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, Problem 2P
Program Plan Intro
To show the sorting of string written in English-language dictionaries order takes time of
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Draw the portion of the state space tree generated by LCBB for the following instances. n = 4, m = 15, (P₁, ..., P) = (10, 10, 12, 18) (w₁,..... W 4) = (2, 4, 6, 9).
Let S : S --> {true, false} be a function that inputs a full binary search tree and outputs either true or false, defined as follows.
B. If X is a binary search tree consisting only of a single vertex, then S(X ) = true if this vertex equals pork and S(X ) = false otherwise.
R. If X is a binary tree with root v, left subtree Xl and right subtree Xr , then S(X ) = true if v equals pork. Otherwise, if v < pork, then S(X ) = S(Xr ). If v > pork, then S(X ) = S(Xl ).
Given a set of 10 letters { I, D, S, A, E, T, C, G, M, W }, answer the following:
a) With the given letters above, we can construct a binary search tree (based on alphabetical ordering) and the sequence < C, D, A, G, M, I, W, T, S, E > is obtained by post-order traversing this tree. Construct and draw such a tree. NO steps of construction required.b) The letter S is first removed from the binary search tree determined above, followed by inserting a new letter R. Draw the updated binary search tree after removal and insertion. Choose the logical predecessor (not successor) of the removing node if necessary
c) Determine and list the sequence of elements obtained by pre-order traversing the updated binary search tree after removal and insertion above. No steps required.
d) Suppose we are given the first six elements < I, D, S, A, E, T > and their frequencies of occurrence < 5, 6, 2, 4, 5, 2 >, construct and draw the Huffman Tree based on these elements above and their…
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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- Write a program that will undertake a range search of all elements lying within limits a, b along a dimension i of the multidimensional data set, represented as a k-d tree. For example, given the data set {(7, 5), (4, 2), (6, 8), (1, 4), (3, 5), (2, 4), (3, 7), (9, 1), (6, 6), (5, 1)} represented as a k-d tree (k = 2), a range search of data elements lying within (a = 3, b = 7) along dimension i = 2 yields {(7, 5), (1, 4), (3, 5), (3, 7), (6, 6)}arrow_forwardConsider the Binary Search Tree method DepthEqual(T) that outputs all node values in a binary search tree T such that the value stored in the node is equal to the depth of the node in the tree. The values can be any integer, including negative numbers. Show pseudocode for an efficient implementation of DepthEqual(T). Hint: For a node n at depth d, whose value is v. if d >= v. then there is no need to search n's left subtreearrow_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
- Suppose you started with an empty binary search tree. We've seen previously that inserting the keys 1, 2, 3, 4, 5, 6, 7 (in that order) would lead to a binary search tree whose shape we called degenerate. Propose a second ordering of the same keys that would also lead to a degenerate-shaped binary search tree. If possible, propose a third ordering of the same keys that would also lead to a degenerate-shaped binary search tree. If there are no more orderings other than the one we provided and the second one you proposed, briefly explain why there are no more.arrow_forwardDraw an ordered tree representing the formula 'q(X - Y * cos(Z), log(K), A / (B – C)'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
- Your colleague proposed a different definition of a binary search tree: it is such binary tree with keys in the nodes that for each node the key of its left child (if exists) is bigger than its key, and the key of its right child (if exists) is less than its key. Is this a good definition for a binary search tree? A. Yes B. Noarrow_forward2. Show the B+‐tree of order three (namely each node has a maximum of three keys/descendents) that result from loading the following sets of keys in order: a. M, I, T b. M, I, T, Q, L, H, R, E, K c. M, I, T, Q, L, H, R, E, K, P d. M, I, T, Q, L, H, R, E, K, P, C, Aarrow_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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