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 13.4, Problem 5E
Program Plan Intro
To verify that each count remains the same after the transformation operation.
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Suppose that we have an estimate ahead of time of how often search keys areto be accessed in a BST, and the freedom to insert them in any order that we desire.Should the keys be inserted into the tree in increasing order, decreasing order of likely frequency of access, or some other order? Explain your answer.
If index entries were placed in sorted order, what would the occupancy of each leaf node of a B+-tree be? Justify your reasoning.
What is the level of node 17, and node 11, respectively? Insert integers 11, 22, 15, 44, 6, 9, 3, 33, 17, 5 into a BST in that order. Draw the final tree. From the BST, delete node with 3, and then delete the node with 22, respectively; Then insert a node with 18 into the BST. Draw the final tree after above operations are finished.
Chapter 13 Solutions
Introduction to Algorithms
Ch. 13.1 - Prob. 1ECh. 13.1 - Prob. 2ECh. 13.1 - Prob. 3ECh. 13.1 - Prob. 4ECh. 13.1 - Prob. 5ECh. 13.1 - Prob. 6ECh. 13.1 - Prob. 7ECh. 13.2 - Prob. 1ECh. 13.2 - Prob. 2ECh. 13.2 - Prob. 3E
Ch. 13.2 - Prob. 4ECh. 13.2 - Prob. 5ECh. 13.3 - Prob. 1ECh. 13.3 - Prob. 2ECh. 13.3 - Prob. 3ECh. 13.3 - Prob. 4ECh. 13.3 - Prob. 5ECh. 13.3 - Prob. 6ECh. 13.4 - Prob. 1ECh. 13.4 - Prob. 2ECh. 13.4 - Prob. 3ECh. 13.4 - Prob. 4ECh. 13.4 - Prob. 5ECh. 13.4 - Prob. 6ECh. 13.4 - Prob. 7ECh. 13 - Prob. 1PCh. 13 - Prob. 2PCh. 13 - Prob. 3PCh. 13 - Prob. 4P
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- Suppose that we have an estimate ahead of time of how often search keys areto be accessed in a BST, and the freedom to insert them in any order that we desire.Should the keys be inserted into the tree in increasing order, decreasing order of likelyfrequency of access, or some other order? Explain your answerarrow_forwardThis exercise is about drawing BSTs. You are asked to: Show the result of inserting 3, 1, 4, 6, 9, 2, 5, and 7 in an initially empty binary search tree. Then show the result of deleting the root. Draw all binary search trees that can result from inserting permutations of 1, 2 and 3. How many types of trees are there? What are the probabilities of each type of tree’s occurring if all permutations are equally likely Given the input {4371, 1323, 6173, 4199, 4344, 9679, 1989}, a fixed table size of 10, and a hash function H(X) = X mod 10, show the resulting Linear probing hash table Separate chaining hash tablearrow_forwardConsider a B Tree used as an index that has THREE levels including the root node. If a newkey is inserted in this index, then what shall be the maximum number of nodes? Explain youranswer with an example.arrow_forward
- Show the results of inserting the keys: F, L, O, R, I, D, A, U, N, V, M, Y, C, S in order into an empty B-tree with minimal degree 2. Draw only the configurations of the tree just before some node being split, and also draw the final configuration. Please make sure your tree configurations are drawn neatly and are easy to read and understand.arrow_forwardShow the results of inserting the keys: F, L, O, R, I, D, A, U, N, V, M, Y, C, S in order into an empty B-tree with minimal degree 2. Draw only the configurations of the tree just before some node being split, and also draw the final configuration. Please make sure your tree configurations are drawn neatly and are easy to read and understand. See attached image for how the tree configurations should be presentedarrow_forwardPlease answer the following questions:1. Which node has the imbalance (i.e., the difference of heights > 1 between its left and right subtree)?Please write the key of that node: 2. Which three nodes are involved? With only one space between nodes, e.g., 1 2 3 3. Which type of imbalance (RR, LL, RL, LR)?arrow_forward
- Consider the array t = [1, 2, 3, 4, 5, 8, 0 , 7, 6] of size n = 9, . a) Draw the complete tree representation for t. b) What is the index of the first leaf of the tree in Part a (in level order)? In general, give a formula for the index of the first leaf in the corresponding complete binary tree for an arbitrary array of size n. c) Redraw the tree from Part a after each call to fixheap, in Phase 1 of heapsort. Remember, the final tree obtained will be a maxheap. d) Now, starting with the final tree obtained in Part c, redraw the tree after each call to fixheap in Phase 2 of heap sort. For each tree, only include the elements from index 0 to index right (since the other elements are no longer considered part of the tree). e) For the given array t, how many calls to fixheap were made in Phase 1? How many calls to fixheap were made in Phase 2? f) In general , give a formula for the total number of calls to fixheap in Phase 1, when heapsort is given an arbitrary array of size n. Justify…arrow_forwardShow that, with n relations, there are (2(n−1))! / (n−1)! different join orders.Hint: A complete binary tree is one where every internal node has exactly two children. Use the fact that the number of different complete binary trees with n leaf nodes is:1/n (2(n − 1)(n − 1) )If you wish, you can derive the formula for the number of complete binary treeswith n nodes from the formula for the number of binary trees with n nodes.The number of binary trees with n nodes is:1/n + 1(2n n) This number is known as the Catalan number, and its derivation can be found in any standard textbook on data structures or algorithms.arrow_forwardUse the given AVL tree to answer each of the following questions: (a) Which is the deepest unbalanced node after inserting key 5, but before re-balancing, and what is its balance factor? (b) After the insertion, perform all the necessary rotations required to re-balance the tree, and after each rotation,state the type of rotation done and list the keys of the resulting tree in the form: After Left/Right rotation: e1, e2, e3, ...., en.arrow_forward
- Consider the sequence of keys (5,16,22,45,2,10,18,30,50,12,1, 77, 66, 55). Draw the result of inserting entries with these keys (in the given order) into an initially empty red-black tree.arrow_forwardShow that, with n relations, there are (2(n−1))!∕(n−1)! different join orders.Hint: A complete binary tree is one where every internal node has exactly two children. Use the fact that the number of different complete binary trees with n leaf nodes is:If you wish, you can derive the formula for the number of complete binary trees with n nodes from the formula for the number of binary trees with n nodes. The number of binary trees with n nodes is:arrow_forwardGiven the following AVL tree, if node 83 is deleted, how many rotations are needed to obtain an AVL tree again? Draw the tree after each rotation. An AVL tree may become unbalanced when a node is deleted. In general, how many sub-trees will be re-balanced in the worst case when an item is deleted from an AVL tree? Explain your answer. An AVL tree may become unbalanced when a node is inserted. In general, what is the maximum number of rotations that must be performed to rebalance the tree due to an insertion? Explain your answer.arrow_forward
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