lecture 13
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The document discusses red-black trees, which are binary search trees augmented with node colors to guarantee a height of O(log n). It describes the properties that red-black trees must satisfy, including that every node is red or black, leaves are black, and if a node is red its children are black. It then proves that these properties ensure the height is O(log n) by showing a subtree has at least 2^bh - 1 nodes, where bh is the black-height. Finally, it notes that common operations like search, insert and delete run in O(log n) time on red-black trees.
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The document discusses algorithms and data structures, focusing on binary search trees (BSTs). It provides the following key points:
- BSTs are an important data structure for dynamic sets that can perform operations like search, insert, and delete in O(h) time where h is the height of the tree.
- Each node in a BST contains a key, and pointers to its left/right children and parent. The keys must satisfy the BST property - all keys in the left subtree are less than the node's key, and all keys in the right subtree are greater.
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1.5 binary search tree
1) Tree data structures involve nodes that can have zero or more child nodes and at most one parent node. Binary trees restrict nodes to having zero, one, or two children.
2) Binary search trees have the property that all left descendants of a node are less than the node's value and all right descendants are greater. This allows efficient searching in O(log n) time.
3) Common tree operations include insertion, deletion, and traversal. Balanced binary search trees use rotations to maintain balance during these operations.
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3) Binary Tree
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9) Evaluation of postfix expression
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11) Infix to Postfix using stack
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13) AVL-Tree
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The document discusses algorithms and data structures, focusing on binary search trees (BSTs). It provides the following key points:
- BSTs are an important data structure for dynamic sets that can perform operations like search, insert, and delete in O(h) time where h is the height of the tree.
- Each node in a BST contains a key, and pointers to its left/right children and parent. The keys must satisfy the BST property - all keys in the left subtree are less than the node's key, and all keys in the right subtree are greater.
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1) Tree data structures involve nodes that can have zero or more child nodes and at most one parent node. Binary trees restrict nodes to having zero, one, or two children.
2) Binary search trees have the property that all left descendants of a node are less than the node's value and all right descendants are greater. This allows efficient searching in O(log n) time.
3) Common tree operations include insertion, deletion, and traversal. Balanced binary search trees use rotations to maintain balance during these operations.
Tree
1) Tree
2) General Tree
3) Binary Tree
4) Full Binay Tree, Complete Binay Tree
5) Binary Tree Traversal (DFS & BFS)
6) Binary Search Tree
7) Reconstruction of Binay Tree
8) Expression Tree
9) Evaluation of postfix expression
10) Infix to Prefix using stack
11) Infix to Postfix using stack
12) Threaded Binary Tree
13) AVL-Tree
14) AVL-Tree Rotation
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Binary search trees (BSTs) are data structures that allow for efficient searching, insertion, and deletion. Nodes in a BST are organized so that all left descendants of a node are less than the node's value and all right descendants are greater. This property allows values to be found, inserted, or deleted in O(log n) time on average. Searching involves recursively checking if the target value is less than or greater than the current node's value. Insertion follows the search process and adds the new node in the appropriate place. Deletion handles three cases: removing a leaf, node with one child, or node with two children.
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- Common operations like insertion, deletion, and searching on a red-black tree work similarly to a binary search tree but may require rotations and color changes to maintain the red-black properties.
- Red-black trees are widely used in applications that require efficient search structures for frequently updated data like process memory management and functional programming data structures.
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2) The balance factor of a node is defined as the height of its left subtree minus the height of its right subtree, and must be -1, 0, or 1 in an AVL tree.
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- A linked list contains nodes that have a data field and a pointer to the next node.
- Common operations on linked lists include insertion, deletion, searching, and traversing the list.
- The time complexity of these operations depends on whether it's at the head, tail, or interior node.
- Linked lists can be implemented using classes for the node and linked list objects.
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The document discusses binary trees and binary search trees. It defines key terms like root, child, parent, leaves, height, depth. It explains tree traversal methods like preorder, inorder and postorder. It then describes binary search trees and how they store keys in a way that searching, insertion and deletion can be done efficiently in O(log n) time. It discusses implementation of operations like search, insert, delete on BSTs. It introduces balanced binary search trees like AVL trees that ensure height is O(log n) through rotations during insertions and deletions.
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The document discusses red-black trees, which are binary search trees augmented with node colors to guarantee a height of O(log n). It reviews the properties of red-black trees, including that every node is red or black, leaves are black, red nodes have black children, and all paths from root to leaf contain the same number of black nodes. It then proves that these properties ensure a height of O(log n) and thus search, minimum, maximum, successor and predecessor operations take O(log n) time. Insert and delete operations also take O(log n) time but require special handling to maintain the red-black tree properties during modification.
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Balanced Tree (AVL Tree & Red-Black Tree)
- A red-black tree is a self-balancing binary search tree where each node is colored red or black. It maintains the properties that the black height of each path is equal and there are no adjacent red nodes, ensuring O(log n) time for operations.
- Common operations like insertion, deletion, and searching on a red-black tree work similarly to a binary search tree but may require rotations and color changes to maintain the red-black properties.
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This document discusses binary search trees and balanced binary search trees like AVL trees. It begins with an overview of binary search trees and their properties. It then introduces the concept of balance factors and describes how AVL trees maintain a balance factor between -1 and 1 for every node through rotations. The document provides examples of single and double rotations performed during insertion to rebalance the tree. It concludes with an algorithm for inserting a node into an AVL tree that searches back up the tree to perform necessary rotations after an insertion causes imbalance.
Lecture3
This document discusses AVL trees, which are self-balancing binary search trees. It begins by explaining that binary search trees have O(log N) best case time for operations but can degrade to O(N) in the worst case if the tree becomes unbalanced. The document then introduces AVL trees, which maintain balance by ensuring that the heights of any node's two child subtrees differ by at most one. This guarantees search, insertion, and deletion operations take O(log N) time. The document explains the node height calculation and covers insertion and deletion algorithms in AVL trees through examples, including single and double rotations to rebalance the tree.
CS-102 BST_27_3_14v2.pdf
Binary search trees have the following key properties:
1. Each node contains a value.
2. The left subtree of a node contains only values smaller than the node's value.
3. The right subtree of a node contains only values larger than the node's value.
Binary search trees allow for efficient insertion and search operations in O(log n) time due to their structure. Deletion may require rebalancing the tree to maintain the binary search tree properties.
UNIT II.ppt
Binary search trees allow searching, insertion, and deletion operations in O(h) time where h is the height of the tree. They represent dictionaries and priority queues. Keys in a binary search tree must satisfy the binary search tree property: all keys in the left subtree are less than the root key and all keys in the right subtree are greater. Red-black trees are a variation that ensures the tree remains balanced during operations to keep h proportional to the logarithm of the number of nodes.
BST.pdf
Binary search trees (BSTs) are binary trees where the value of each internal node is greater than all values in its left subtree and less than all values in its right subtree. This ordering property allows efficient search, insertion, and deletion operations that take O(log n) time on average for a tree with n nodes. Common BST operations include searching for a value, inserting a new value, deleting a value, and finding the minimum, maximum, successor, or predecessor of a given node.
Trees - Non Linear Data Structure
Slides cover definition of tree data structure with examples, related terminologies, accessors methods, query methods, generic methods, traversal algorithms (preorder, postorder, inorder) traversal, Binary tree, Binary tree implementation using linked list and array, Binary search
Lecture notes data structures tree
The document discusses different types of tree data structures, including general trees, binary trees, binary search trees, and their traversal methods. General trees allow nodes to have any number of children, while binary trees restrict nodes to having 0, 1, or 2 children. Binary search trees organize nodes so that all left descendants are less than the parent and all right descendants are greater. Common traversal orders for trees include preorder, inorder, and postorder, which differ in whether they process the root node before or after visiting child nodes.
Trees
The document discusses binary search trees and their operations. It covers definitions of binary search trees, how to search, find the minimum/maximum keys, insert and delete nodes. It then discusses AVL trees, which are self-balancing binary search trees where the heights of the left and right subtrees of every node differ by at most one. It explains rotations needed during insertions and deletions to maintain balance in AVL trees.
Sorting
Insertion sort and merge sort are discussed. Insertion sort works by inserting elements in the proper position one by one. Its worst case time is O(N^2). Merge sort uses a divide and conquer approach to sort elements. It divides the list into halves, recursively sorts the halves, and then merges the sorted halves. Merging two sorted lists takes linear time. The overall time for merge sort is O(N log N). Heaps are discussed as a way to implement priority queues. A heap has the heap property where a node is always less than its children. This allows finding the minimum element and deleting it to take O(log N) time.
210 trees
This document discusses trees and binary trees. It covers tree definitions, properties, traversals, and common algorithms. Specifically for binary trees, it discusses their implementation, operations like insertion and deletion, and binary search trees. BSTs allow for efficient search, insertion, and deletion in O(h) time where h is the tree height. Balanced BSTs like red-black trees maintain h as O(log n) to guarantee efficient performance.
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lecture 30
The document reviews concepts related to NP-completeness, including reductions between problems. It provides examples of reducing the directed Hamiltonian cycle problem to the undirected version. It also reduces 3-SAT to the clique problem by transforming a Boolean formula to a graph, then further reduces clique to vertex cover. Hundreds of problems have been shown to be NP-complete through relatively simple reductions like these that leverage previous results.
lecture 29
The document discusses reductions between problems to prove NP-completeness. It first reviews P, NP, reductions, NP-hard and NP-complete problems. It then walks through reducing directed Hamiltonian cycle to undirected Hamiltonian cycle to traveling salesman problem (TSP) to prove that TSP is NP-complete in 3 steps or less.
lecture 28
This document discusses NP-completeness and algorithms. It begins by reviewing tractable versus intractable problems, and the complexity classes P and NP. P contains problems solvable in polynomial time, while NP includes problems verifiable in polynomial time. The document then discusses NP-complete problems, which are the hardest problems in NP. It notes that reducing one problem to another shows the first problem is no harder than the second. The document concludes by outlining the steps to prove a problem is NP-complete: choosing a known NP-complete problem, reducing it to the target problem in polynomial time, and showing the target is in NP.
lecture 27
The document discusses NP-completeness and algorithms. It introduces dynamic programming and greedy algorithms. It then discusses the activity selection problem and shows it can be solved greedily by choosing the activity with the earliest finish time at each step. Finally, it defines the classes P and NP, introduces the concept of reductions to show problems are NP-complete, and states that if any NP-complete problem can be solved in polynomial time, then P=NP.
lecture 26
The document discusses various algorithms techniques including greedy algorithms, dynamic programming, and their application to problems like the activity selection problem and knapsack problem. It provides examples of optimal subproblems, overlapping subproblems, and how dynamic programming and greedy algorithms can be used to solve problems exhibiting these properties.
lecture 25
The document describes the 0-1 knapsack problem and how to solve it using dynamic programming. The 0-1 knapsack problem involves packing items of different weights and values into a knapsack of maximum capacity to maximize the total value without exceeding the weight limit. A dynamic programming algorithm is presented that breaks the problem down into subproblems and uses optimal substructure and overlapping subproblems to arrive at the optimal solution in O(nW) time, improving on the brute force O(2^n) time. An example is shown step-by-step to illustrate the algorithm.
lecture 24
The document discusses dynamic programming and its application to solving the longest common subsequence (LCS) problem. It presents the LCS algorithm, which uses dynamic programming to find the length and sequence of the longest subsequence common to two strings X and Y in O(mn) time, where m and n are the lengths of X and Y, respectively. It provides an example running the LCS algorithm on strings X="ABCB" and Y="BDCAB" to determine their longest common subsequence is "BCB".
lecture 23
The document discusses dynamic programming and amortized analysis. It reviews how dynamic tables use amortized analysis to achieve an overall cost of O(1) per insertion by occasionally doubling the table size and reinserting all elements. This results in a worst case cost of O(n) for a single insertion but averages to O(1) over many insertions. It also discusses using an accounting method with a $3 charge per insertion to pay for future table resizes, achieving an amortized cost of O(1) per operation. Finally, it introduces dynamic programming and uses the longest common subsequence problem to illustrate how it breaks problems into optimal subrecurring subproblems.
lecture 22
The document discusses algorithms and data structures covered in a CS algorithms course, including Dijkstra's algorithm, Bellman-Ford algorithm, shortest paths in DAGs, Kruskal's algorithm for minimum spanning trees, and disjoint-set union data structures. It provides examples and explanations of how each algorithm works.
lecture 21
The document discusses algorithms for solving the single-source shortest path problem, including Dijkstra's algorithm, Bellman-Ford algorithm, and shortest paths in directed acyclic graphs (DAGs). It reviews the key ideas behind relaxation and how algorithms like Bellman-Ford and Dijkstra's use relaxation to iteratively improve the upper bound on shortest path distances. Running times of the algorithms are analyzed, with Dijkstra's algorithm taking O(ElogV) time and Bellman-Ford taking O(VE) time.
lecture 20
The document describes Prim's algorithm for finding a minimum spanning tree (MST) of a weighted undirected graph. It shows the pseudocode for Prim's algorithm and step-by-step workings on an example graph to build up the MST. It notes that the key operation is decreasing the key value, which is called once for each edge in the MST, and the running time depends on the priority queue implementation, being O(ElgV) for a binary heap and O(VlgV+E) for a Fibonacci heap.
lecture 19
The document discusses depth-first search (DFS) graph algorithms. DFS explores a graph by going as deep as possible at each step and backtracking when no further progress can be made. It distinguishes between different types of edges discovered: tree edges connect newly found nodes, back edges connect to ancestors, and cross/forward edges connect between subtrees or ancestors and descendants. DFS runs in O(V+E) time and can detect cycles in O(V) time by checking for back edges.
lecture 18
The document discusses the algorithms of breadth-first search (BFS) and depth-first search (DFS) on graphs. It provides pseudocode for BFS and DFS, and examples of running the algorithms on sample graphs. Key points include:
- BFS uses a queue to explore all neighbors of a vertex before moving to the next level. It finds the shortest paths from the source.
- DFS uses recursion to explore as deep as possible before backtracking. It identifies tree edges, back edges, and forward edges.
- Both BFS and DFS run in O(V+E) time on a graph with V vertices and E edges.
lecture 17
The document discusses interval trees and breadth-first search (BFS) algorithms. Interval trees are used to maintain a set of intervals and efficiently find overlapping intervals given a query. BFS is a graph search algorithm that explores all neighboring vertices of a starting node before moving to neighbors of neighbors. BFS builds a breadth-first tree and calculates the shortest path distances from the source node in O(V+E) time and space.
lecture 16
The document discusses algorithms for finding minimum spanning trees in graphs. It describes two common algorithms: Kruskal's algorithm and Prim's algorithm. Kruskal's algorithm sorts the edges by weight and builds up the spanning tree by adding edges that do not create cycles. Prim's algorithm maintains a set of vertices in the spanning tree and iteratively adds the lowest weight edge connecting a vertex outside the set to one inside. Both run in O((V+E)logV) time using efficient data structures.
lecture 15
The document discusses two algorithms for matrix multiplication and finding the median of an unsorted list:
1) Strassen's algorithm improves on the traditional O(n^3) matrix multiplication algorithm by using divide and conquer to achieve O(n^lg7) time complexity.
2) Finding the median can be done in expected O(n) time using quickselect, or deterministically in O(n) time by choosing the median of medians as the pivot.
lecture 14
The document discusses red-black trees and their operations. It reviews the properties of red-black trees that guarantee logarithmic height. It then describes that inserting nodes can violate properties and requires recoloring nodes or rotating the tree. The key steps of insertion involve adding the new node as red, then fixing any violations by moving them up the tree through recoloring and rotations.
lecture 10
The document discusses various sorting algorithms and their time complexities:
1. Comparison sorts like merge sort and quicksort have a best case time complexity of O(n log n).
2. Counting sort runs in O(n+k) time where k is the range of input values, and is not a comparison sort.
3. Radix sort treats input as d-digit numbers in some base k and uses counting sort to sort on each digit, achieving O(dn+dk) time which is O(n) when d and k are constants.
4. A randomized selection algorithm finds the ith order statistic in expected O(n) time using randomized partition.
lecture 9
The document discusses various sorting algorithms and their time complexities, including:
- Insertion sort runs in O(n^2) time in the worst case.
- Merge sort and heap sort run in O(nlogn) time in the worst case.
- Any comparison-based sorting algorithm requires Ω(nlogn) time.
- Counting sort and radix sort can run in O(n) time by avoiding comparisons, but have additional requirements on the key range.
lecture 8
The document discusses analyzing the average case runtime of quicksort. It shows that:
1) The average case runtime of quicksort is O(n log n) due to partitions being randomly balanced on average.
2) This is proved rigorously by modeling quicksort as alternating best and worst case partitions, showing the cost is absorbed by subsequent partitions.
3) A recurrence is written and solved to formally prove the average case runtime is O(n log n).
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