5 AVL trees: deletion

Transcription

1 5 AVL trees: deletion

2 Definition of AVL trees Definition: A binary search tree is called AVL tree or height-balanced tree, if for each node v the height of the right subtree h(t r ) of v and the height of the left subtree h(t l ) of v differ by at most 1. Balance factor: Theory 1 - AVL trees: deletion 2

3 Deletion from an AVL tree We roceed similarly to standard search trees: 1. search for the key to be deleted. 2. if the key is not contained, we are done. 3. otherwise we distinguish three cases: (a) the node to be deleted has no internal nodes as its children. (b) the node to be deleted has exactly one internal child node. (c) the node to be deleted has two internal children. After deleting a node the AVL roerty may be violated (similar to insertion). This must be fixed aroriately Theory 1 - AVL trees: deletion 3

4 Examle Theory 1 - AVL trees: deletion 4

5 Node has only leaves at children Done! Call uout()! Call uout()! height {1, 2} Theory 1 - AVL trees: deletion 5

6 Node has only leaves at children Case1: height = 1, bal() = Done! Theory 1 - AVL trees: deletion 6

7 Node has only leaves at children Case 2.1: height = 2, bal() = 1, height(tl) =1, height(tr) =1 Left rotation tr tl tr tl done! Theory 1 - AVL trees: deletion 7

8 Node has only leaves at children Case 2.2: height = 2, bal() = 1, height(tl) =, height(tr) =1 Left rotation bal() = tl tr tl tr Call uout()! Case 2.3: height = 2, bal() = 1, height(tl) =1, height(tr) = w x Right-left rotation w y x y tr Call uout()! tl Theory 1 - AVL trees: deletion 8

9 Node has one internal node as a child Call uout! Theory 1 - AVL trees: deletion 9

10 Node has two internal nodes as children First we roceed just like we do in standard search trees: 1. Relace the content of the node to be deleted by the content of its symmetrical successor q. 2. Then delete node q. Since q can have at most one internal node as a child (the right one), cases 1 and 2 aly for q Theory 1 - AVL trees: deletion 1

11 The method uout The method uout works similarly to uin. It is called recursively along the search ath and adjusts the balance factors via rotations and double rotations. When uout is called for a node, we have (see above): 1. bal() = 2. The height of the subtree rooted in has decreased by 1. uout will be called recursively as long as these conditions are fulfilled (invariant). Again, we distinguish 2 cases, deending on whether Is the left or the right child of its arent φ. Since the two cases are symmetrical, we only consider the case where is the left child of φ Theory 1 - AVL trees: deletion 11

12 Examle Theory 1 - AVL trees: deletion 12

13 Case 1.1: is the left child of φ and bal(φ)= -1 φ -1 φ uout(φ) uout() Since the height of the subtree rooted in has decreased by 1, the balance factor of φ changes to. By this, the height of the subtree rooted in φ has also decreased by 1 and we have to call uout(φ) (the invariant now holds for φ!) Theory 1 - AVL trees: deletion 13

14 Case 1.2: is the left child of φ and bal(φ)= φ φ 1 uout() done! Since the height of the subtree rooted in has decreased by 1, the balance factor of φ changes to 1. Then we are done, because the height of the subtree rooted in φ has not changed Theory 1 - AVL trees: deletion 14

15 Case 1.3: is the left child of φ and bal(φ)= +1 φ 1 uout() q Then the right subtree of φ was higher (by 1) than the left subtree before the deletion. Hence, in the subtree rooted in φ the AVL roerty is now violated. We distinguish three cases according to the balance factor of q Theory 1 - AVL trees: deletion 15

16 Case 1.3.1: bal(q) = φ +1 v w -1 u q w v +1 left rotation done! u 1 h 1 h 1 2 h h + 1 h h 1 2 h +1 3 h Theory 1 - AVL trees: deletion 16

17 Case 1.3.2: bal(q) = +1 φ +1 v r w uout(r) uout() u q w 1 v left rotation u 1 h 1 h 1 2 h 3 h + 1 h h 1 2 h 3 h + 1 Again, the height of the subtree has decreased by 1, while bal(r) = (invariant). Hence we call uout(r) Theory 1 - AVL trees: deletion 17

18 Case 1.3.3: bal(q) = -1 φ v +1 r z uout(r) uout() u q w -1 double rotation right-left v w z 1 u h 1 h h h - 1 h 1 h Since bal(q) = -1, one of the trees 2 or 3 must have height h. Therefore, the height of the comlete subtree has decreased by 1, while bal(r) = (invariant). Hence, we again call uout(r) Theory 1 - AVL trees: deletion 18

19 Observation Unlike insertions, deletions may cause recursive calls of uout after a double rotation. Therefore, in general a single rotation or double rotation is not sufficient to rebalance the tree. There are examles where for all nodes along the search ath rotations or double rotations must be carried out. Since h log 2 (n) + 1, we may conclude that the deletion of a key form an AVL tree with n keys can be carried out in at most O(log n) stes. AVL trees are a worst-case efficient data structure for finding, inserting and deleting keys Theory 1 - AVL trees: deletion 19