**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.

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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

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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.

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

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