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
Concept explainers
Question
Chapter 19.4, Problem 1E
Program Plan Intro
Show that for any positive integer n, a sequence of Fibonacci-heap operations that creates a Fibonacci heap consisting of one tree is a linear chain of n nodes.
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
Show that by adding values to a skew heap in the right sequence, any binary tree that possesses the heap property can be created. (This understanding is crucial to comprehending why an amortised accounting approach is required.)
Consider the problem of determining the smallest element in a maxheap. The smallest elements of a max heap must be one of the n/2 leaves. (Otherwise,there must be a nonleaf that is smaller than one of its descendants, which means thetree is not a max heap.) Thus, it is sufficient to search all leaves. Prove a lower boundthat searching all the leaves is necessary
Suppose we generalize the “cut rule” (in the implementation of decrease-key operation for a Fibonacci heap) to cut a node x from its parent as soon as it loses its kth child, for some integer constant k. (The rule that we studied uses k = 2.) For what values of k can we upper bound the maximum degree of a node of an n-node Fibonacci heap with O(log n)?
Chapter 19 Solutions
Introduction to Algorithms
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, computer-science and related others by exploring similar questions and additional content below.Similar questions
- For the given complete binary tree (given in an array): 1 2 3 4 5 6 7 8 9 10 11 J B K A O T U X R Q D Convert it into a max heap and provide your answer by dragging and dropping the given alphabets on correct positions in array.arrow_forwardConsider a rooted n node binary tree represented using pointers. The best upper bound on the time required to determine the number of subtrees having exactly 4 nodes is O(na logbn). Then the value of a + 10b is.arrow_forwardDemonstrate that any binary tree that has the heap property can be generated by inserting values into a skew heap in an appropriate order. (This realization is important to understanding why an amortized accounting scheme is necessary.)arrow_forward
- Draw the portion of the state space tree generated by LCBB for the following instances. n = 4, m = 15, (P₁, ..., P) = (10, 10, 12, 18) (w₁,..... W 4) = (2, 4, 6, 9).arrow_forwardLet's assume that a binary heap is represented using a binary tree such that each node may have a left child node and a right child node. For this type of representation, we can still label the nodes of the tree in the same way as we label the nodes for an array representation. That is, the root node has a label 1. In general, for a node with label i, its left child node will have a label 2i and the right child has a label 2i+1. For any i with 1 <= I <= n , Terry says that the following easy algorithm will walk you from the root node to the node with label i: First find the binary representation P of i. Start with the rightmost bit (least significant bit) of P, walk down from the root as follow: For a 0 bit, walk to the left child, for a 1 bit walk to the right child. At the end, you’ll reach the node with label i. Which of the following is the most appropriate? A. Terry’s algorithm is wrong and not fixable. B. Terry’s algorithm is right. C. Terry's algorithm can be…arrow_forwardApply the exhaustive parsing algorithm to determine whether or not bbaaaba ∈ L(G5), where G5 is this CFG: S ⟶ bSR | a (1) (2) R ⟶ aRb | a (3) (4) To illustrate the workings of the algorithm, show the breadth-first tree that it, in effect, traversed during its execution. (Each node of that tree is labeled by a sentential form.)arrow_forward
- Using Java Design an algorithm for the following operations for a binary tree BT, and show the worst-case running times for each implementation:preorderNext(x): return the node visited after node x in a pre-order traversal of BT.postorderNext(x): return the node visited after node x in a post-order traversal of BT.inorderNext(x): return the node visited after node x in an in-order traversal of BT.arrow_forwardWrite an algorithm that gets a table with shortest-path tree of G rooted at s (result of Dijkstra algo, for every vertex we have the previous vertex in SPT). The complexity should be at most O(n), where n is the number of vertices. The algorithm should return the corresponding shortest-path tree with root s- an arborescence A with root s s.t. every path in A is a shortest path in G.arrow_forwardConsider the array t = [1, 2, 3, 4, 5, 8, 0 , 7, 6] of size n = 9, . a) Draw the complete tree representation for t. b) What is the index of the first leaf of the tree in Part a (in level order)? In general, give a formula for the index of the first leaf in the corresponding complete binary tree for an arbitrary array of size n. c) Redraw the tree from Part a after each call to fixheap, in Phase 1 of heapsort. Remember, the final tree obtained will be a maxheap. d) Now, starting with the final tree obtained in Part c, redraw the tree after each call to fixheap in Phase 2 of heap sort. For each tree, only include the elements from index 0 to index right (since the other elements are no longer considered part of the tree). e) For the given array t, how many calls to fixheap were made in Phase 1? How many calls to fixheap were made in Phase 2? f) In general , give a formula for the total number of calls to fixheap in Phase 1, when heapsort is given an arbitrary array of size n. Justify…arrow_forward
- 4. SHORT ANSWERS:i. Suppose tree T is a min heap of height 3. - What is the largest number of nodes that T can have? _____________________- What is the smallest number of nodes that T can have? ____________________ii. The worst case complexity of deleting any arbitrary node element from heap is ___________arrow_forwardGiven a binary tree T and a source node s in it, provide the pseudocode for an iterative algorithm to traverse T starting from s using breadth-first traversal, also known as level-order traversal. Each node in T contains an integer key that can be accessed. Each time a node is visited, its key should be printed. Note: You do not have to implement your algorithm.arrow_forwardGive an argument for why the Prim's algorithm will always return a Minimum Spanning Tree?arrow_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 Education
Starting Out with Python (4th Edition)Computer ScienceISBN:9780134444321Author:Tony GaddisPublisher:PEARSON
Digital Fundamentals (11th Edition)Computer ScienceISBN:9780132737968Author:Thomas L. FloydPublisher:PEARSON
C How to Program (8th Edition)Computer ScienceISBN:9780133976892Author:Paul J. Deitel, Harvey DeitelPublisher:PEARSON
Database Systems: Design, Implementation, & Manag...Computer ScienceISBN:9781337627900Author:Carlos Coronel, Steven MorrisPublisher:Cengage Learning
Programmable 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