computer notes - Data Structures - 20
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The document discusses AVL trees and balanced binary search trees. It provides the following key points: 1) An AVL tree is a self-balancing binary search tree where the height of the two child subtrees of every node differ by at most one. 2) A balanced binary search tree is one where the height of the left and right subtrees of each node differ by no more than one. 3) Inserting new nodes can cause the tree to become unbalanced if the insertion breaks the height balance property. Rotations may be needed to rebalance the tree.
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Trees, Binary Search Tree, AVL Tree in Data Structures
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computer notes - Data Structures - 30
computer notes - Data Structures - 30ecomputernotes The document discusses different techniques for improving the efficiency of union-find algorithms, including union-by-size, path compression, and union-by-height. Union-by-size works by making the root of the smaller tree the child of the larger tree during a union operation, keeping tree heights small. Path compression further optimizes find operations by updating the parent pointers along the search path. Together these optimizations allow union-find algorithms to run in almost linear time for practical purposes.
computer notes - Data Structures - 31
computer notes - Data Structures - 31ecomputernotes Union-find data structures can be used to efficiently generate random mazes. A maze can be represented as a grid of cells where each cell is initially isolated by walls. Removing walls corresponds to union operations, joining the cells' sets. A maze is generated by randomly performing unions until the entrance and exit cells are in the same set, connected by a path through the maze.
computer notes - Data Structures - 4
computer notes - Data Structures - 4ecomputernotes The document discusses different types of linked lists, including singly linked lists, doubly linked lists, and circularly linked lists. It provides code examples for implementing linked lists in C++ and compares the time complexity of different linked list operations. It also describes how a circularly linked list can be used to solve the Josephus problem of eliminating people seated in a circle.
computer notes - Data Structures - 22
computer notes - Data Structures - 22ecomputernotes The document discusses deletion in AVL trees and outlines 5 cases to consider when deleting nodes from an AVL tree. It also discusses expression trees and parse trees, providing examples of an expression tree for a mathematical expression and a parse tree for an SQL query. Other uses of binary trees mentioned include their use in compilers for expression trees, parse trees, and abstract syntax trees.
computer notes - Data Structures - 35
computer notes - Data Structures - 35ecomputernotes The document discusses various data structures including skip lists, AVL trees, and hashing. It explains that skip lists allow for logarithmic-time operations and are simple to implement. Hashing provides constant-time operations by mapping keys to array indices via a hash function, but collisions must be handled. Common hash functions discussed include summing character codes or converting to a number in a prime base.
computer notes - Data Structures - 14
computer notes - Data Structures - 14ecomputernotes The document discusses recursion and different traversal methods for binary trees. It explains how the call stack is used during recursive function calls and shows examples of preorder and inorder recursive tree traversals. It then describes how to perform non-recursive inorder traversal using an explicit stack. Finally, it introduces level-order traversal, which visits all nodes at each level from left to right before proceeding to the next level.
computer notes - Data Structures - 29
computer notes - Data Structures - 29ecomputernotes The document discusses using trees and parent pointers to represent sets and solve the dynamic equivalence problem. Each element begins in its own set, represented as a tree with that element as the root. The union operation merges two trees by making one root point to the other. Find returns the root of the tree an element belongs to by traversing parent pointers up the tree. This can be optimized using a technique called union by rank to keep tree heights small.
computer notes - Data Structures - 38
computer notes - Data Structures - 38ecomputernotes The document discusses three elementary sorting algorithms: selection sort, insertion sort, and bubble sort. It explains their basic mechanisms, provides examples, and analyzes their time complexities. Selection sort and bubble sort have a worst-case time complexity of O(n2) because they require iterating through the entire array multiple times. Insertion sort's worst-case is also O(n2) as it may require shifting over elements each iteration. The document then introduces the idea of divide and conquer algorithms like merge sort which can achieve optimal O(n log n) time complexity.
computer notes - Data Structures - 25
computer notes - Data Structures - 25ecomputernotes The document describes the build heap algorithm for creating a min heap data structure from an unordered array of keys. The algorithm iterates from the last parent node to the first, calling a percolateDown function to shift nodes and maintain the min heap property as it builds the heap structure from the initial array. It also provides code snippets in C++ for implementing common heap operations like insertion, deletion, and accessing the minimum element.
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computer notes - Data Structures - 20
- 1. Class No.20 Data Structures http://ecomputernotes.com
- 2. AVL Tree AVL (Adelson-Velskii and Landis) tree. An AVL tree is identical to a BST except height of the left and right subtrees can differ by at most 1. height of an empty tree is defined to be (–1). http://ecomputernotes.com
- 3. AVL Tree An AVL Tree 5 8 2 4 3 1 7 height 0 1 2 3 http://ecomputernotes.com
- 4. AVL Tree Not an AVL tree 6 8 1 4 3 1 5 height 0 1 2 3 http://ecomputernotes.com
- 5. Balanced Binary Tree The height of a binary tree is the maximum level of its leaves (also called the depth). The balance of a node in a binary tree is defined as the height of its left subtree minus height of its right subtree. Here, for example, is a balanced tree. Each node has an indicated balance of 1, 0, or –1. http://ecomputernotes.com
- 6. Balanced Binary Tree -1 0 1 0 0 0 0 -1 0 1 0 0 0 0 0 0 0 http://ecomputernotes.com
- 7. Balanced Binary Tree Insertions and effect on balance -1 0 1 0 0 0 0 -1 0 1 0 0 0 0 0 0 0 U 1 U 2 U 3 U 4 U 5 U 6 U 7 U 8 U 9 U 10 U 11 U 12 B B B B B B http://ecomputernotes.com
- 8. Balanced Binary Tree Tree becomes unbalanced only if the newly inserted node is a left descendant of a node that previously had a balance of 1 (U 1 to U 8 ), or is a descendant of a node that previously had a balance of –1 (U 9 to U 12 ) http://ecomputernotes.com
- 9. Balanced Binary Tree Insertions and effect on balance -1 0 1 0 0 0 0 -1 0 1 0 0 0 0 0 0 0 U 1 U 2 U 3 U 4 U 5 U 6 U 7 U 8 U 9 U 10 U 11 U 12 B B B B B B http://ecomputernotes.com
- 10. Balanced Binary Tree Consider the case of node that was previously 1 -1 0 1 0 0 0 0 -1 0 1 0 0 0 0 0 0 0 U 1 U 2 U 3 U 4 U 5 U 6 U 7 U 8 U 9 U 10 U 11 U 12 B B B B B B http://ecomputernotes.com
- 11. Inserting New Node in AVL Tree 1 0 A B T 1 T 3 T 2 1 http://ecomputernotes.com
- 12. Inserting New Node in AVL Tree 2 1 A B T 1 T 3 T 2 new 1 2 http://ecomputernotes.com
- 13. Inserting New Node in AVL Tree 2 1 A B T 1 T 3 T 2 new 1 2 0 0 A B T 1 T 3 T 2 new Inorder: T 1 B T 2 A T 3 Inorder: T 1 B T 2 A T 3 http://ecomputernotes.com
- 14. AVL Tree Building Example Let us work through an example that inserts numbers in a balanced search tree. We will check the balance after each insert and rebalance if necessary using rotations. http://ecomputernotes.com
- 15. AVL Tree Building Example Insert(1) 1 http://ecomputernotes.com
- 16. AVL Tree Building Example Insert(2) 1 2 http://ecomputernotes.com
- 17. AVL Tree Building Example Insert(3) single left rotation 1 2 3 -2 http://ecomputernotes.com
- 18. AVL Tree Building Example Insert(3) single left rotation 1 2 3 -2 http://ecomputernotes.com
- 19. AVL Tree Building Example Insert(3) 1 2 3 http://ecomputernotes.com