AVL Trees Outline and Required Reading: AVL Trees ( 11.2) CSE 2011, Winter 2017 Instructor: N. Vlajic

Transcription

1 1 AVL Trees Outline and Required Reading: AVL Trees ( 11.2) CSE 2011, Winter 2017 Instructor: N. Vlajic

2 AVL Trees 2 Binary Searc Trees better tan linear dictionaries; owever, te worst case performance of find, insert, remove operations could still be linear (O(n)) Heigt-Balance Property for every internal node v of T, te eigts of te cildren of v can differ at most by 1; if v as only external-node cildren, its eigt is 1 (according to te textbook) AVL Tree any binary searc tree T tat satisfies te eigt-balance property is said to be an AVL Tree If an AVL tree as n nodes, wat is te cost of find, insert, remove?? 51 1

3 ttp://robinsswei.gitub.io/visgraps/avltree.tml 3

4 ttps:// 4

5 AVL Trees (cont.) 5 Balance Factor te difference in eigts of its two subtrees ( R - L ) of a Node BF = R - L L R Balanced Node node wit BF 1; if BF >1, te node is unbalanced tree is left-eavy if BF -1 tree is equal-eigt if BF = 0 tree is rigt-eavy if BF +1 Balance Factor BF of eac node in an AVL tree {-1, 0, 1} of AVL Trees

6 AVL Trees (cont.) 6 Example 1 [ Testing for AVL-tree property. ] Is tis an AVL tree? Balance factor for eac node must be {-1, 0, 1} BF=

7 AVL Trees (cont.) 7 Propos. 1.A: In an AVL tree T storing n items te tree eigt () is O(log 2 n). Propos. 1.B: In an AVL tree of eigt te number of nodes (n) is Ω(2 ). Proof: An AVL is of eigt. How many nodes does te tree ave? An AVL tree of eigt will ave at least one subtree of eigt (-1) and one of eigt (-2). Hence, te number of nodes n AVL in tat tree is: n AVL () 1 + n AVL (-1) + n AVL (-2) > n AVL (-2) n(-2) n(-1) n(-1) n(-1) AVL tree of eigt : possibility 1 AVL tree of eigt : possibility 2

8 AVL Trees (cont.) 8 n AVL () > 2 n AVL (-2) n AVL () at least doubles eac time increases by 2. n AVL () > 2 n AVL (-2) > 2 n AVL (-4) > 2 2 n AVL (-4) > 2 3 n AVL (-6) > 2 i n AVL ( -2 i ) = = 2 (-1)/2 n AVL ( 1 ) Base case: n AVL (1) = 1-2 i = 1 i = (-1)/2 n AVL () > 2 (-1)/2 n AVL () = Ω(2 ) AVL (n) = O(log(n))

9 AVL Trees: Update Operations 9 AVL Trees BST tat guarantee eigt-balance property, i.e. searc time complexity of O(log(n)) if an insert / delete destroys te balance, te balance must be restored troug one or more rotations NOTE: insertion in an AVL tree is performed using te BST s insert metod (wit expandexternal); removal from an AVL tree is performed using te BST s removeelement metod (wit removeaboveexternal) Proposition 2: An AVL tree wit BF=0 for all nodes, cannot be become unbalanced after a single insert / delete operation. Proof: left for exercise

10 AVL Trees: Update Operations (cont.) 10 Rotation restructuring of te tree tat: 1) restores balance property and 2) maintains binary searc tree property inorder traversal must produce te same results on bot original and restructured tree Types of Rotation (1) Single-Left Rotation - on a node and 2 cildren (2) Single-Rigt Rotation - on a node and 2 cildren (3) Double Left-Rigt Rotation - on a node, cild and grandcild (4) Double Rigt-Left Rotation - on a node, cild and grandcild

11 AVL Trees: Update Operations (cont.) 11 Definition of Single-Left and Single-Rigt Rotations Original Tree Single-Left Rotation around node (parent, rigt cild and its subtrees affected) Single-Rigt Rotation around node (parent, left cild and its subtrees affected) R L L R L R L R L,, R,, L,, R L R L R L,, R,, L,, R L,, R,, L,, R Single Left and Rigt Rotation preserve te inorder traversal ordering, i.e. tey preserve binary searc tree property.

12 AVL Trees: Update Operations (cont.) 12 Definition of Double Left-Rigt Rotation Original Tree After (1 st ) Left-Rotation around node After (2 nd ) Rigt-Rotation around node R R L R L L R R L R L L L,, L,, R,, R L,, L,, R,, R L,, L,, R,, R Double Left-Rigt Rotation preserves te inorder traversal ordering. Wat as tis double rotation acieved wit respect to te balance property?

13 AVL Trees: Update Operations (cont.) 13 Definition of Double Rigt-Left Rotation Original Tree After (1 st ) Rigt-Rotation around node After (2 nd ) Left-Rotation around node y L L R L L L R R L R R R L,, L,, R,, R L,, L,, R,, R L,, L,, R,, R Double Rigt-Left Rotation preserves te inorder traversal ordering. Wat as tis double rotation acieved wit respect to te balance property?

14 AVL Trees: Insertion 14 Insertion Scenarios tat Require Reconstruction BF=1 (a) BF()=1 and new node is inserted in R after insertion: BF(y)=2, BF()= +1 BF=0 BF=0 out of balance node and node first affected by insertion are bot rigt-eavy L R L R (b) BF()=1 and new node is inserted in L after insertion: BF()=2, BF()= -1 BF=0 BF=-1 BF=0 (c) BF()=-1 and new node is inserted in L after insertion: BF()=(-2), BF()= -1 out of balance node and node first affected by insertion are bot left-eavy L R L R (d) BF(y)=-1 and new node is inserted in R after insertion: BF()=(-2), BF()= +1

15 AVL Trees: Update Operations on Insertion 15 Types of Imbalance due to Insertion and Corresponding Fixes cause of imbalance BF() BF(RC) required rotation new node inserted in te rigt-subtree of s rigt cild (RC) single-left new node inserted in te left-subtree of s rigt cild (RC) +2-1 double rigt-left new node inserted in te left-subtree of s left cild (LC) -2-1 single-rigt new node inserted in te rigt-subtree of s left cild (LC) double left-rigt

16 AVL Trees: Update Operations on Insertion (cont.) 16 (a) Insertion in Rigt Subtree of Rigt Cild BF of te rigt cild canges form 0 to +1 BF=+1 BF=+2 BF=0 1 L L BF=0 R 2 L BF=+1 +1 L R BF=0 L N R L +1 Original AVL Tree Unbalanced AVL Tree (after insertion) N Restored AVL Tree (after single-left rotation around imbalanced node) Tree eigt is decreased!!! Similarly, in case (c), imbalance caused by insertion in te left subtree of te left cild, can be restored wit a single-rigt rotation.

17 See wat appens for 17 Input: 5, 3, 9, 8, 11 Insert: 15

18 AVL Trees: Update Operations on Insertion (cont.) 18 (b) Insertion in Left Subtree of Rigt Cild BF of te rigt cild canges form 0 to -1 BF=+1 BF=+2 BF=0 BF=0 BF=-1 BF=0 BF=1 L BF=0 L BF= R L R Original AVL Tree 2 N L R R Unbalanced AVL Tree (after insertion) L N L R R Restored AVL Tree (after rigt-left rotation) Tree eigt is decreased!!! Similarly, case (d), imbalance caused by insertion in te rigt subtree of te left cild can be restored wit a left-rigt rotation.

19 See wat appens for 19 Input: 5, 3, 9, 8, 11 Insert: 7

20 AVL Trees: Update Operations on Insertion (cont.) 20 Could case (b) be fixed wit a single-left rotation around??? BF=+2 BF=-2 L 2 L N BF=-1 BF=-1 R R BF=+1 BF=-1 L R L N R 2 Unbalanced AVL Tree (after insertion) AVL Tree after single-left rotation

21 See wat appens for 21 Input: 15, 10, 17, 12, 9 Insert: 5

22 See wat appens for 22 Input: 15, 10, 17, 12, 9 Insert: 11

23 AVL Trees: Update Operations on Insertion (cont.) 23 Rotation-Based Algoritm for AVL-Tree Reconstruction if root R of a subtree becomes unbalanced to te rigt (+2), after insertion of a new node if BF(rigtCild(R))=+1 ten { single-left rotation around R } if BF(rigtCild(R))=-1 ten { single-rigt rotation around rigtcild(r); single-left rotation rotation around R } if root R of a subtree becomes unbalanced to te left (-2), after insertion of a new node if BF(leftCild(R))=-1 ten { single-rigt rotation around R } if BF(leftCild(R))=+1 ten { single-left rotation around leftcild(r); single-rigt rotation rotation around R }

24 AVL Trees: Update Operations on Insertion (cont.) 24 Restructure Algoritm AVL-Tree Reconstruction (from textbook!) Part 1: Insertion 1.1 perform a BST insertion of element into Tree (T), and let N be te newly-created node containing element 1.2 let C be eiter N or an ancestor of N suc tat grandparent(c) is te nearest eigt-unbalanced ancestor of N 1.3 let rotated-st be te subtree obtained by rotating C, parent(c) and grandparent(c) 1.4 let rotated-st replace te subtree of T rooted at grandparent(c) BF =2 GP P BF =1 C BF =1 T 1 T 2 T 3 T 4 N unbalanced AVL tree

25 AVL Trees: Update Operations on Insertion (cont.) 25 Part 2: Rotation 2.1 Let (,, ) be a left-to-rigt (inorder) enumeration of C, parent(c), grandparent(c) 2.2 Let (T 1, T 2, T 3, T 4 ) be a left-to-rigt (inorder) enumeration of te four subtrees of,, (tese subtrees are not rooted at,, ) 2.3 Let rotated-st be constructed as follows: (a) its top node is (b) s left and rigt cildren are and (c) s left and rigt subtrees are T 1 and T 2 (d) s left and rigt subtrees are T 3 and T Return rotated-st. GP P T 1 T 2 T 3 C T 4 N T 3 T 4 T 1 T 2 rotated-st N

26 AVL Trees: Update Operations on Insertion (cont.) 26 Example 2 [ reconstruction after insertion using restructure algoritm ] GP BF =2 P BF =1 C BF =1 T 1 T 2 T 3 T 4 T 1 T 2 T 3 T 4 N N unbalanced AVL tree reconstructed AVL tree

27 AVL Trees: Update Operations on Insertion (cont.) 27 Example 3 [ reconstruction after insertion using restructure algoritm ] GP BF =2 C P BF =1 GP P C BF =1 T 1 T 2 T 3 T 2 T 3 T 4 T 1 N T 4 N AVL after insertion reconstructed AVL tree

28 AVL Trees: Update Operations on Insertion (cont.) 28 Remarks on restructure Algoritm Unifies all 4 types of rotation into a single restructure operation. Preserves te inorder traversal ordering of all te nodes in T. Restores eigt-balance property locally, at te tree involved nodes (,, ). Restores eigt-balance property globally. (Tis property applies to eac of te 4 rotations, as well.) new subtree is sorter by 1, ence all ancestors tat became unbalanced after insertion, are now balanced A BF =2 B BF <=1 T +1 T

29 AVL Trees: Update Operations on Insertion (cont.) 29 Example 4 [ AVL tree rebalancing ] For te given unbalanced AVL tree, restore its balance by using one of te four rotations or restructure algoritm

30 AVL Trees: Update Operations on Insertion (cont.) 30 =6 bf=2 7 =1 =2 1 2 =1 =3 4 =1 = =3 =1 = bf=-2 =2 bf=+1 nearest unbalanced 8 =5 13 bf=-2 =4 bf=-2 = =1 new 15 =1 In tis case, insertion of a new node as caused imbalance of all its ancestors (starting from te grandparent)!!!

31 AVL Trees: Update Operations on Insertion (cont.) 31 =5 bf=1 7 =3 4 =4 bf=-1 13 =2 2 =2 6 =3 bf=-1 11 =2 15 =1 1 =1 =1 3 5 =1 = = After only one rotation/restructuring, te eigt of te unbalanced subtree as restored its old value all ancestors ave restored teir balance. In te case of node insertion, only single rotation restores te balance in te entire AVL tree!!! Wat sould we do in case of deletion!?

32 BF= BF= BF=1 7 7 BF=2!!! BF= BF= after deletion -2 after restructure In te case of node deletion, by restoring te balance of one node we could destroy te balance of one of its ancestors!!!

33 AVL Trees: Removal 33 AVL-Tree Reconstruction after Deletion using restructure Algoritm Let us assume tat an AVL Tree as become unbalanced after performing removeaboveexternal(w). 1. Let be te first eigt-unbalanced node encountered wile moving up te tree from W. Let be te cild of wit te larger eigt. Let be te cild of wit te larger eigt. (May not be unique.) 2. Do restructure on (,, in order!) to restore balance at te subtree rooted at. 3. Continue cecking for unbalanced nodes iger in te tree! (NOTE: restructure operation on lower nodes can destroys te balance of iger nodes.)

34 AVL Trees: Removal (cont.) 34 Example 5 [ reconstruction after deletion using restructure algoritm ] T original AVL tree T2 T3 unbalanced AVL tree T4

35 AVL Trees: Removal (cont.) T T3 T4 T2 reconstructed AVL tree

36 AVL Trees: Implementation 36 Link-Structure vs. Array link-structure better - restructure operation can be performed in O(1) time restructure in array implementation very costly if index-updates are required Computation of BFs find & store eac node s eigt in te node itself; BF(v) = BF(rigtCild(v).) BF(leftCild(v).) cost of finding all BFs: O(n) explain! AVLItem class extends Item class (as used in BSTs) defines additional instance variable eigt AVL Tree class extends BinarySearcTree class inerits: size, isempty, find, findall, remove, removeall overrides: insertitem, removeelement

37 AVL Trees: Implementation (cont.) 37 Link Structure for AVL Trees - Performance Metod size, isempty find, insert, remove Time O(1) O(log(n)) find - O(log(n)) as down-pase (searc ) only insert - O(log(n)) initial findelement runs in O(log(n)) restructuring up te tree runs in O(log(n)) down pase up pase

38 AVL Trees: Exercise 38 Example 6 [ AVL tree rebalancing ] For te given unbalanced AVL tree, restore its balance by using one of te four rotations or restructure algoritm newly inserted node

39 39 =1 =2 1 2 =1 =5 bf= =2 bf=-1 =4 bf=+1 unbalanced 7 6 =3 15 bf=-1 STEP 1: identify te newly inserted node tat as caused imbalanced STEP 2: ceck BF of all nodes from te newly inserted one up to te root (only!) =1 5 =1 =2 bf= =1 STEP 3: identify te lowest unbalanced node and perform rotation/restructuring around tat node new

40 AVL Trees: Questions 40 Q.1 Assume tat te following tree is a standard binary searc tree (sorted lexicograpically). Draw tis tree after te insertion of a new node wit element=r. G B S P Now, assume tat te tree in te picture is an AVL tree. Draw te AVL tree tat will result wen you insert a new node wit element=r. Finally, assume tat te pictured above is an AVL tree. Draw te AVL tree tat will result wen you insert a new node wit element=t.