**Transcription:**Explain why the decision tree of any sorting algorithm on \( n \) elements must have at least \( n! \) leaf nodes.---**Explanation for Educational Context:**In the context of sorting algorithms, the decision tree is a conceptual tool used to understand the comparisons made by the algorithm to determine a sorted order. Each internal node of the tree represents a comparison between elements, and each leaf node represents a possible outcome (i.e., a sorted permutation).For a complete sorting of \( n \) elements, there are \( n! \) possible permutations — each a different way to arrange the elements in order. To account for all possible input sequences and outcomes, the decision tree must have at least \( n! \) leaf nodes, where each leaf node corresponds to a unique permutation. This ensures that for any initial arrangement of the elements, the algorithm can lead to the correct sorted order.This concept underscores the lower bound of the complexity of comparison-based sorting algorithms, which is \( O(n \log n) \), as the structure of the decision tree implies a minimum number of comparisons required.

Answered: Explain why the decision tree of any…

Engineering

Computer Science

Explain why the decision tree of any sorting algorithm on n elements must have at least n! leaf nodes.

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

See similar textbooks

Related questions

Q: What is the output from the pre-order, in-order, and post-order traversal of the following binary…

Q: Which of the following statement(s) about search algorithms and binary search trees are true? You…

A: As per the question statement, We need to find the statements which are true.

Q: Consider the following binary search tree Any random element from the given Binary Search Tree is to…

A: Binary search tree is a data structure that allows to maintain a sorted list of numbers.Binary…

Q: How will you construct an optimal binary search tree?

A: Answer is

Q: (0 | 1) (10) * (01 | 1) + 0 ? Make a Context Free Grammar (CFG) form of the RE.

A: Answer: We need to write the context free grammar from Regular Expression so we will see in the more…

Q: If a node in the above binary search tree is to be located by binary tree search operation, what is…

A: Introduction: Here we are required to explain what will be the average number of comparisons that…

Q: Still slower than merge sort, binary tree sort implemented using a self-balancing binary search tree…

A: A binary program search ranking is the basis for the ordering method identified as Binary Tree Sort.…

Q: If a binary search tree has n nodes, what are the lowest and highest nodes? Provide a mathematical…

A: In a binary search tree, there left child of a node has a lower value than the parent, and the right…

Q: Given a sorted (increasing order) array with unique integer elements, write an algorithm to create a…

A: Introduction : A binary search tree (BST) is a type of binary tree in which each node has at most…

Q: What is the time complexity of searching an element in a binary search tree? Can it ever be worse…

A: Binary search tree is a data structure that is commonly used for searching elements efficiently. In…

Q: Which of the three classic traversal algorithms yields a sorted list if applied to a binary search…

A: The correct answer for the above mentioned question is given in the following steps for your…

Q: Which of the algorithms defined in this chapter would be appropriate for this problem?Should the…

A: Vacuum World problem:- The depth-first search technique is best suited to the vacuum-world issue.…

Q: The umber of 1's in the binary representation 4096-15 256+5*16+3) is

A: The answer for the number of 1's in binary representation is

Q: Heapsort. Arrange the following elements in ascending order. Show the state of the tree at every…

A: The program is written in Java. The below program is sorted based on then Heapsort. Please check the…

Q: In the following array representation of the Binary tree, the degree of node C is: Tree А В С 0 1 2…

A: A binary tree can be represented using an array.

Q: What is the time complexity of traversing a balanced binary search tree with n nodes? O(1) O Olm)

A: The correct answer is O(n).

Q: What is the minimum number of nodes that a complete binary search tree of height 4 can have

A: Binary search tree: A complete binary search tree is a binary tree in which every level is…

Q: In a FULL binary search tree where every non lead node has two children, what is the worst case and…

A: The average case and worst case complexity of operations is O(logn)

Q: Given a sorted (increasing order) array with unique integer elements, write an algorithm to create a…

A: ANSWER:-

Q: Even in the worst case, a self-balancing binary search tree takes O(n log n) time to sort a binary…

A: Sorting is a fundamental operation in computer science, and various algorithms have been developed…

Q: Please find the correct total number of possible binary search trees with N keys ? Due to the…

A: Introduction Binary search tree: An advanced approach for analyzing nodes & their left and right…

Q: Code the following using JavaScript: Create your own red-black tree and verify that binary search…

A: Here i explain it: ==============================================================================…

Q: binary search tree, binary tree sort ime in the worst case.

A: A binary program search ranking is the basis for the ordering method identified as Binary Tree…

Q: Subj - Analysis of Algorithms/Design & Anal Algorithms. Please complete thre question together You…

A: Introduction: The analysis of the complexity of a recurrence relation involves finding the…

Q: Consider the following array of numbers A = {15, 4, 21, 12, 10, 0, 21, 12, 3, 14}. Show the result…

Q: What is the best-case scenario in terms of search complexity for binary search trees? O(N-1). N is…

A: 1) The best-case scenario in terms of search complexity for binary search trees is O(log N), where N…

Q: uppose that you have an initially empty binary search tree T that you populate by traversing n array…

A: We need to discuss the time complexity of sorting using inorder traversal of BST.

Q: What is the average case complexities of a Binary Search Tree ? O 0(1) O(log n) O O(n) O O(n log n)

Q: With the Binary Search Tree shown below, what is the sixth node visited using a pre-order traversal?…

A: I have given an answer in step 2.

Q: In a binary search tree, to remove a node N that has left child C1 and right child C2, we do the…

A: The binary search tree is a node-based binary tree data structure whose left node elements are…

Q: Write a function (pseudocode is OK) to traverse binary search tree and return the ith node in sorted…

A: #include <iostream>using namespace std; // perform bubble sortvoid bubbleSort(int array[], int…

Q: CKTRACKING

A: Backtracking is an algorithmic method for solving a problem in a different way. To explain the…

Q: r words, assume that the first comparison a1 : a2 always satisfies a1 > a2.

A: Lets first see what is Insertion sort algorithm . This algorithm is used to insert the element to…

Concept explainers

Merge sort

Merge Sort recursively sorts a given collection of numbers or elements by using the divide and conquer strategy in less time. It is one of the most used algorithms, with the worst-case time complexity of O (n log n). It was invented by John Von Neumann. The algorithm continuously divides an array into sub-arrays until each sub-array consists of a single element. It then merges those sub-arrays to produce a sorted array.

Question

**Transcription:**

Explain why the decision tree of any sorting algorithm on \( n \) elements must have at least \( n! \) leaf nodes.

---

**Explanation for Educational Context:**

In the context of sorting algorithms, the decision tree is a conceptual tool used to understand the comparisons made by the algorithm to determine a sorted order. Each internal node of the tree represents a comparison between elements, and each leaf node represents a possible outcome (i.e., a sorted permutation).

For a complete sorting of \( n \) elements, there are \( n! \) possible permutations — each a different way to arrange the elements in order. To account for all possible input sequences and outcomes, the decision tree must have at least \( n! \) leaf nodes, where each leaf node corresponds to a unique permutation. This ensures that for any initial arrangement of the elements, the algorithm can lead to the correct sorted order.

This concept underscores the lower bound of the complexity of comparison-based sorting algorithms, which is \( O(n \log n) \), as the structure of the decision tree implies a minimum number of comparisons required.

Expert Solution

Step 1

The decision tree for any sorting algorithm must have n! permutations on n elements to sort properly.

Step by step

Solved in 4 steps with 1 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, computer-science and related others by exploring similar questions and additional content below.

Similar questions

Generate the decision tree for sorting the following sequence 50, 100, 150 with all its possible permutations
Even in the worst case scenario, using a self-balancing binary search tree to do a binary tree sort takes O(n log n) time, which is much longer than the time required for merge sort.
Data Structure and algorithms
- The search for an element on a binary search tree has an average complexity which is A Linear B Logarithmic C Polynomial D Exponential
Using a self-balancing binary search tree, binary tree sort is nevertheless slower than merge sort, taking O(n log n) time in the worst case.
Data Structure and algorithms
Binary tree sort employing a self-balancing binary search tree takes O(n log n) time in the worst scenario, slower than merge sort.
Even when employing a self-balancing binary search tree to construct binary tree sort, the worst-case time required is O(n log n), making it slower than merge sort.
By moving through an array from left to right and adding each entry, a binary search tree was made. Print every conceivable array that may have resulted to a binary search tree with unique elements.
In the worst-case situation, a binary tree sort that employs a self-balancing binary search tree requires temporal complexity of the kind O(n log n). Is this approach much more sluggish than merge sort?
Given a sorted array of n comparable items A, create a binary search tree from the items in A which has height h <= log: n. Your algorithm must create the tree in O(n) time.