Binary Search Tree and AVL Trees. Binary Search Tree. Binary Search Tree. Binary Search Tree. Techniques: How does the BST works?

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1 Binary Searc Tree and AVL Trees Binary Searc Tree A commonly-used data structure for storing and retrieving records in main memory PUC-Rio Eduardo S. Laber Binary Searc Tree Binary Searc Tree A commonly-used data structure for storing and retrieving records in main memory A commonly-used data structure for storing and retrieving records in main memory It guarantees logaritmic cost for various operations as long as te tree is balanced It guarantees logaritmic cost for various operations as long as te tree is balanced It is not surprising tat many tecniques tat maintain balance in BSTs ave received considerable attention over te years Tecniques: How does te BST works? AVL Trees Splay Trees

2 How does te BST works? Fundamental Property: x How does te BST works? Fundamental Property: x y < x How does te BST works? Example: 50, 0, 9,, 79,, 5, 5,,, 5, 9, 7,, 5. Fundamental Property: x y < x x < z eigt of a binary tree eigt of a binary tree At eac level te number of nodes may double, suc tat for a binary tree wit eigt we ave at most: = + nodes

3 eigt of a binary tree At eac level te number of nodes may double, suc tat for a binary tree wit eigt we ave at most: = nodes Or equivalently: eigt of a binary tree At eac level te number of nodes may double, suc tat for a binary tree wit eigt we ave at most: = nodes Or equivalently: A binary searc tree wit n nodes can ave mininum eigt log n BST Te eigt of a binary tree is a limit for te time to find out a given node BST Te eigt of a binary tree is a limit for te time to find out a given node BUT... BST Te eigt of a binary tree is a limit for te time to find out a given node BUT... It is necessary tat te tree is balanced BST Te eigt of a binary tree is a limit for te time to find out a given node BUT... It is necessary tat te tree is balanced ( almost every internal node as cildren)

4 BST Algoritm Complexity of Seacing in balanced BST Algoritm BST(x) If x = root ten element was found Else if x < root ten searc in te left subtree else searc in te rigt subtree O(log n) Including a node in a BST Including a node in a BST Add a new element in te tree at te correct position in order to keep te fundamental property. Add a new element in te tree at te correct position in order to keep te fundamental property. Algoritm Insert(x, T) If x < root ten Insert (x, left tree of T) else Insert (x, rigt tree of T) Removing a node in a BST Removing a node in a BST SITUATIONS: SITUATIONS: Removing a leaf Removing an internal node wit a unique cild Removing an internal node wit two cildren Removing a leaf Removing an internal node wit a unique cild Removing an internal node wit two cildren

5 Removing a Leaf Removing a Leaf Removing a Leaf Removing a node in a BST SITUATIONS: Removing a leaf Removing an internal node wit a unique cild Removing an internal node wit two cildren Removing an internal node wit a unique cild Removing an internal node wit a unique cild It is necessary to correct te pointer, jumping te node: te only grandcild becomes te rigt son. 5

6 Removing an internal node wit a unique cild Removing an internal node wit a unique cild Removing an internal node wit a unique cild Removing a node in a BST SITUATIONS: Removing a leaf Removing an internal node wit a unique cild Removing an internal node wit two cildren Removing an internal node wit two cildren Removing an internal node wit two cildren Find te element wic preceeds te element to be removed considering te ordering (tis corresponds to remove te element most to te rigt from te left subtree)

7 Removing an internal node wit two cildren Removing an internal node wit two cildren Removing an internal node wit two cildren Removing an internal node wit two cildren Find te element wic preceeds te element to be removed considering te ordering (tis corresponds to remove te element most to te rigt from te left subtree) Switc te information of te node to be removed wit te node found Removing an internal node wit two cildren Removing an internal node wit two cildren Find te element wic preceeds te element to be removed considering te ordering (tis corresponds to remove te element most to te rigt from te left subtree) Switc te information of te node to be removed wit te node found Remove te node tat contains te information we want to remove 7

8 Removing an internal node wit two cildren Removing an internal node wit two cildren Removing an internal node wit two cildren Te tree may become unbalanced Te tree may become unbalanced Remove: node Te tree may become unbalanced Remove: node

9 Te tree may become unbalanced Te tree may become unbalanced Remove: node Remove node Remove: node Remove node Te tree may become unbalanced Te tree may become unbalanced Te binary tree may degenerate after operations of insertion and remotion: becoming a list, for example. Te binary tree may degenerate after operations of insertion and remotion: becoming a list, for example. Te access time becomes no longer logaritmic HOW TO SOLVE THIS PROBLEM??? Balanced Trees: Balanced Trees: AVL Trees Splay Trees Treaps Skip Lists AVL Trees Splay Trees Treaps Skip Lists 9

10 AVL TREES (Adelson-Velskii and Landis 9) BST trees tat maintain a reasonable balance all te time. Key idea: if insertion or deletion get te tree out of balance ten fix it immediately All operations insert, delete, can be done on an AVL tree wit N nodes in O(log N) time AVL TREES (Adelson-Velskii and Landis) AVL Tree Property: It is a BST in wic te eigts of te left and rigt subtrees of te root differ by at most and in wic te rigt and left subtrees are also AVL trees AVL TREES (Adelson-Velskii and Landis) AVL TREES Example: AVL Tree Property: It is a BST in wic te eigts of te left and rigt subtrees of te root differ by at most and in wic te rigt and left subtrees are also AVL trees 7 5 Heigt: lengt of te longest pat from te root to a leaf. Let r be te root of an AVL tree of eigt Let N denote te minimum number of nodes in an AVL tree of eigt 0

11 Let r be te root of an AVL tree of eigt Let N denote te minimum number of nodes in an AVL tree of eigt T r Let r be te root of an AVL tree of eigt Let N denote te minimum number of nodes in an AVL tree of eigt T r T e T d T e T d - Let r be te root of an AVL tree of eigt Let N denote te minimum number of nodes in an AVL tree of eigt T r Let r be te root of an AVL tree of eigt Let N denote te minimum number of nodes in an AVL tree of eigt T r N + N - + N - T e T d T e T d - - ou ou - N + N - + N - N - + N + N - + N - N - + N -

12 N + N - + N - N - + N + N - + N - N - + N - (N - ) N - (N - ) (N - ) N + N - + N - N - + N + N - + N - N - + N - (N - ) (N - ) ( N - ) N - (N - ) (N - ) ( N - ) N - N + N - + N - N - + N - N + N - + N - N - + N - Cases: = N = = N = (N - ) (N - ) ( N - ) N - i N -i (N - ) (N - ) ( N - ) N - i N -i

13 N + N - + N - Cases: N + N - + N - Cases: N - + = N = = N = N - + = N = = N = N - N - (N - ) (N - ) ( N - ) N - (N - ) (N - ) ( N - ) N - i N -i Solving te base case we get: N > /- Tus te eigt of an AVL tree is O(log n) i N -i Solving te base case we get: N > / Tus te eigt of an AVL tree is O(log n) Heigt of AVL Tree Insertion in an AVL Tree Tus, te eigt of te tree is O(logN) Were N is te number of elements contained in te tree Insertion is as in a binary searc tree (always done by expanding an external node) Tis implies tat tree searc operations Find(), Max(), Min() take O(logN) time. Insertion in an AVL Tree Insertion in an AVL Tree Insertion is as in a binary searc tree (always done by expanding an external node) Example: Insertion is as in a binary searc tree (always done by expanding an external node) Example: Insert node

14 Insertion in an AVL Tree Insertion in an AVL Tree Insertion is as in a binary searc tree (always done by expanding an external node) Example: Insert node Insertion is as in a binary searc tree (always done by expanding an external node) Example: Insert node Insertion in an AVL Tree Insertion in an AVL Tree Insertion is as in a binary searc tree (always done by expanding an external node) Example: Insert node Insertion is as in a binary searc tree (always done by expanding an external node) Example: Insert node Insertion in an AVL Tree How does te AVL tree work? Insertion is as in a binary searc tree (always done by expanding an external node) Example: Insert node Unbalanced!! 5

15 How does te AVL tree work? How does te AVL tree work? After insertion and deletion we will examine te tree structure and see if any node violates te AVL tree property After insertion and deletion we will examine te tree structure and see if any node violates te AVL tree property If te AVL property is violated, it means te eigts of left(x) and rigt(x) differ by exactly How does te AVL tree work? Rotations After insertion and deletion we will examine te tree structure and see if any node violates te AVL tree property If te AVL property is violated, it means te eigts of left(x) and rigt(x) differ by exactly If it does violate te property we can modify te tree structure using rotations to restore te AVL tree property Two types of rotations Single rotations two nodes are rotated Double rotations tree nodes are rotated Localizing te problem Case Analysis Two principles: Case Case Imbalance will only occur on te pat from te inserted node to te root (only tese nodes ave ad teir subtrees altered - local problem) + + Rebalancing sould occur at te deepest unbalanced node (local solution too) 5

16 Case Analysis Case Analysis Case. Case. + + T T Te new node was inserted in T Te new node was inserted in T Case Analysis Case Analysis Case. Case Not a possible case oterwise te tree would be unbalanced before te insertion of te new node Not a possible case oterwise te tree would be unbalanced before te insertion of te new node Case.: Single Rotation (Rigt) Case.: Single Rotation (Left) Te eigt of te tree rooted at x decreases by one unit Te eigt of te tree rooted at x decreases by one unit

17 Single Rotation - Example Example + + Node 0 added Tree is an AVL tree by definition. Tree violates te AVL definition! Perform rotation. Example Example After Rotation x y y x + C B A A B C Tree as tis form. Tree as tis form. Case.: Single Rotation fails Case.: Double Rotations Sometimes a single rotation fails to solve te problem k k + X k Y Z X Y k Z + In suc cases, we need to use a double-rotation Te eigt of te tree rooted at x decreases by one unit 7

18 Double Rotation - Example Example + + Delete node 9 Tree is an AVL tree by definition. AVL tree is violated. Example After Double Rotation y x y z x z C A B B A B B C Tree as tis form. Tree as tis form Correctness of rebalancing procedure Te nodes involved in rotations: {x,y} in case. and nodes {x,y,z} in case. ave te AVL property after rebalancing te tree. If a node x is rebalanced ten te eigt of te te subtree rooted at x decreases by one unit Tis guarantees tat all ancestors of x become balanced again Insertion Algoritm Use an extra field per node to keep te eigt of eac subtree Step. Traverse te pat from te new leaf to te root of tree updating every node s eigt Step. Traverse te pat from te new leaf to te root to find te deepest ancestor u of te new leaf tat became unbalanced Step. Determine te current case and rebalance u accordingly Step. Traverse te pat from u to te root updating te subtree s eigt

19 Insertion Deletion Te time complexity to perform a rotation is O() Perform normal BST deletion Te time complexity to find a node tat violates te AVL property is dependent on te eigt of te tree, wic is O( log N ) Perform exactly te same cecking as for insertion to restore te tree property Summary AVL Trees Maintains a Balanced Tree Modifies te insertion and deletion routine Performs single or double rotations to restore structure Guarantees tat te eigt of te tree is O(logn) Te guarantee directly implies tat functions find(), min(), and max() will be performed in O(logn) 9