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 12.3, Problem 6E
Program Plan Intro
To describe the TREE-DELETE procedure to implement a fair strategy such that choose node y as its predecessor rather than its successor.
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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.
What is the time complexity for these operation in average case (balanced tree), respectively? What is the time complexity for these operation in average case (balanced tree), respectively? In a BST with n nodes, what is the time complexity for searching, insertion, and deletion in worst case? What is the height of node 3, and node 9, respectively? What is the depth of node 3, and node 9, respectively? 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.
That we are given the option to put the search keys in whatever order we like and that we are aware of how frequently a BST will need to access them.Should the keys be added to the tree in a different sequence, or in order of decreasing likelihood of access? Describe your response.
Chapter 12 Solutions
Introduction to Algorithms
Ch. 12.1 - Prob. 1ECh. 12.1 - Prob. 2ECh. 12.1 - Prob. 3ECh. 12.1 - Prob. 4ECh. 12.1 - Prob. 5ECh. 12.2 - Prob. 1ECh. 12.2 - Prob. 2ECh. 12.2 - Prob. 3ECh. 12.2 - Prob. 4ECh. 12.2 - Prob. 5E
Ch. 12.2 - Prob. 6ECh. 12.2 - Prob. 7ECh. 12.2 - Prob. 8ECh. 12.2 - Prob. 9ECh. 12.3 - Prob. 1ECh. 12.3 - Prob. 2ECh. 12.3 - Prob. 3ECh. 12.3 - Prob. 4ECh. 12.3 - Prob. 5ECh. 12.3 - Prob. 6ECh. 12.4 - Prob. 1ECh. 12.4 - Prob. 2ECh. 12.4 - Prob. 3ECh. 12.4 - Prob. 4ECh. 12.4 - Prob. 5ECh. 12 - Prob. 1PCh. 12 - Prob. 2PCh. 12 - Prob. 3PCh. 12 - Prob. 4P
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- This 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_forwardIf index entries were placed in sorted order, what would the occupancy of each leaf node of a B+-tree be? Justify your reasoning.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.arrow_forward
- Show the results of inserting the keys Q,S,F,K,H,I,F,T,V,P,M,R,N,W,A in oder to empty an B- tree with minimum degree 2. Only draw the configuration ot the tree just before some node must split and also draw the final configuration.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. (breadth-first order)arrow_forwardRefine the mathematical model to better explain the experimental results in the table given in the text. Specifically, show that the average numberof compares for a successful search in a tree built from random keys approaches thelimit 2 ln N 2 – 3 1.39 lg N – 1.85 as N increases, where .57721... is Euler’sconstant. Hint : Referring to the quicksort analysis in Section 2.3, use the fact that theintegral of 1/x approaches ln N .arrow_forward
- Let T1, T2 be 2-3-4 trees and let x be a key [in neither T1, T2], such that for each pair of keys x1 ∈ T1 and x2 ∈ T2, we have x1 < x < x2. Thus, every key in T1 is smaller than every key in T2. Devise an algorithm that constructs the union of x with these two trees. Of course, the trees T1, T2 may be of different height, and the result must be a valid 2-3-4 tree. Your algorithm should run in O(1 + |h(T1) − h(T2)|).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_forwardLet T be an arbitrary splay tree storing n elements A1, A2, . An, where A1 ≤ A2 ≤ . . . ≤ An. We perform n search operations in T, and the ith search operation looks for element Ai. That is, we search for items A1, A2, . . . , An one by one. What will T look like after all these n operations are performed? For example, what will the shape of the tree be like? Which node stores A1, which node stores A2, etc.? Prove the answer you gave for formally. Your proof should work no matter what the shape of T was like before these operations.arrow_forward
- Draw the BST constructed by inserting the values [53, 25, 11, 63, 4, 88,59, 3, 15, 82, 92, 27, 55, 14] in the order shown, into an initially empty tree. 2. Using the tree traversal algorithms and the BST from above, show theoutput sequence for a (a) Preorder traversal(b) Postorder traversal 3. Delete the node with value 25 and draw the resultant tree.(a) Use two methods to delete the node with value 25 and draw the resul-tant tree. also describe both methods that you have usedarrow_forwardCreate a binary search tree as discussed in class, using the given numbers in the order they’re presented. State if the resulting tree is has the attributes of being FULL, BALANCED, or COMPLETE. 37, 20, 18, 56, 40, 42, 12, 5, 6, 77, 21, 54arrow_forwardIs there any work that a red-black tree could execute with a time complexity of O(log n)? Pick one: To find a predecessor or a successor; to insert and delete; or to do all three togetherarrow_forward
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