) 2004 Goodrich, Tamassia, Dickerson 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 T 2 T 2 (structure of tree above is not modified) T 2 (structure of tree above is not modified) T Goodrich, Tamassia, Dickerson 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 es no no ig stop is a left-left grandchild? is a right-right grandchild? is a right-left grandchild? is a left-right grandchild? right-rotate about g, right-rotate about p left-rotate about g, left-rotate about p left-rotate about p, right-rotate about g right-rotate left-rotate about es right-rotate about p, about the root the root left-rotate about g 2004 Goodrich, Tamassia, Dickerson Spla Trees 6 es es es ig-ig ig-ig ig-ag ig-ag

7 Visualiing the Splaing Cases T 4 ig-ag T 4 T 2 T 4 ig-ig T 2 T 2 ig w T 4 w T 2 T 4 T 2 T 2 T Goodrich, Tamassia, Dickerson Spla Trees 7

8 Splaing Eample let = is the right child of its parent, which is the left child of the grandparent left-rotate around p, then rightrotate around g g p (1,Q) (1,C) (5,H) (7,P) 1. (before rotating) (5,I) (6,Y) g p p g (1,C) (1,Q) (5,H) (7,P) 2. (1,C) (5,H) (after first rotation) 3. (5,I) (6,Y) (5,I) (6,Y) is not et the root, so we spla again 2004 Goodrich, Tamassia, Dickerson Spla Trees 8 (1,Q) (7,P) (after second rotation)

9 Splaing Eample, Continued now is the left child of the root right-rotate around root (1,Q) (7,P) (1,C) (5,H) (5,I) (6,Y) 1. (before appling rotation) (1,Q) (1,C) (5,H) (7,P) 2. (after rotation) (5,I) (6,Y) is the root, so stop 2004 Goodrich, Tamassia, Dickerson Spla Trees 9

10 Eample Result of Splaing tree might not be more balanced e.g. spla before (1,Q) 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) (5,H) (7,P) (5,I) (6,Y) (1,Q) (1,Q) (1,C) (5,H) (7,P) (1,C) (5,H) (7,P) (5,I) (6,Y) after first spla 2004 Goodrich, Tamassia, Dickerson Spla Trees 10 (5,I) (6,Y) after second spla

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) 2004 Goodrich, Tamassia, Dickerson Spla Trees 11

12 Spla Trees & Ordered Dictionaries which nodes are splaed after each operation? method get(k) if ke found, use that node spla node if ke not found, use parent of ending eternal node put(k,v) use the new node containing the entr inserted erase(k) 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) 2004 Goodrich, Tamassia, Dickerson 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) Goodrich, Tamassia, Dickerson Spla Trees 13

14 Cost per ig ig w T 4 w T 2 T 2 T 4 Doing a ig at costs at most rank () - rank(): cost = rank () + rank () - rank() - rank() < rank () - rank() Goodrich, Tamassia, Dickerson Spla Trees 14

15 Cost per ig-ig and ig-ag T 4 ig-ig T 2 T 2 Doing a ig-ig or ig-ag at costs at most 3(rank () - rank()) - 2 T 4 ig-ag T 4 T 2 T 4 T Goodrich, Tamassia, Dickerson 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 i1 d / 2 i1 cost (3(rank 3(rank( r) i ( ) rank rank ( )) 2004 Goodrich, Tamassia, Dickerson Spla Trees 16 i 0 i1 3(rank( r) rank( )) d ( )) 2( d 2. 2) / 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 2004 Goodrich, Tamassia, Dickerson Spla Trees 17