Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 6.5, Problem 1E
Program Plan Intro
To describe the operation of HEAP-EXTRACT-MAX on the heap
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A 4-ary max heap is like a binary max heap, but instead of 2 children, nodes have 4 children. A 4-ary heap can be represented by an array as shown in Figure (up to level 2). Write down MAX_HEAPIFY(A,i) function for a 4-ary max-heap that restores the heap property for ith node.
Ilustrate an operation of MAX-HEAP-INSERT(A,10) on heap A = .
Please assume that you have been given an implementation of a stack that supports both push and pop in O(1) time. With this information, you would like to implement a queue with these stacks.
(a) In what way can you efficiently implement a queue using two of these stacks? “Efficiently” in this case means in a way which will allow you to do part B.
(b) Please prove that the amortized cost of each dequeue and enqueue operation is O(1) for your stack-based queue by using the aggregate amortized analysis technique.
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
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- Develop a priority-queue implementation that uses a dway heap. Find the best value of d for various edge-weighted digraph models.Develop a priority-queue implementation that uses a dway heap. Find the best value of d for various edge-weighted digraph models.arrow_forward2. Implementation of the operations for Min Heap is already given, create a program for the implementation of the operations of Max Heap. It should include operations insert(), extractMax() & delete().arrow_forwardProblem 2 (. ). Consider the following heap, which shows integer keys in the nodes: 22 15 34 27 28 37 49 56 66 30 41 34 Suppose that you execute the removeMin( ) operation on the above tree. Show the resulting tree. You need to describe, step by step, how the resulting tree is generated.arrow_forward
- 3. Неаps In PS1, we worked with an Extract-Insert-Stable (EIS) min heap which was defined as a min heap with no duplicate elements where the result of calling ExtractMin and immediately re-inserting the same element was the original heap. In this question we want to consider a related concept Insert-Extract-Stability IES. The pair (H, x) is the combination of min heap H with no duplicate elements and an element x. The pair is together considered IES if the result of inserting x into H and immediately calling ExtractMin results in the original heap H. (a) Formally describe the relationship between the elements of H and also the new element x so that (H, x) is IES. (b) Prove that your description holds by showing that it applies to all IES (H,x) pairs and does not hold for any (H, x) which is not IES.arrow_forwardUse a triply linked structure as opposed to an array for implementing a priority queue using a heapordered binary tree. Each node will require three links: two to move up the tree and one to move down it. Even if the maximum size of the priority queue is unknown at the outset, your solution should nonetheless provide logarithmic running times for each operation.arrow_forwardAn input array B contains 2k-1 distinct elements (k is a positive integer). Allelements in B are arranged in ascending order. The constructHeap algorithm inHeapsort is applied to B to construct a maximizing heap. Analyze how many keycomparisons are performed in this heap construction. Your result must be in termsof k.arrow_forward
- Given an array out of order, BUILD-MAX-HEAP() turns it to a max-heap by repeating MAX-HEAPIFY from the last parent down to the first parent, root. For example, A = [4,1,3,2,16,9,10,14,8,7] and BUILD-MAX-HEAP(A) will make change per iteration as below. [4, 1, 3, 2, 16, 9, 10, 14, 8, 7] after MAX-HEAPIFY for the parent 16 [4, 1, 3, 14, 16, 9, 10, 2, 8, 7] after MAX-HEAPIFY for the parent 2 [4, 1, 10, 14, 16, 9, 3, 2, 8, 7] after MAX-HEAPIFY for the parent 3 [4, 16, 10, 14, 7, 9, 3, 2, 8, 1] after MAX-HEAPIFY for the parent 1 [16, 14, 10, 8, 7, 9, 3, 2, 4, 1] after MAX-HEAPIFY for the parent 4 Suppose A = [5, 1, 9, 6, 2, 3, 8, 7, 4]. Show how A will change for each MAX-HEAPIFY.arrow_forward2. Consider the heap implementation of a priority queue. Let H be a heap on n ele- ments. Delete(H, i) deletes the element in position i. Construct an example heap with at least 15 elements, and identify a position i with the property that Delete(H, i) results in the execution of calls to Heapify-up. You must draw your example heap as a binary tree and identify the element at position i. Then justify that your example satisfies the required property.arrow_forwardAn input array B contains 2k-1 distinct elements (k is a positive integer). All elements in B are arranged in ascending order. The constructHeap algorithm in Heapsort is applied to B to construct a maximizing heap. Analyze how many key comparisons are performed in this heap construction. Your result must be in terms of k. Please do not copy the answers from chegg as it is wrong. Thank you very much.arrow_forward
- // Suppose we have the following min-heap:// 2, 3, 4, 16, 7, 11, 9, 31, 18, 21, 12// After calling delFront(), the min-heap now looks as follows:// 3, 7, 4, 16, 12, 11, 9, 31, 18, 21 In java language in a min heap create a method that deletes the min element all details under method with examples please and thank you essentially just a method that deletes the min number and resorts the heaparrow_forwardDo B,C,D ASAP!! PLEASE Time is short Consider the following series of random numbers:80 35 50 18 36 29 25 17 67 23 12 19 5 3 2a. Create a Priority Queue using an array data structure, draw and explainarray at each stepb. Draw a d-heap, where ? = 4c. Explain steps of removing min i.e. deleteMin(), identify hole location(s),slide-down, bubble-up and last elementd. Explain steps of inserting 1arrow_forwardQuestion 6 The worst-case run-time to change the priority of a heap element is: A. e(n logn) B. O(n) C. e(1) D. e(logn) E. O(n) Question 7 Consider a maximum heap, represented by an array: 52, 42, 32, 22, 27, 28, 29, 20, and 16. Now consider that a value 47 is inserted into this heap. What is the new heap after the insertion? A. 52, 42, 32, 22, 27, 28, 29, 20, 16, 47 В. 52, 47, 32, 42, 27, 28, 29, 22, 16, 20 C. 52, 42, 32, 22, 47, 28, 29, 20, 16, 27 D. 52, 47, 32, 22, 42, 28, 29, 20, 16, 27 E. 52, 47, 32, 42, 27, 28, 29, 22, 20, 16 Question 8 The output of an in-order traversal of a binary tree is 1,2.3,4.5.6, and the output of pre- order traversal of the same binary tree is 6,1.4.2.3,5. what is the output of the post-order traversal of the same binary tree? A. 3,1,4,5,2,6 B. 3,2,4,5,1.6 C. 3,2,5,4,1.6 D. 1,2,4.6,5,3 E. 3,2,5,1.4.6 Question 9 Given an array A storing n elements. Which one of the following statements is true? A. Heap sort is always the slowest algorithm in practice to…arrow_forward
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