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 19.4, Problem 1E
Program Plan Intro
Show that for any positive integer n, a sequence of Fibonacci-heap operations that creates a Fibonacci heap consisting of one tree is a linear chain of n nodes.
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Think about the challenge of finding the smallest element in a maximum heap. A max heap's smallest components must be one of the n/2 leaves. (If not, the tree cannot be a max heap since there must be a nonleaf smaller than one of its descendants.) Therefore, a thorough leaf search is sufficient. Show, at the very least, that leaf-by-leaf searching is required.
Consider the problem of determining the smallest element in a maxheap. The smallest elements of a max heap must be one of the n/2 leaves. (Otherwise,there must be a nonleaf that is smaller than one of its descendants, which means thetree is not a max heap.) Thus, it is sufficient to search all leaves. Prove a lower boundthat searching all the leaves is necessary
Let'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…
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- 1. Can we perform maximum spanning tree using "Kruskal" ? Give Proper explanation in PDF | Yes, It can be done by modifying the weight values | Yes, It can be done by modifying the direction of the edges |No, it is not possible. Yes, It can be done by using max-heap or sorting the edge weights with decreasing order 1. //Initialise single source(G, s) 2. S=0 3. Q=V[G] 4. While Q != 0 Do u=extract-min(Q) 5. S=S union {u} For each vertex v in adj[u] 6. 7. 8. Do relax(u, v, w) 3. What happens when while loop in line 4 is changed to while Q>1? There can be several Minimum costs of spanning tree in one graph While loop gets executed for v-1 times While loop gets executed only once |While loop does not get executed 2. Which of the following is/are not false? Prims' never accepts cycles in the MST There can be several Minimum costs of spanning tree in one graph Complexity of Dijstra algorithm can be (V*V) Kruskal can work in weighted directed sparse graph |Weighted interval scheduling requires…arrow_forwardGiven a binary tree T and a source node s in it, provide the pseudocode for an iterative algorithm to traverse T starting from s using breadth-first traversal, also known as level-order traversal. Each node in T contains an integer key that can be accessed. Each time a node is visited, its key should be printed. Note: You do not have to implement your algorithm.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
- Using Java Design an algorithm for the following operations for a binary tree BT, and show the worst-case running times for each implementation:preorderNext(x): return the node visited after node x in a pre-order traversal of BT.postorderNext(x): return the node visited after node x in a post-order traversal of BT.inorderNext(x): return the node visited after node x in an in-order traversal of BT.arrow_forwardSome implementations of binary search trees allow for duplicates by introducing the following rule: for each node n. the left subtree of n contains only nodes smaller than n, and the right subtree of n contains only nodes greater or equal to n. Implement a successor and predecessor operation on such a tree. Discuss their time complexity as a function of the height of the tree.arrow_forwardFrom the below array representation of a binary tree, find the father of node Q? z E xY A N Node H 5 7 8 9 10 | 11 12 13 Index 1 4 6 O a. Y O b. Z O c. E O d. X 3. 2.arrow_forward
- Recall that the height of a node in a binary tree is the number of edges (links) from the node to the deepest leaf in its subtree (e.g., the height of leaves is 0). To answer the following question, consider a complete binary tree T of size n in which the last level contains all possible nodes from left to right. (a) Specify the number of nodes at height h in T for any h≥ 0 (no proof is needed). (b) Specify the asymptotic running time of Bubble-Down for a node of height ʼn as a function of h. Then show the total time complexity to Bubble-Down all nodes at level h of T when we apply the Heapify operation on T. Justify your answer in one or two sentences. (c) Use your answer in part to prove that the time complexity of the Heapify operation is O(n). You need to sum the total work over all levels. Show your work. Hint: 0(1). x=0 =arrow_forwardUse 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_forwardAgain recall the definition of the set of full binary trees. Suppose that a full binary tree T' is created using the recursive rule applied to full binary trees T1 and T2. Let v(T) denote the number of vertices in tree T. Which expression correctly describes v(T')? Explain your response (no need for a formal proof). a. v(T') = 2 · v(T1) + 2 · v(T2) + 1 . b. v(T') = 2.v(T1) + 2 • v(T2) · c. v(T') = v(T1) + v(T2) + 1 d. v(T') = v(T1) + v(T2)arrow_forward
- Implement a priority queue using a heapordered binary tree, but use a triply linked structure instead of an array. You will needthree links per node: two to traverse down the tree and one to traverse up the tree. Yourimplementation should guarantee logarithmic running time per operation, even if nomaximum priority-queue size is known ahead of time.arrow_forwardDiagrammatically step by step apply Prim’s algorithm to find a minimum spanning tree for the given wcightcd graph G. bt K G Choose the starting node for the algorithm based on the last digit your roll number as v Last digit of your roll number [ 0 [ 12 4]s[6[7]8]9 e Root vertex alb|e e[f[g[h]i]earrow_forwardConsider a random binary tree on which Breadth- First Search is applied starting from the root node. There exist a node A and B at the distance six from the root. If A is the nth node with a maximum possible value of n and B is the nth node with a minimum possible value of n in the Breadth First Search traversal. The value of A + Bis (Root node is 1st node)arrow_forward
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