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 6.3, Problem 3E
Program Plan Intro
To show that the number of nodes of height h in any n -element heap are at most
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
From the below array representation of a binary tree, find the father of node Q?
z E xY
A N
Node
H
5
7
8 9 10
| 11 12 13
Index
1
4
6
O a. Y
O b. Z
O c. E
O d. X
3.
2.
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)?
Determine the height of an n-element heap. Show all work necessary!
Also, you must prove its correctness.
Chapter 6 Solutions
Introduction to Algorithms
Ch. 6.1 - Prob. 1ECh. 6.1 - Prob. 2ECh. 6.1 - Prob. 3ECh. 6.1 - Prob. 4ECh. 6.1 - Prob. 5ECh. 6.1 - Prob. 6ECh. 6.1 - Prob. 7ECh. 6.2 - Prob. 1ECh. 6.2 - Prob. 2ECh. 6.2 - Prob. 3E
Ch. 6.2 - Prob. 4ECh. 6.2 - Prob. 5ECh. 6.2 - Prob. 6ECh. 6.3 - Prob. 1ECh. 6.3 - Prob. 2ECh. 6.3 - Prob. 3ECh. 6.4 - Prob. 1ECh. 6.4 - Prob. 2ECh. 6.4 - Prob. 3ECh. 6.4 - Prob. 4ECh. 6.4 - Prob. 5ECh. 6.5 - Prob. 1ECh. 6.5 - Prob. 2ECh. 6.5 - Prob. 3ECh. 6.5 - Prob. 4ECh. 6.5 - Prob. 5ECh. 6.5 - Prob. 6ECh. 6.5 - Prob. 7ECh. 6.5 - Prob. 8ECh. 6.5 - Prob. 9ECh. 6 - Prob. 1PCh. 6 - Prob. 2PCh. 6 - Prob. 3P
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
- 30. In the implementation for a breadth-first search we studied, a queue was used. The code below replaces the queue with a stack. List the pre-order enumeration that the vertices in the graph below are visited using this modified method, starting from vertex 0. } 3 /** stack-based search */ static void modSearch (Graph G, int start) { Stack S new AStack (G.n()); S.push(start); } 6 G.setMark (start, VISITED); while (S.length) > 0) { } modSearch: intv S.pop(); PreVisit (G, v); for (int w = G.first (v); w < G.n(); w = G.next(v, w)) if (G.getMark (w) == UNVISITED) { G.setMark (w, VISITED); S.push(w);arrow_forwardTo show that an n-element heap has height [Ign].arrow_forwardDescribe a fast recursive algorithm for reversing a singly linked list L, sothat the ordering of the nodes becomes opposite of what it was before. Code and describe each code.arrow_forward
- Question 8 In the worst case, what is the time complexity to implement a get() method in a singly linked list of length n : log 2 n log 2 n – 1 n/2arrow_forwardCreate a min-heap from a binary search tree. Show the technique in action using the following binary search tree.arrow_forwardIn Heapsort, we assume that arrays are indexed from 1 to n. For example, if A = [16,4,10,14,7,9,3,2,8,1], then A[2] = 4 (highlighted in red). %3D %3D The MAX-HEAPIFY(A, i) procedure inputs an array A and an index i. It assumes that the binary trees rooted at LEFT(i) and RIGHT(1) are max heaps, and ensures that A[i] "floats down" so that the output is an array that obeys the max-heap property. An example is provided on pg. 155 of the textbook. In this example, A = [16,4,10,14,7,9,3,2,8,1]. Then MAX-HEAPIFY(A, 2) corrects the 2nd element of A, which is 4, moving this number to the correct position. The output is [16,14,10,8,7,9,3,2,4,1]. Let A = [2,11,10,9,6,7,8,3,5,4,1]. Determine MAX-HEAPIFY(A,1).arrow_forward
- Consider the array with elements: 18, 10, 20, 5, 8, 9, 3. Applying the heapify() to convert the list to a minimum heap rearranges the elements to: 18 10 3 58 9 20 3 10 9 5 8 18 20 359 10 8 18 20 3 10 18 5 8 9 20 Consider DFS in the graph below starting form node E. The next node visited in the partial traversal E H GCFwill be node A. True False Consider the following maximum heap: 20 10 18 5 8 9 4. After inserting node 15, the heap elements will rearrange to 15 20 18 10 8945 15 18 20 4 9 8 10 5 20 15 18 10 894 5 20 10 18 158945arrow_forwardCreate a min-heap from a binary search tree. Use the following binary search tree to demonstrate the method.arrow_forwardDevelop a topological sort implementation thatmaintains a vertex-indexed array that keeps track of the indegree of each vertex. Initialize the array and a queue of sources in a single pass through all the edges Then, perform the following operations until the source queue is empty:■ Remove a source from the queue and label it.■ Decrement the entries in the indegree array corresponding to the destinationvertex of each of the removed vertex’s edges.arrow_forward
- Develop a topological sort implementation thatmaintains a vertex-indexed array that keeps track of the indegree of each vertex. Initialize the array and a queue of sources in a single pass through all the edges. Then, perform the following operations until the source queue is empty:■ Remove a source from the queue and label it.■ Decrement the entries in the indegree array corresponding to the destination vertex of each of the removed vertex’s edges If decrementing any entry causes it to become 0, insert the corresponding vertex onto the source queue.arrow_forwardFun problem. Given a linked list, check in O(n) if it is a palin-drome. For example, [1 →2 →3 →2] is not a palindrome, while[1 →2 →2 →1] and [3 →5 →3] are.1This class has all necessary methods of the stack data structure, e.g., push, pop, peek, isEmpty, etc.arrow_forwardShow the result after a deleteMin on this binary heap. (Show each step). 12 18 16 25 32 24 40 45 58 50 42 54 48 52 49 46 59 57arrow_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