Data Structures and Algorithms in Java
6th Edition
ISBN: 9781118771334
Author: Michael T. Goodrich
Publisher: WILEY
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Chapter 8, Problem 2R
Program Plan Intro
To represent a tree achieving the worst-case running time scenario for
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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.
In the worst-case scenario, binary tree sort employing a self-balancing binary search tree requires O(n log n) time.
E.
D
2. In the binary search tree above, finding node E requires one comparison and finding
node A requires four comparisons. What is the expected number of comparisons
required to find a node chosen at random?
Chapter 8 Solutions
Data Structures and Algorithms in Java
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- Write an algorithm that gets a table with shortest-path tree of G rooted at s (result of Dijkstra algo, for every vertex we have the previous vertex in SPT). The complexity should be at most O(n), where n is the number of vertices. The algorithm should return the corresponding shortest-path tree with root s- an arborescence A with root s s.t. every path in A is a shortest path in G.arrow_forwardFor a 2-3 tree containing real numbers, design an algorithm for computing the range (i.e., the difference between the largest and smallest numbers in the tree) and determine its worst-case efficiency. You may describe the steps of this algorithm in English.arrow_forwardThe result of Prim's method is always a Minimum Spanning Tree, but why is that?arrow_forward
- Suppose you are given an undirected graph G and a start node s. Your task is to design an algorithm that returns FALSE if G is not a tree and returns TRUE and labels each vertex v with the number of nodes in the subtree rooted at v if G is a tree. Note that the orientation of edges is implicit given the start node. Hint: Modify DFS to solve the problem.arrow_forwardGiven an AVL tree containing N positive integers, print out all the even values contained in the tree in descending order (e.g. 12, 8, 6, 2). What will be the time complexity for an efficient algorithm? Please choose the closest bound function.arrow_forwardThe BST remove algorithm traverses the tree from the root to find the node to remove. When the node being removed has 2 children, the node's successor is found and a recursive call is made. One node is visited per level, and in the worst-case scenario, the tree is traversed twice from the root to a leaf. A BST with N nodes has at least log2N levels and at most N levels. Therefore, the runtime complexity of removal is best case O(logN) and worst case O(N). Two pointers are used to traverse the tree during removal. When the node being removed has 2 children, a third pointer and a copy of one node's data are also used, and one recursive call is made. Thus, the space complexity of removal is always O(1)." I have to explain this clearly! and the advantages of the BST algorithimarrow_forward
- Suppose we consider the special case of IS, WIS where the input graph is a tree T (V, E) with an identified root r ≤ V (there are no other limitations on structure). This means that for every v € V(T), the subtree rooted at v is well-defined - we will refer to this subtree as Tv. = In this section, you are asked to develop a polynomial-time algorithm for solving WIS (and including IS, when weights are all 1). For every v € V, we define the following: Kv,o to be the total weight of the maximum-weight Independent set of subtree T, which does not include v itself. • Kv,1 to be the total weight of the maximum-weight Independent set of subtree T with v belonging to the independent set. D(i) Develop a pair of recurrences which express the value of к,0 (and similarly of ×,₁) in terms of the values of the child nodes of v. Include the "base case" whsn v is a leaf, and then describe how we can exploit the recurrences to design a polynomial-time algorithm to solve WIS for a tree. D(ii) In the…arrow_forwardExplain how to use an AVL tree or a red-black tree to sort n comparable elements in O(n log n) time in the worst case.arrow_forwardGive an argument for why the Prim's algorithm will always return a Minimum Spanning Tree?arrow_forward
- An in-order tree walk of an n-node binary search tree can be implemented by finding the minimum elementin the tree with TREE-MINIMUM and then making n-1 calls to TREE-SUCCESSOR. Prove that this algorithm runs inΘ(n) time.arrow_forwardplease do fast As a function of the minimum degree t, what is the upper bound on the minimum number of keys that can be stored in a B-tree of height h?arrow_forwardwrite Parallel lock-free spanning tree algorithm (span-lockfree) for given data.arrow_forward
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