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 18.3, Problem 1E
Program Plan Intro
Show the results of deleting C , P and V in order from the tree of figure 18.8(f).
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Prove the correctness of the RB-INSERT (T, z) algorithm (formal description is given
in Chapter 13.3 of the textbook), i.e., the procedure to insert an element into a red-black tree while
maintaining the binary search tree properties.
(a) Explain (2, 4) tree.
(b) Consider the sequence of keys (7,18,23,45,10,18,30,50,12,1,56,74). Draw the result of
inserting entries with these keys (in the given order) into an initially empty (2, 4) tree.
Suppose we exchange elements a[i] and a[i+k], which were
originally out of order. Prove that at least 1 and at most 2k-1 inversions are removed.
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- Consider the problem of in-order traversals of binary trees. We have seen one way to perform them: recursively, from the top of the tree. Another is to be able to identify, for each node, which node of the tree would follow it in an in-order traversal. Describe, using standard tree terminology, how you would reliably find the "next" after any arbitrary node in a binary tree. Indicate what you would need to know about the current node and how to navigate the tree to get to the next-in-order. You do not need to write code for this problem.arrow_forward(b) Consider the sequence of keys (9, 18, 4, 17, 25, 40, and 80). Draw the result of inserting entries with these keys (in the given order) into an initially empty red-black tree.arrow_forwardGiven the B-tree below, what is the final tree after deleting 5, 7, 23 using inorder successor? Write the answer in a paper and explain every process.arrow_forward
- Show every steps of inserting, into an initially empty order-5 B-tree, entries with keys (4, 40, 23, 50, 11, 34, 62, 78, 66, 22, 90, 59, 25, 72, 64, 77, 39, 12, 41, 73).arrow_forwardIn the BinaryTree abstract data structure, there is a remove() function. a. Explain briefly the purpose of the remove() function. b. The remove() function runs differently depending on the number of subtree on a node. i. Explain briefly, how to estimate the number of substrees given a binary tree node. ii. Give an example in a single sentence to justify that the worst-case time complexity for the remove() is O( n).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_forward
- Show the results of deleting C, P and V, in order, from the B-tree with minimum degree 3 given below : ELPTX AC JK NO QRS UV YZarrow_forwardQuestion 2: (a) Explain (2, 4) tree. (b) Consider the sequence of keys (4,17,22,45,10,18,30,50,12,1,56,81). Draw the result of inserting entries with these keys (in the given order) into an initially empty (2, 4) tree.arrow_forwarda. Given the definition of a binary tree, Insert, into an empty binary search tree, entries with keys {55, 34, 19, 78, 65, 8, 22, 97, 53, 26,76, 39, 93, 69, 11 ,83,2,35,70,77} (in this order). Draw the tree after each insertionarrow_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_forwardGiven an AVL tree T storing n keys, what are the asymptotic running times, using O notation, for each dictionary operation (i.e., search, insertion, and deletion) on T in the worst-case?arrow_forwardGive the uppor-bound for the recurrence T(n)=2π(n/2)+n∧2, using the Recursion Tree method. You must show at least 3 levels of the tree, and give the explicit log base when using the tree height!arrow_forward
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