Heaps and Bounds! Consider the following operation for a maximum heap: removeMax: Remove and return the element whose key has maximum value from the binary heap. In class, we discussed how this operation can be performed in O(log, n) time for a binary heap. Prove, for a binary heap, that any algorithm for the removeMax operation requires (log₂ n) operations in the worst case. Hint: Prove the claim via reduction (using contradiction). Is there some problem you could solve faster (than what we know to be impossible) if you could accomplish, obtaining a o(log, n)-time algorithm in the worst case here?

Q: removeMax: Remove and return the element whose key has maximum value from the binary heap. In class,…

A: Rapidly, or in fast progress, as in The corn is growing by leaps and bounds, or School enrollment is…

Q: Let H1 and H2 be two (binary) max-heaps with n1 and n2 elements respectively. If every element in H1…

A: Answer: To merge two binary max-heaps H1 and H2 with n1 and n2 elements respectively, such that…

Q: Question 10 Consider the following weighted graph. Weights of edges are given in parentheses. h(6)…

A: As per  our guidelines, we only have answer the first question   It is a subset of edges of a…

Q: Given any max binary heap with k distinct items, will you end up with the same heap after you insert…

A: You can remove the smallest or largest object from the min-heap or the max-heap, respectively. All…

Q: Show the results of inserting the key set {12,1,3,7,4,5,15} in order into an initially empty binary…

A: Array Implementation of binary max-heap 1. Use an array to store the data. 2. Start storing from…

Q: lets say these are are a representation of max-heap 21 14 7 9 12 5 what do we get after one call of…

A: Given keys are, 21, 14, 7, 9, 12, 5 Heap is almost a binary tree, in this tree every node contains 2…

Q: Suppose that we would like to verify if the elements in array A satisfy the heap property, assuming…

A: Max heap is a data structure in which max element present on root of tree.

Q: pseudo code for checking whether an array H[1..n] is a heap and determine its

A: SUMMARY  Simple Result is to first check root if it’s lesser than all of its descendants. Also…

Q: What the tightest asymptotic upper bound for the worst-case running time of heap sort? Select one:…

A: In order to perform heap sort, 2 operations are needed: insertion and root deletion. The time taken…

Q: In Heapsort, we assume that arrays are indexed from 1 to n. For example, if A =…

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Q: Apply your HeapMax class to solve the following problem: In a research project your teammate who is…

A: import java.util.*; class Main{     public static void main(String[] args) {         int…

Q: Consider an unsorted array of n elements. Recall that the Heapsort Algorithm consists of two parts:…

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Q: A heap with n nodes contains only one node where the max-heap property is violated. To use the…

A: Heapify: Heapify: Creating a heap from a binary tree represented with the help of an array is the…

Q: Think about the challenge of finding the smallest element in a maximum heap. A max heap's smallest…

A: A lower bound on the number of leaves that must be searched to locate the smallest element. Let B…

Q: 4. You've probably seen in previous courses how to build a min-heap on n elements in O(n) time and…

A: Note - As per the guidelines we are only allowed to answer 1 question at a time. Build_Heap takes…

Q: draw minimum heap after applying heapify to array P = {8,3,5,20,10,1,2}

A: Given array 8,3,5,20,10,1,2 We have to build min heap from given array.  Min heap property:-  Parent…

Q: When using a t-ary heap in heapsort, find the value of t that minimises the coefficient of N lg N in…

A: Solution When using t he t-ary heap is heapsort, the value of t that minimizes the coefficient N lg…

Q: In case the heap data structure is represented by arrays, write the algorithm that removes v from…

A: Given that in case the heap data structure is represented by arrays, write the algorithm that…

Q: eng at n

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Q: For starting array 25 22 24 28 26 29 21 23 27, tell, for each element: what is the FIRST other…

A: A heap is a specialized data structure that is a complete binary tree (a tree structure in which…

Q: Argue the correctness of HEAP-INCREASE-KEY using the following loop invari- ant: At the start of…

A: Pseudo code for HEAP-INCREASE-KEY is reproduced for convenience HEAP-INCREASE-KEY(A, i, key):    if…

Q: Consider an unsorted array of n elements. Recall that the Heapsort Algorithm consists of two parts:…

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Q: write pseudo-code of an algorithm that finds the maximum value in a binary min-heap what is the…

A: Binary tree: It is a non linear data structure. Here data elements is called node of tree. These are…

Q: You are given a Heap A[1, ..., n]. Describe an algorithm to delete an element at index 'k' (1 <= k…

A: Deletion of Element from any intermediary position is costly.so, it is suggested to replace the…

Q: Build a min_heap from an array of n items: CO(1) OO( log n) OO(n) (n log n ) * CO( n? )

A: Find the time complexity of min_heap from an array of n items: O(1) O(log n) O(n) O(n log n) O(n^2)

Q: As we already studied different algorithms associated with binary heap. In place of binary heap, now…

A: K-ary heaps are a generalization of binary heap(K=2) in which each node have K children instead of…

Q: You have been given an array of integers that have just been sorted using the QuickSort algorithm.…

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Q: Perform removeMin twice on the following binary min heap, show the heap as arrays after each…

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Q: Suppose that H is a min-heap, and we have a pointer (index of the array representation) to an…

A: In thе contеxt of a binary hеap, a hеap propеrty is maintained such that for еvеry nodе i othеr than…

Q: are these true or false   heaps can be constructed from n un ordered keys in O(n) time.   huffmans…

A: - We need to highlight the validity of the provided statements.

Q: Show the resulting binary max heap after removing the root from the max heap in the picture.

A: Here when we delete the root ie 94,it will be removed and it will be replaced by last element of…

Q: Consider an unsorted array of n elements. Recall that the Heapsort Algorithm consists of two parts:…

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Q: To create a heap of n elements, we need to call heapify function

A: Question is "To create a heap of n elements, we need to call heapify function how many times?"

Q: Show that the problem of finding the kth smallest element in a heap takes at least Ω(k) time in the…

A: We have to explain that problem of finding the kth smallest element in a heap takes at least Ω(k)…

Q: An insert adds a new single-node tree to the forest. So a sequence of n inserts into an initially…

A: algorithm for given statement is given in next step:-

Q: Show how you can implement insert in O(log n) time in a binary heap.

A: Here is the Answer and Explanation

Q: 4. Heaps. Let T be a min-heap (whose root contains the minimal element) storing n keys. Give an…

A: We know that a min heap is  a heap where root contains the minimal element so in this question we…

Q: Consider two max-heaps A and B, where A heap-size - 100, B.heap-size - n, and n is asymptoticall…

A: Hello studentGreetingsIn the realm of data structures and algorithms, the efficient merging of data…

Q: In the best case, the time complexity of heap sort is O (n)

A: Sorting is a process of following an Algorithm to arrange all the values in either Ascending order…

Q: Suppose node i is removed from a max heap. We can't just remove it because we will no longer have a…

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Q: Consider a heap of size n stored as an array. If the node i has no child, then which of the…

A: The heap is a tree based data structure which follows some property.   It's divided into max heap…

Q: Q4. Consider Following 4 lists lists: L1 45 12 06 34 02 18 09 L2 35 22 05 14 08 22 30 Apply Heap…

A: Heap sort:- For L1: heap of the array is  [45, 34, 18, 12, 2, 6, 9]Step  0 : [45, 34, 18, 12, 2, 6,…

Q: Suppose you are given an array A[1..n] with a max-heap structure. If you run HEAPSORT on the array,…

A: Suppose that you are given an array A[1..n] with a max-heap structure. If you run HEAPSORT on the…

Q: Q1. Implement in Java the greedy algorithm for giving change with minimum number of coins: while…

A: ''Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Consider an unsorted array of n elements. Recall that the Heapsort Algorithm consists of two parts:…

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Q: Suppose we have p max heaps, with q elements in each. We wish to combine these into a single heap.…

A: Insertion and Deletion in Heap: The standard deletion operation on Heap is to delete the element…

Q: draw minimum heap after applying heapify to array P = {8,3,5,20,10,1,2}

A: Here in this question we have given an array and we have asked to convert it into minimum heap.