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 4P
(a)
Program Plan Intro
Give theimplementation of the MINIMUM operation on the heap and describe the cost of implementation.
(b)
Program Plan Intro
Write the implementation steps of operation DECREASE-KEY.
(c)
Program Plan Intro
Write the implementation steps of INSERT operation.
(d)
Program Plan Intro
Write the steps of DELETE operation to delete a given leaf nodex without changing the cost of implementation.
(e)
Program Plan Intro
Implement the EXTRACT-MIN operation to extracts the leaf with the smallest key.
(f)
Program Plan Intro
Write the implementation steps of UNION operation without changing the original time taken for implementation.
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1-Given the binary min-heap A = [1, 60, 18, 83, 97, 68, 47, 99] do the following:
(a) Draw the tree representation.
(b) Show the steps involved when inserting the element 8.
(c) Show the steps involved when deleting the smallest element from A (note that we are deleting from the original heap A, not from the outcome of question 1b).
(d)Given the array [5, 93, 68, 91, 37, 11, 8, 54, 84, 19, 10, 80, 68], create a binary min-heap using the procedure MakeHeap from the lectures.
Give a sequence of arrays highlighting the two elements that get swapped in the array and draw a tree representation of the final heap.
(e)Describe (with some pseudocode) how to write an ADT that imple- ments a priority queue using a binary min-heap , and which ALSO allows changing the priority of a key.
8. Given a sequence of numbers: 19, 6, 8, 11, 4, 5(I) Draw a binary min-heap (in a tree form) by inserting the above numbers readingthem from left to right.
(II) Show a tree that can be the result after the call to deleteMin() on the above heap.(III) Show a tree after another call to deleteMin().
.1. Draw the binary min-heap tree that results from inserting 65, 12, 73, 36, 30, 55, 24, 92 in that order into an initially empty binary min heap. BOX IN YOUR FINAL BINARY MIN HEAP.
.2. Using the final binary min-heap above, show the result of performing a dequeue( ) operation, assuming a priority queue is implemented using a binary min-heap. BOX IN YOUR FINAL BINARY MIN HEAP.
Chapter 19 Solutions
Introduction to Algorithms
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Similar questions
- The heap presented in the text is also known as a max-heap, in whicheach node is greater than or equal to any of its children. A min-heap is a heap inwhich each node is less than or equal to any of its children. Min-heaps are oftenused to implement priority queues. Revise the Heap class toimplement a min-heap.arrow_forwarda. b. c. d. e.3. Show(a) (b)Is (a) a heap? Motivate your answer.Tree (b) is a complete binary tree. Change the tree into a min heap and show every step.the results of the following operations on an initially empty max heap: insert 2, 3, 4, 1, 9, one item at a time;delete one item from the heap;insert the item 7 and then the item 6;delete one item from the heapinsert the item 5. [7]the array presentation of tree that resulted from Question 2.arrow_forward2. Show the results of the following operations on an initially empty max heap: a. insert 2, 3, 4, 1, 9, one item at a time; b. delete one item from the heap; c. insert the item 7 and then the item 6; d. delete one item from the heap e. insert the item 5. [7] 3. Show the array presentation of tree that resulted from Question 2.arrow_forward
- 6.Code_ 6.Idea: Maintain a max heap of k elements.We can iterate through all points.If a point p has a smaller distance to the origin than the top element of aheap, we add point p to the heap and remove the top element.After iterating through all points, our heap contains the k closest points tothe origin.""" from heapq import heapify, heappushpop def k_closest(points, k, origin=(0, 0)): # Time: O(k+(n-k)logk) # Space: O(k) """Initialize max heap with first k points. Python does not support a max heap; thus we can use the default min heap where the keys (distance) are negated. """ heap = [(-distance(p, origin), p) for p in points[:k]] heapify(heap) """ For every point p in points[k:], check if p is smaller than the root of the max heap; if it is, add p to heap and remove root. Reheapify. """ for point in points[k:]: dist = distance(point, origin) heappushpop(heap, (-dist, point)) # heappushpop does conditional check…arrow_forwardCircle the following statement that is false regarding binary heaps.a. Binary heaps are not always complete binary trees.b. Binary heaps can be used for sorting in O(n log n) time.c. Binary heaps can be reheapified after a pop in worst case O(log n) time.arrow_forwardDATA STRUCTURES AND ALGORITHMS C++ Build a heap tree from the given information or values. Write an algorithm that ‘SORT’ the data maintained in heap such that when data is retrieved from the heap, it should be sorted in descending order. Also show its step by step working. 50 45 65 60 75 90 15 35 70 45 55arrow_forward
- 1. a)Show the result of inserting the following values one at a time into an initially empty binary heap. (Show the heap after each insert). Use trees to illustrate each heap. 42, 11, 28, 8, 13, 61, 18 b) Show how the final heap created in the previous problem would be stored in an array.arrow_forwardFig 1.a : Insert 6 and redraw AVL tree in figure 1 Fig 1.b : After that, insert 9 and redraw also new AVL Tree---Fig 2.a Insert 1 and redraw Heap in fig 2.Fig 2.b : After that, remove a value from heap and redraw it.arrow_forwardDescribe the concept of a Binary heap and the heap order property. [10 marks] Explain the use of the binary heap as an effective implementation for a priority queue [10 MARKS]arrow_forward
- Answer the following questions on binomial min-heap. Insert the values in set A into an initially empty binomial min-heap Show only the final tree. A: 13 62 98 43 16 24arrow_forwardProgramming Language: Dr. Racket (R5RS) Question: (a) Define a SCHEME procedure, named (heap-insert f x H), which adds element x to heap H using the first-order relation f to determine which element belongs at the root of each (sub)tree.For instance, if we wanted the same behavior as the heaps in the lecture slides (min-heap), we would use the “less than” function as our first-order relation: (heap-insert < 100 (heap-insert < 10 (list))) (10 () (100 () ()))arrow_forwardObject is name(age,time) For example : apple (1,11) the Binary tree order depend on age Write a java code that would add a new node to the tree in the leaf position where binary search determines a node for d should be inserted. To build fix the trees so that the properties of the max heap are maintained (i.e. the parent node must have higher adoption priority than its children). This means that we need to perform upheap if needed. Note that since this is also a binary search tree, we need to make sure that when performing upheap we don’t break the properties of the binary search tree. To ensure this, instead of performing upheap as seen in class, we will need to implement a tree rotation that reverses the parent-child relationship whenever necessary. Depending if the child that has to be swap in the parent position is the left or the right child, we will need to perform a right rotation or a left rotationarrow_forward
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