CS2210 Data Structures and Algorithms. Lecture 9: AVL TREES definition, properties, insertion

Transcription

1 CS2210 Data Structures and Algorithms Lecture 9: AVL TREES definition, roerties, insertion v

2 BST Performance BST ith n nodes and of height h methods find, insert and remove take O(h) time h is O(n) in the orst case and O(log n) in the best case orst case best case 2

3 Balanced Tree Motivation Need to insure height h of BST is O(log n) then orst case comleit for find, insert, remove is O(log n) height h is O(log n) for a balanced tree Informall, a tree is balanced, if for an node the sie of its left subtree not too different from sie of the right subtree or, equivalentl, left and right subtree heights are not too different More formall, a tree is balanced if its height is O(log n) 3

4 Balanced Trees Comlete tree is an eamle of a balanced tree used for a riorit queue hea suorts insert and removemin, but it does not suort find and remove thus hea does not ork for an ordered dictionar There are other eamles of balanced trees that suort find, insert, remove AVL trees 4

5 Definition: Height of a Node height a tree node v is the height of the subtree rooted at node v recall that tree height is the maimum over tree node deths

6 Definition: AVL Tree Inventors: Adel'son-Vel'skii and Landis (1962) "An algorithm for organiation of information", Doklad Akademii Nauk USSR AVL Tree is BST that satisfies the height-balance roert for ever node v, the heights of its left and right children differ b at most 1 Height-balance roert ensures that height is logarithmic in the number of nodes 6

7 Height of an AVL Tree Theorem: Height h of AVL tree storing n kes is O(log n) Proof: Let n(h) be number of internal nodes in smallest AVL tree of height h n(1) = 1 and n(2) = 2 8 n(h) = 1 + n(h-1) + n(h-2) 2 4 h-2 h-1 smallest AVL tree of height h > 1 + n(h-2) + n(h-2) > 2n(h-2) solving: n(h) > 2n(h-2), n(h) > 4n(h-4), n(h) > 8n(n-6),,n(h) > 2 i n(h-2i) base case: h-2i=1, solving for i, e get i=(h-1)/2 therefore n(h) > 2 (h-1)/2 n(1)=2 (h-1)/2 take logarithms of both sides: h < 2log n(h) + 1 h

8 Oerations in AVL Tree The height of an AVL tree is O(log n) Thus find is O(log n) erformed just like in BST since AVL tree is BST Have to sho ho to insert and remove in AVL trees, hile maintaining 1. the height-balance roert 2. the binar search tree order 8

9 Insertion in an AVL Tree Insertion starts as in BST eanding at eternal node Eamle: insert 54 before inserting after inserting insertion node 9

10 Insertion in an AVL Tree After inserting at eternal, heightbalance roert is likel lost Need to restructure the tree Will use ictorial notation

11 Rebalancing After Insertion Node is unbalanced if difference in heights of its left and right children is more than 1 After insertion, onl heights of ancestors of insertion node could change if change, increase b eactl 1 Need to check onl ancestors of Follo the ath from to the root, udating the heights and correcting an unbalanced nodes in about 50% of the cases, insertion causes no unbalance 11

12 Rebalancing After Insertion Suose is the first unbalanced node on the ath from to the root Height difference beteen the left and the right child of is more than 1 tree as balanced before the insertion insertion can change height onl b 1 Thus this height difference is eactl 2 One subtree has height, other height +2 as inserted into the higher subtree the higher subtree could be on the left as ell +2

13 Rebalancing After Insertion Let S be the higher subtree could be either on the left or on the right of as balanced before insertion, so height of S as +1 before insertion S +2 S had at least one internal node before insertion, since its height is +1 Therefore S has a non-leaf root Let be the root of S +1 S before insertion

14 Rebalancing After Insertion Let be the root of S balanced after insertion Eand structure of S before insertion Case 1: both subtrees of S have height +1 S before insertion Case 2: one subtree of S has height, another height 1 ill be inserted in higher subtree -1 imossible, since ould become unbalanced after insertion

15 Before/After Insertion Structure S before insertion S after insertion +1 could have been inserted into the right subtree of 15

16 Eanded Picture after Insertion S S S could be either the left or right subtree of could have been inserted either in left or right subtrees of Let R be the higher subtree of

17 Rebalancing After Insertion Let R be the higher subtree of R contains, hich is an internal node therefore R has at least one internal node Let be the root of R +1 height of R ent from before insertion to +1 after insertion as balanced before insertion and stas balanced after insertion both subtrees of had height -1 before insertion After insertion, one subtree of has height, the other height -1 R R before insertion R after insertion

18 Eanded Picture after Insertion +2-1 could be either left or right child of could be either left or right child of

19 Before Insertion vs after Insertion

20 Rebalance BST order is reserved all kes in gre subtree larger than all kes in ello subtree smaller than

21 Rebalance This case is hen to the right of, to the left of Three other cases to the right of, to the right of to the left of, to the left of to the left of, to the right of

22 Rebalance, other cases to the right of, is to the right of

23 Rebalance, other cases to the left of, is to the left of

24 Rebalance, other cases to the left of, is to the right of

25 Trinode Restructuring trinode restructuring handles all cases Call node as the middle (has middle ke) node as the smallest (has smallest ke) nodes as the largest (has largest ke) Make middle node ne subtree arent smallest node its left child largest node its right child subtrees of middle node are orhaned left subtree, if resent, goes to the ne left child right subtree, if resent, goes to the ne right child

26 PseudoCode for Trinode Restructuring Algorithm TriNodeRestructure(,,) Inut: Unbalanced node, is the taller child of, is the taller child of Outut: osition of the node that goes in the lace of in the tree structure if.right() = and.left() = then a = ; b = ; c = if.right() = and.right() = then a = ; b = ; c = if.left() = and.left() = then a = ; b = ; c = if.left() = and.right() = then a = ; b = ; c = if ( = root) then root = b; //root changes after trinoderestructure b.arent = null else // reconnect arent of to the node relacing if.arent.left() = then MakeLeftChild(.arent, b); else MakeRightChild(.arent, b); if b.left() and b.left() // left orhan resent MakeRightChild(a, b.leftchild); if b.right() and b.right() //right orhan resent MakeLeftChild(c, b.rightchild); MakeLeftChild(b,a); MakeRightChild(b,c); return b

27 PseudoCode for Trinode Restructuring Algorithm TriNodeRestructure(,,) Inut: Unbalanced node, is the taller child of, is the taller child of Outut: osition of the node that goes in the lace of in the tree structure if.right() = and.left() = if.right() = and.right() = if.left() = and.left() = if.left() = and.right() = then a = ; b = ; c = then a = ; b = ; c = then a = ; b = ; c = then a = ; b = ; c = a b c -1

28 PseudoCode for Trinode Restructuring Algorithm TriNodeRestructure(,,) Inut: Unbalanced node, is the taller child of, is the taller child of Outut: osition of the node that goes in the lace of in the tree structure if ( = root ) then root = b; //root changes after trinoderestructure b.arent = null else // reconnect arent of to the node relacing if.arent.left() = then MakeLeftChild(.arent, b) else MakeRightChild(.arent,b) a b c -1 root a b c -1 root

29 Trinode Restructuring Algorithm TriNodeRestructure(,,) Inut: Unbalanced node, is the taller child of, is the taller child of Outut: osition of the node that goes in the lace of in the tree structure a b d c if ( = root ) then -1 root = b; //root changes after trinoderestructure b.arent = null eventuall ant this else // reconnect arent of to the node relacing if.arent.left() = then MakeLeftChild(.arent, b) else MakeRightChild(.arent, b) a d b c -1 a b d c -1

30 Trinode Restructuring a b d c Algorithm TriNodeRestructure(,,) Inut: Unbalanced node, is the taller child of, is the taller child of Outut: osition of the node that goes in the lace of in the tree structure if b.left() and b.left() // left orhan resent MakeRightChild(a, b.leftchild); if b.right() and b.right() //right orhan resent MakeLeftChild(c, b.rightchild); -1 eventuall ant this a b d c a b d c -1-1

31 Trinode Restructuring a b d c Algorithm TriNodeRestructure(,,) Inut: Unbalanced node, is the taller child of, is the taller child of Outut: osition of the node that goes in the lace of in the tree structure if b.left() and b.left() // left orhan resent MakeRightChild(a, b.leftchild); if b.right() and b.right() //right orhan resent MakeLeftChild(c, b.rightchild); -1 eventuall ant this a b d c a b d c -1-1

32 PseudoCode for Trinode Restructuring Algorithm TriNodeRestructure(,,) Inut: Unbalanced node, is the taller child of, is the taller child of Outut: osition of the node that goes in the lace of in the tree structure MakeLeftChild(b,a) MakeRightChild(b,c) return b a b d c a b d c -1-1

33 PseudoCode for Trinode Restructuring MakeLeftChild(a,b) makes node b left child of node a a.leftchild = b b.arent = a MakeRightChild(a,b) makes node b right child of node a a.rightchild = b b.arent = a Trinode restructuring is O(1) constant number of comarisons and assignments 33

34 Insertion and Trinode Restructuring before insert after insert, before trinode restructure after insert and trinode restructure After one trinode restructure, height-balance roert is restored globall can sto checking ancestor ath after trinode restructure

35 Insertion into AVL Tree Algorithm AVLtreeInsert(k,o) Inut: ke k and value o; Outut: node here the entr as inserted = TreeInsert(k,o,T.root) // holds osition of ne entr (k,o) // no need to check and if needed, restore height-balance roert = hile ( null) // traverse u the tree, checking for imbalance setheight() // reset the height of since it ma have changed if getheight(.left)-getheight(.right) > 1 then =TriNodeRestructure(tallerChild(tallerChild()),tallerChild(),) setheight(.left); setheight(.right); setheight(); break // eit hile loo, tree is balanced after 1 trinoderestructure = arent() return setheight() = 1+ma(.left.height,.right.height) tallerchild(v) returns the child of v ith larger height

36 Insertion Eamle Continued after inserting after inserting

37 Inserting in AVL Tree, Summar After inserting into at node, go u the tree, folloing ancestor ath from, checking if an node on this ath has become unbalanced If an unbalanced node is found, erform tri-node restructuring tree becomes balanced and and can return from insert right after trinode restructuring Trace at most one ath in the tree, erforminc constant number of oerations at each node, thus insert is O(log n) 37