Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
expand_more
expand_more
format_list_bulleted
Question
Chapter 21, Problem 3P
(a)
Program Plan Intro
To argue that line 10 executes exactly once for each pair
(b)
Program Plan Intro
To argue that the number of sets in the disjoint-set data structure is equals to the depth of
(c)
Program Plan Intro
To prove the LCA procedure correctly prints the least common ancestor of
(d)
Program Plan Intro
To analyzes the running time of LCA procedure for disjoint-set data structure.
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solutionStudents have asked these similar questions
If n1, n, ., nk is a sequence of nodes in the tree such that n; is the parent of
n41 for 1
Draw the binary search tree if the in-order traversal of that binary search tree are given as below:
In-order traversal: 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 19, 20
The BST remove algorithm traverses the tree from the root to find the node to remove. When the node being removed has 2 children, the node's successor is found and a recursive call is made. One node is visited per level, and in the worst-case scenario, the tree is traversed twice from the root to a leaf. A BST with N nodes has at least log2N levels and at most N levels. Therefore, the runtime complexity of removal is best case O(logN) and worst case O(N). Two pointers are used to traverse the tree during removal. When the node being removed has 2 children, a third pointer and a copy of one node's data are also used, and one recursive call is made. Thus, the space complexity of removal is always O(1)."
I have to explain this clearly! and the advantages of the BST algorithim
Knowledge Booster
Similar questions
- Kruskal's Algorithm: Kruskal's algorithm is a minimum-spanning-tree algorithm which finds an edge of the least possible weight that connects any two trees in the forest. This algorithm is a Greedy Algorithm. The steps to find a MST using this algorithm are as follows: Sort all the edges in non-decreasing order of their weight. Pick the smallest edge. Check if it forms a cycle with the spanning tree formed so far. If cycle is not formed, include this edge. Else, discard it. Repeat step2 until there are (V-1) edges in the spanning tree. Example with 10 vertices and 20 weighted edges:arrow_forwardComputer Science Consider a left-child, right-sibling tree T containing n nodes, in which each node stores, among other information, a real number >0, called the node’s score. Find a subset S of all nodes such that the sum of the scores of nodes in S is maximized subject to the constraint that a node and its child cannot both be in S. 1. In C++, write a function to solve the max sum for the nodes with the conditions given. 2. Analyze the running timearrow_forwardYou are given a weighted tree T.(As a reminder, a tree T is a graph that is connected and contains no cycle.) Each node of the tree T has a weight, denoted by w(v). You want to select a subset of tree nodes, such that weight of the selected nodes is maximized, and if a node is selected, then none of its neighbors are selected.arrow_forward
- Computer Science A left-child, right-sibling tree T with n nodes holds a score, a real integer >0, in each node. Find a subset S of all nodes that maximizes the sum of their scores while ensuring that node and child cannot be in S. 1. Write a C++ method to solve the max sum for nodes with provided criteria. 2. Assess runtimearrow_forwardConsider a random binary tree on which Breadth- First Search is applied starting from the root node. There exist a node A and B at the distance six from the root. If A is the nth node with a maximum possible value of n and B is the nth node with a minimum possible value of n in the Breadth First Search traversal. The value of A + Bis (Root node is 1st node)arrow_forwarda) Given a depth-first search tree T, the set of edges in T are referred to as "tree edges" while those not in T are referred to as "back edges". Modify the implementation of the Depth-First Search algorithm to print out the set of tree edges and the set of back edges for the following graph. 1(0 1 1 0 0 1 o) 210 10 0 0 31 10 1 0 0 1 40 0 10 0 0 0 50 0 0 0 0 1 1 6 10 001 70 0 10 1 1 0 0 1arrow_forward
- Computer Science Each node in a left-child, right-sibling tree T has a real integer >0 score. Find a subset S of nodes that maximises their scores while excluding node and child. 1. Write a C++ function to solve the greatest sum for given nodes. Runtime evaluationarrow_forward4-Clique Problem The clique problem is to find cliques in a graph. A clique is a set of vertices that are all adjacent - connected - to each other. A 4-clique is a set of 4 vertices that are all connected to each other. So in this example of the 4-Clique Problem, we have a 7-vertex graph. A brute-force algorithm has searched every possible combination of 4 vertices and found a set that forms a clique: https://en.wikipedia.org/wiki/Clique_problem You should read the Wikipedia page for the Clique Problem (and then read wider if need be) if you need to understand more about it. Note that the Clique Problem is NP-Complete and therefore when the graph size is large a deterministic search is impractical. That makes it an ideal candidate for an evolutionary search. For this assignment you must suppose that you have been tasked to implement the 4-clique problem as an evolutionary algorithm for any graph with any number of vertices (an n-vertex graph). The algorithm succeeds if it finds a…arrow_forwardLet T be a binary search tree with n nodes, in which n > 0. When T is linear, the search algorithm makes key comparisons, in the unsuccassful case.arrow_forward
- What is the rationale behind the fact that Prim's algorithm always produces a Minimum Spanning Tree?arrow_forwardKruskal's algorithm is a minimum-spanning-tree algorithm which finds an edge of the least possible weight that connects any two trees in the forest. This algorithm is a Greedy Algorithm. The steps to find a MST using this algorithm are as follows: Sort all the edges in non-decreasing order of their weight. Pick the smallest edge. Check if it forms a cycle with the spanning tree formed so far. If cycle is not formed, include this edge. Else, discard it. Repeat step2 until there are (V-1) edges in the spanning tree. The graph must have 10 vertices and 20 weighted edges.arrow_forward4. Construct a binary tree whose nodes in preorder and inorder traversals are given as follows: Preorder Travesal: G, B, Q, A, C, K, F, P, D, E, R, H Inorder Travesal: Q, B, K, C, F, A, G, P, E, D, H, Rarrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
- Database System ConceptsComputer ScienceISBN:9780078022159Author:Abraham Silberschatz Professor, Henry F. Korth, S. SudarshanPublisher:McGraw-Hill EducationStarting Out with Python (4th Edition)Computer ScienceISBN:9780134444321Author:Tony GaddisPublisher:PEARSONDigital Fundamentals (11th Edition)Computer ScienceISBN:9780132737968Author:Thomas L. FloydPublisher:PEARSON
- C How to Program (8th Edition)Computer ScienceISBN:9780133976892Author:Paul J. Deitel, Harvey DeitelPublisher:PEARSONDatabase Systems: Design, Implementation, & Manag...Computer ScienceISBN:9781337627900Author:Carlos Coronel, Steven MorrisPublisher:Cengage LearningProgrammable Logic ControllersComputer ScienceISBN:9780073373843Author:Frank D. PetruzellaPublisher:McGraw-Hill Education
Database System Concepts
Computer Science
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:McGraw-Hill Education
Starting Out with Python (4th Edition)
Computer Science
ISBN:9780134444321
Author:Tony Gaddis
Publisher:PEARSON
Digital Fundamentals (11th Edition)
Computer Science
ISBN:9780132737968
Author:Thomas L. Floyd
Publisher:PEARSON
C How to Program (8th Edition)
Computer Science
ISBN:9780133976892
Author:Paul J. Deitel, Harvey Deitel
Publisher:PEARSON
Database Systems: Design, Implementation, & Manag...
Computer Science
ISBN:9781337627900
Author:Carlos Coronel, Steven Morris
Publisher:Cengage Learning
Programmable Logic Controllers
Computer Science
ISBN:9780073373843
Author:Frank D. Petruzella
Publisher:McGraw-Hill Education