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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H.W: Show the result of inserting 10, 12, 1, 14,6, 5,8, 15,3,9,7, 4, 11, 13, and 2, one at a time, into an initially empty binary heap?
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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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- Suppose we have to do m inserts and n deletions, starting from an empty heap. When would it be more beneficial to use a d-heap rather than a binary heap?arrow_forwardGiven a max-priority queue A = <8, 5, 7, 3, 2, 4, 1>, consider calling Max-Heap-Insert(A, 6). Note: Array index starts from 1. (1) What will be the resulting max-priority queue A? Please write your answer as a sequence of numbers separated by ", ". For example: 1, 2, 3, 4, 5, 6, 7, 8arrow_forwardGiven a max-heap H with n elements and a value x, write the pseudocode of most efficientpossible algorithm that prints all keys in H that are greater than x (i.e., use the command print).The max-heap H should remain intact. Please indicate the worst-case running time of your algorithmand justify your answer. (Note: A procedure that always traverses the whole heap, regardless of thevalue of x and the elements in the heap, is considered an inefficient algorithm.)arrow_forward
- What does the initially empty max-heap h contain after the following sequence of pseudo-code operations? Show at least 5 binary heaps as stated in Figure 1 to show the changes after a sequence of heap operations.arrow_forwardThe largest item in a heap must appear in position 1, and the second largest mustbe in position 2 or position 3. Give the list of positions in a heap of size 31 where thekth largest (i) can appear, and (ii) cannot appear, for k=2, 3, 4 (assuming the values tobe distinct).arrow_forwarda. Show how to implement a queue with two ordinary stacks so that the amortized cost of each Enqueue and Dequeue operation is O(1). b. Consider an ordinary min-heap supporting operations Insert and Extract-min that each costs O(lg n) worst-case time complexity when there are n items in the heap. Give a potential function Φ such that the amortized cost of Insert is O(lg n) and the amortized cost of Extract-min is O(1). Show Φ yields these amortized time bounds (n is the number of items currently in the heap and the size of the heap is unknown). Note: “Show” means to provide a proof with logical and precise arguments and/or derivations.arrow_forward
- 1. Assume we build a binary max-on-top heap from keys 10, 12, 1, 14, 6, 5, 8, 15, 3, 9, 7, 4, 11, 13, 2. a. Show the result of building this heap by inserting the above keys one at a time in the order given (from left to right), into an initially empty binary heap. Please show key steps and short illustrations if necessary. b. Show the heap from problem (a) above after executing three deleteMaximum operations on this heap. Show each step. 2. Use the same key set given Q1, write a program to output the median using heap structure. The median is the value separating the higher half from the lower half of a data sample. For example, the median of input is 8. 3. Use the same input stream of Q1, implement HeapSort in ascending order.arrow_forwardAlgorithm Demonstrate, step by step, how to construct a max-heap for the list of integers A[1..8] = (12, 8, 15, 5, 6, 14, 1, 10) (in the given order), following Example 4.14.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_forward
- What precisely is a pile? "A heap operation has a temporal complexity of O(nlogn)" - give an appropriate illustration to explain the statement.arrow_forwardA d-heap is similar to a binary heap, except that nodes can have up to d children. Suppose d = 3, insert the following items into a max 3-heap: 10, 2, 6, 20, 9, 25, 12, 19. Show all the steps.arrow_forwardplease illustrate the process of HEAPSORT withA = [50, 30, 37, 22, 7, 9, 5]. Here, BUILD-MAX-HEAP has already been executed, and A is a max-heap.arrow_forward
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