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 18.2, Problem 6E
Program Plan Intro
To explain the CPU time of B-tree search if it uses BINARY-SEARCH instead of LINEAR-SEARCH.
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Illustrate that via AVL single rotation, any binary search tree T1 can be transformed into another search tree T2 (with the same items).Give an algorithm to perform this transformation using O(N log N) rotation on average. Use c++.
Use the procedure TREE-SUCCESSOR and TREE-MINIMUM to write a function of x, x is a node in a binary search tree, to produce the output that INORDERTREE-WALK function would produce. Determine the upper bound running time complexity of F(x) and prove its correctness.
Suppose the we build a binary search tree from input data in which the distribution of keys is uniformly random. What can we expect about the height of this tree and the running time of subsequent SEARCH and INSERT operations on this tree?
Your response should use asymptotic notation to support your answer(s).
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- Suppose that we implement a union-find structure by representing each set using a balanced search tree. Describe and analyze algorithms for each of the methods for a union-find structure so that every operation runs in at most O(logn) time in worst case.arrow_forwardConsider 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_forwardIn the worst-case scenario, binary tree sort employing a self-balancing binary search tree requires O(n log n) time, which is slower than merge sort.arrow_forward
- Prove that, if values are distinct, any binary search tree can be constructed by appropriately ordering insertion operations.arrow_forwardIn the worst case, binary tree sort utilising a self-balancing binary search tree requires O(n log n) time, but it is still slower than merge sort.arrow_forward1. Find the Big-O time complexity for the following:-Inserting an element in an (unbalanced) binary search tree? And what is the best case?-Best case for the height of an (unbalanced) binary search tree? And what is the average case big-O given the avg height if elements are inserted in random order?2. In a binary search tree of N nodes, how many subtrees are there?arrow_forward
- Hey, Kruskal's algorithm can return different spanning trees for the input Graph G.Show that for every minimal spanning tree T of G, there is an execution of the algorithm that gives T as a result. How can i do that? Thank you in advance!arrow_forwardFor any given AVL tree T of n elements, we need to design an algorithm that finds elements x in T such that a <= x <= b, where a and b are two input values. Assume that T has m elements x satisfying a <= x <= b. What is the best time complexity for such an algorithm? A. O(m) B. O(m log n) C.O(m + log n) D.O(log m + log n)arrow_forwardConsider the Fibonacci sequence F(0)=0, F(1)=1 and F(i)=F(i-1)+F(i-2) for i > 1. For the sake of this exercise we define the height of a tree as the maximum number of vertices of a root-to-leaf path. In particular, the height of the empty tree is zero, and the height of a tree with a single vertex is one. Prove that the number of nodes of an AVL tree of height h is at least F(h) and this inequality is tight only for two values of h.arrow_forward
- Computer Science Consider a left-child, right-sibling tree T containing n nodes, in which each node stores, among other information, a real number >0, called the node’s score. Find a subset S of all nodes such that the sum of the scores of nodes in S is maximized subject to the constraint that a node and its child cannot both be in S. 1. In C++, write a function to solve the max sum for the nodes with the conditions given. 2. Analyze the running timearrow_forwardConsider a sort of items, according to their keys, that inserts all the items one at a time into an initially empty regular binary search tree and then applies an in-order traversal to complete the sort. Assume that all items have distinct keys. Using big-Theta notation, what is the worst-case complexity of the sort? What is the average-case complexity of the sort? Now answer the same two questions if an AVL tree is used instead of a regular binary search tree.arrow_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
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