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 6.1, Problem 3E
Program Plan Intro
To show that the root of the subtree contains the largest value occurring anywhere in that subtree for a max-heap.
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Heap is a special tree-based non-linear data structure in which the tree is a complete binary tree. GIVE the TWO (2) types of Heap data structure.
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The depth of a heap or B-tree is never more than O(logN), so operations on these structures are also O(logN). True or false?
True or False:
1. In a mini binary heap, each parent must be >= to its children
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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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- Given a Binary Tree (BT), convert it to a Doubly Linked List(DLL) in-place. The left and right pointers in nodes are to be used as previous and next pointers respectively in converted DLL. The order of nodes in DLL must be the same as in Inorder for the given Binary Tree. The first node of Inorder traversal (leftmost node in BT) must be the head node of the DLL.arrow_forwardShow the binary tree that corresponds to the following array-based implementation of a binary tree. Does this represent a heap and/or binary search tree (i.e., exhibits heap property and/or binary search property)? 15 8 10 7 6 1 5 3arrow_forwardThe Heapify Problem: make algorithm for Heapify(r ) pre-cond: The balanced binary tree rooted at A[r ] is such that its left and right subtrees are heaps. post-cond: Its values are rearranged in place to make it complete heap.arrow_forward
- Use 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_forwardAnswer 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_forwardWrite algorithm Heapify(r ) pre-cond: The balanced binary tree rooted at A[r ] is such that its leftand right subtrees are heaps. post-cond: Its values are rearranged in place to make it completeheap.arrow_forward
- Consider Figure 2 and answer the following. Give reason of your answer. a)Is this a complete binary tree? b) Is this a binary search tree? c) Is this a max heap? d) Is this a graph? e) Write the post-order traversal of the tree presented in Figure 2.arrow_forwardTrue or false I) It is possible that a best case scenario in linear search can be a worst case scenario in binary search. ii) A priority queue is a First In First Out (FIFO) structure. iii) For every node in a heap, its left child must be smaller than its right child. iv) Any complete binary tree can also be considered as a full tree.arrow_forwardIn this task, a binomial heap should be implemented. A binomial heap is implemented byan array with its elements being binomial trees: on the first place there is the binomial treeB0 with one element, on the second place there is the binomial tree B1 with two elements,on the third place there is the binomial tree B2 with four elements,. . . Each binomial treeis implemented recursively using the class BinomialNode. In this class there is nothing toimplement. Methods in this class are needed for the implementation of the methods in theclass BinomialHeap. More precisely, in the class BinomialHeap you should implement thefollowing methods:(i) insert, which takes an integer (a key) and inserts it in the binomial heap. Themethod returns true, if the key is successfully inserted and false otherwise. In thecase, if the array should be resized, use the method resizeArray (see below).(ii) getMin, which returns the minimal key in binomial heap. The implementation willbe also efficient if you…arrow_forward
- Write a function cumulative_mul that mutates the Tree t so that each node's label becomes the product of its label and all labels in the subtrees rooted at the node. def cumulative_mul(t): "" "Mutates t so that each node's label II I| || the corresponding subtree rooted at t. >>> t = Tree(1, [Tree(3, [Tree(5)]), Tr >>> cumulative_mul(t) >>> t Tree(105, [Tree(15, [Tree(5)]), Tree(7) II I| || "*** YOUR CODE HERE ***"arrow_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_forwardDraw a single tree whose inorder traversal is and whose postorder traversal is f, a, g, b, h, d, i, c, j, e f.g, a, h, i, d. j, e, c, barrow_forward
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