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 19, Problem 3P
(a)
Program Plan Intro
To analyse the amortized running time of FIB-HEAP-CHANGE-KEY when k is greater than or equal to
(b)
Program Plan Intro
Give an efficient implementation of FIB-HEAP-PRUNE( H,r ) and also describe the amortized running time of Implementation.
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a. 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.
An 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.
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.
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- An 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_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_forwardHeaps 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 N(log, 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?arrow_forward
- Determine the height of an n-element heap. Show all work necessary! Also, you must prove its correctness.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_forward1.For a full binary tree, the number of leaf-nodes is more than non-leaf nodes. True False 2.For a Max-Heap, the functions Max and Extract-Max have same runtime complexity. True False 3.Heap-increase-Key and Heap-Decrease-Key both have same runtime complexity because both call Heapify function. True False 4.A sorted linked-list has fast insertion but slow extraction. True False 5.Don’t use Max-Heap in case you often perform search operation. Use sorted linked-list inserted. True Falsearrow_forward
- 4. Heaps. Let T be a min-heap (whose root contains the minimal element) storing n keys. Give an efficient algorithm for reporting all the keys in T that are less than or equal to a given key x (which may or may not be in T). What is the Big Oh() performance of this algorithm?arrow_forwardAn insert adds a new single-node tree to the forest. So a sequence of n inserts into an initially empty heap will simply create n single-node trees. The cost of an insert is clearly O(1).A deleteMin operation removes the node indicated by minPtr. This turns all children of the removed node into roots. We then scan the set of roots (old and new) to find the new minimum, a potentially very costly process. We also perform some rebalancing, i.e., we combine trees into larger ones. The details of this process distinguish different kinds of addressable priority queue and are the key to efficiency.turn now to decreaseKey(h,k) which decreases the key value at a handle h to k. Of course, k must not be larger than the old key stored with h. Write algorithm for given statementarrow_forwardConstruct a priority queue using a heapordered binary tree, but instead of an array, use a triply linked structure. Each node will require three links: two to go down the tree and one to traverse up the tree. Even if no maximum priority-queue size is specified ahead of time, your solution should ensure logarithmic running time per operation.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_forwardDevelop a priority-queue implementation that uses a dway heap. Find the best value of d for various edge-weighted digraph models.arrow_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_forward
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