CE 221 Data Structures and Algorithms

Transcription

1 CE Data Structures and Algoritms Capter 4: Trees (AVL Trees) Text: Read Weiss, 4.4 Izmir University of Economics

2 AVL Trees An AVL (Adelson-Velskii and Landis) tree is a binary searc tree wit a balance condition. It must be easy to maintain, and it ensures tat te dept of te tree is O (log N). Idea : left and rigt subtrees are of te same eigt (not sallow). Izmir University of Economics

3 Balance Condition for AVL Trees Idea : Every node must ave left and rigt subtrees of te same eigt. Te eigt of an empty tree is - (Only perfectly balanced trees wit k - nodes would satisfy). AVL Tree = BST wit every node satisfying te property tat te eigts of left and rigt subtrees can differ only by one. Te tree on te left is an AVL tree. Te eigt info is kept for eac node. Izmir University of Economics 3

4 Te Heigt of an AVL Tree - I For an AVL Tree wit N nodes, te eigt is at most.44log(n+) In practice it is sligtly more tan log N. Example: An AVL tree of eigt 9 wit fewest nodes (43). Izmir University of Economics 4

5 Te Heigt of an AVL Tree - II Te minimum number of nodes, S(), in an AVL tree of eigt is given by S(0)=, S()=, S() = S(-) + S(-) + Recall Fibonacci Numbers? (F(0) = F() =, F(n)=F(n-) + F(n-)) Claim: S() = F(+) - for all 0. Proof: By induction. Base cases; S(0)==F()-=-, S()==F(3)-=3-. Assume by inductive ypotesis tat claim olds for all eigts k. Ten, S(+) = S() + S(-) + = F(+) - +F(+) - + (by inductive ypotesis) = F(+3) We also know tat F( ) Izmir University of Economics

6 Tus, all te AVL tree operations can be performed in O (log N) time. We will assume lazy deletions. Except possibly insertion (update all te balancing information and keep it balanced) Izmir University of Economics 6 Te Heigt of an AVL Tree - III ) ( ) ( ) ( ) ( ) ( ) ( S S F S F F ) (log 0.37 ).44 log( *log log log ) log( )log ( log ) log( ) ( n O N N N N S N

7 AVL Tree Insertion Example: Let s try to insert 6 into te AVL tree below. Tis would destroy te AVL property of te tree. Ten tis property as to be restored before te insertion step is considered over. It turns out tat tis can always be done wit a simple modification to te tree known as rotation. After an insertion, only te nodes tat are on te pat from te insertion point to te root migt ave teir balance altered. As we follow te pat up to te root and update te balancing information tere may exist nodes wose new balance violates te AVL condition. We will prove tat our rebalancing sceme performed once at te deepest suc node works. 6 Izmir University of Economics 7

8 AVL Tree Rotations Let s call te node to be balanced α. Since any node as at most cildren, and a eigt imbalance requires tat α s subtrees eigt differ by. Tere are four cases to be considered for a violation: ) An insertion into left subtree of te left cild of α. (LL) ) An insertion into rigt subtree of te left cild of α. (LR) 3) An insertion into left subtree of te rigt cild of α. (RL) 4) An insertion into rigt subtree of te rigt cild of α. (RR) Cases and 4 are mirror image symmetries wit respect to α, as are Cases and 3. Consequently; tere are basic cases. Case I (LL, RR) (insertion occurs on te outside) is fixed by a single rotation. Case II (RL, LR) (insertion occurs on te inside) is fixed by double rotation. Izmir University of Economics 8

9 Single Rotation (LL) Let k be te first node on te pat up violating AVL balance property. Figure below is te only possible scenario tat allows k to satisfy te AVL property before te insertion but violate it afterwards. Subtree X as grown an extra level ( levels deeper tan Z now). Y cannot be at te same level as X (k ten out of balance before insertion) and Y cannot be at te same level as Z (ten k would be te first to violate). Izmir University of Economics 9

10 Single Rotation (RR) Note tat in single rotation inorder traversal orders of te nodes are preserved. Te new eigt of te subtree is exactly te same as before. Tus no furter updating of te nodes on te pat to te root is needed. Izmir University of Economics 0

11 Single Rotation-Example I AVL property destroyed by insertion of 6, ten fixed by a single rotation. BST node structure needs an additional field for eigt. Izmir University of Economics

12 Single Rotation-Example II Start wit an initially empty tree and insert items troug 7 sequentially. Dased line joins te two nodes tat are te subject of te rotation. Izmir University of Economics

13 Single Rotation-Example III Insert 6. Balance problem at te root. So a single rotation is performed. Finally, Insert 7 causing anoter rotation. Izmir University of Economics 3

14 Double Rotation (LR, RL) - I Te algoritm tat works for cases and 4 (LL, RR) does not work for cases and 3 (LR, RL). Te problem is tat subtree Y is too deep, and a single rotation does not make it any less deep. Te fact tat subtree Y as ad an item inserted into it guarantees tat it is nonempty. Assume it as a root and two subtrees. Izmir University of Economics 4

15 Double Rotation (LR) - II Below are 4 subtrees connected by 3 nodes. Note tat exactly one of tree B or C is levels deeper tan D (unless all empty). To rebalance, k 3 cannot be root and a rotation between k and k 3 was sown not to work. So te only alternative is to place k as te new root. Tis forces k to be k s left cild and k 3 to be its rigt cild. It also completely determines te locations of all 4 subtrees. AVL balance property is now satisfied. Old eigt of te tree is restored; so, all te balancing and and eigt updating is complete. Izmir University of Economics

16 Double Rotation (RL) - III In bot cases (LR and RL), te effect is te same as rotating between α s cild and grandcild and ten between α and its new cild. Every double rotation can be modelled in terms of single rotations. Inorder traversal orders are always preserved between k, k, and k 3. Double RL = Single LL (α->rigt)+ Single RR (α) Double LR = Single RR (α->left)+ Single LL (α ) Izmir University of Economics 6

17 Double Rotation Example - I Continuing our example, suppose keys 8 troug are inserted in reverse order. Inserting is easy but inserting 4 causes a eigt imbalance at node 7. Te double rotation is an RL type and involves 7,, and 4. Izmir University of Economics 7

18 Double Rotation Example - II insert 3: double rotation is RL tat will involve 6, 4, and 7 and will restore te tree. Izmir University of Economics 8

19 Double Rotation Example - III If is now inserted, tere is an imbalance at te root. Since is not between 4 and 7, we know tat te single rotation RR will work. Izmir University of Economics 9

20 Double Rotation Example - IV Insert : single rotation LL; insert 0: single rotation LL; insert 9: single rotation LL; insert 8: witout a rotation. Izmir University of Economics 0

21 Double Rotation Example - V Insert 8½: double rotation LR. Nodes 8, 8½, 9 are involved. Izmir University of Economics

22 Implementation Issues - I To insert a new node wit key X into an AVL tree T, we recursively insert X into te appropriate subtree of T (let us call tis T LR ). If te eigt of T LR does not cange, ten we are done. Oterwise, if a eigt imbalance appears in T, we do te appropriate single or double rotation depending on X and te keys in T and T LR, update te eigts (making te connection from te rest of te tree above), and are done. Izmir University of Economics

23 Implementation Issues - II Anoter efficiency issue concerns storage of te eigt information. Since all tat is really required is te difference in eigt, wic is guaranteed to be small, we could get by wit two bits (to represent +, 0, -) if we really try. Doing so will avoid repetitive calculation of balance factors but results in some loss of clarity. Te resulting code is somewat more complicated tan if te eigt were stored at eac node. Izmir University of Economics 3

24 Implementation Issues - III First, te declarations. Also, a quick function to return te eigt of a node dealing wit te annoying case of a NULL pointer. Izmir University of Economics 4

25 Implementation - LL /* Tis function can be called only if k as a left */ /* cild. Perform a rotate between k and its left */ /* cild k. Update eigts, ten return new root */ Izmir University of Economics

26 Implementation - RR /* Tis function can be called only if k as a rigt */ /* cild. Perform a rotate between k and its rigt */ /* cild k. Update eigts, ten return new root */ private AvlNode<AnyType> rotatewitrigtcild(avlnode<anytype> k) { AvlNode<AnyType> k = k.rigt; k.rigt = k.left; k.left = k; k.eigt = Mat.max(eigt(k.left), eigt(k.rigt))+; k.eigt = Mat.max(eigt(k.rigt), k.eigt)+; return k; } Izmir University of Economics 6

27 Implementation - LR /* Double LR = Single RR (α->left)+ Single LL (α ) */ /* Tis function can be called only if k3 as a left */ /* cild and k3's left cild k as a rigt cild k */ /* single rotation between k and k followed by */ /* single rotation between k3 and k */ Izmir University of Economics 7

28 Implementation - RL /* Double RL = Single LL (α->rigt)+ Single RR (α) */ /* Tis function can be called only if k as a rigt */ /* cild k3 and k's rigt cild as a left cild k */ private AvlNode<AnyType> doublewitrigtcild(avlnode<anytype> k) { k.rigt = rotatewitleftcild( k.rigt ); return rotatewitrigtcild( k ); } Izmir University of Economics 8

29 AVL Tree Insertion Izmir University of Economics 9

30 Balancing Izmir University of Economics 30

31 AVL Tree Deletion Izmir University of Economics 3

32 Homework Assignments 4.8, 4.9, 4. You are requested to study and solve te exercises. Note tat tese are for you to practice only. You are not to deliver te results to me. Izmir University of Economics 3