Three Sorting Algorithms Using Priority Queues 1st edition by Amr Elmasry ISBN 3540206958Â 9783540206958
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Authors:Amr Elmasry , Tags:Algorithms and Computation , Author sort:Elmasry, Amr , Languages:Languages:eng , Published:Published:Oct 2003
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ISBN 10: 3540206958
ISBN 13: 978-3540206958
Author: Amr Elmasry
We establish a lower bound of B(n) = n⌈logn ⌉ − 2⌈logn⌉ + 1 on the number of comparisons performed by any algorithm that uses priority queues to sort n elements. Three sorting algorithms using priority queues are introduced. The first algorithm performs the same comparisons as the classical Mergesort algorithm, but in a different order. The second algorithm performs at most 2nlogn + O(n) comparisons, with the advantage of being adaptive; meaning that it runs faster when the input sequence has some presortedness. In particular, we show that this algorithm sorts an already sorted sequence in linear time; a fact that is not obvious since there is no special checks to guarantee this behavior. The third algorithm is almost implicit; it can be implemented using the input array and less than n extra bits. The number of comparisons performed by this algorithm is at most B(n)+2.5n. The three algorithms have the advantage of producing every element of the sorted output, after the first, in O(log n), and can be implemented to be practically efficient.
Three Sorting Algorithms Using Priority Queues 1st Table of contents:
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Introduction
- 1.1 Overview of Sorting Algorithms
- 1.2 The Role of Priority Queues in Sorting
- 1.3 Motivation for Using Priority Queues in Sorting Algorithms
- 1.4 Structure of the Paper
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Background and Preliminaries
- 2.1 Definition and Properties of Priority Queues
- 2.2 Types of Priority Queues: Binary Heaps, Fibonacci Heaps, and Others
- 2.3 Comparison of Sorting Algorithms: An Introduction
- 2.4 Time and Space Complexity Considerations in Sorting
- 2.5 Previous Work on Sorting with Priority Queues
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Sorting Algorithm 1: Heap Sort
- 3.1 Introduction to Heap Sort
- 3.2 How Priority Queues Are Used in Heap Sort
- 3.3 Detailed Algorithm for Heap Sort Using Priority Queues
- 3.4 Time Complexity Analysis of Heap Sort
- 3.5 Space Complexity of Heap Sort
- 3.6 Example Walkthrough of Heap Sort
- 3.7 Limitations and Performance Considerations
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Sorting Algorithm 2: Merge Sort with a Priority Queue
- 4.1 Introduction to Merge Sort
- 4.2 Integrating a Priority Queue with Merge Sort
- 4.3 Detailed Algorithm for Merge Sort Using Priority Queues
- 4.4 Time Complexity Analysis of Merge Sort with Priority Queue
- 4.5 Space Complexity of Merge Sort Using Priority Queues
- 4.6 Example Walkthrough of Merge Sort with Priority Queues
- 4.7 Advantages and Disadvantages of Using a Priority Queue
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Sorting Algorithm 3: k-Way Merge Sort Using Priority Queues
- 5.1 Introduction to k-Way Merge Sort
- 5.2 Role of Priority Queues in k-Way Merge Sort
- 5.3 Detailed Algorithm for k-Way Merge Sort with Priority Queues
- 5.4 Time Complexity Analysis of k-Way Merge Sort
- 5.5 Space Complexity of k-Way Merge Sort Using Priority Queues
- 5.6 Example Walkthrough of k-Way Merge Sort with Priority Queues
- 5.7 Performance Evaluation and Use Cases
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Comparison of the Three Sorting Algorithms
- 6.1 Time Complexity Comparison: Heap Sort, Merge Sort, and k-Way Merge Sort
- 6.2 Space Complexity Comparison
- 6.3 Practical Considerations for Choosing Between the Algorithms
- 6.4 Real-World Applications of Sorting with Priority Queues
- 6.5 Trade-offs Between Simplicity, Efficiency, and Space Usage
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Optimizations and Variants
- 7.1 Optimizing Priority Queue Operations for Sorting
- 7.2 Use of Fibonacci Heaps and Other Advanced Priority Queues
- 7.3 Parallelization of Sorting Algorithms Using Priority Queues
- 7.4 Adaptive Sorting with Priority Queues
- 7.5 Hybrid Sorting Algorithms Incorporating Priority Queues
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Applications of Sorting with Priority Queues
- 8.1 Sorting Large Datasets and External Sorting
- 8.2 Real-Time Data Processing and Streaming Algorithms
- 8.3 Sorting in Distributed Systems and MapReduce
- 8.4 Applications in Computational Biology and Bioinformatics
- 8.5 Use of Priority Queues in Database Management Systems (DBMS)
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Challenges and Open Problems
- 9.1 Limitations of Priority Queues in Sorting Algorithms
- 9.2 Scaling Sorting Algorithms for Very Large Datasets
- 9.3 Open Problems in Optimizing Sorting Algorithms with Priority Queues
- 9.4 Exploring New Priority Queue Variants for Sorting
- 9.5 Future Research Directions in Sorting with Priority Queues
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Conclusion
- 10.1 Summary of Key Insights from the Three Sorting Algorithms
- 10.2 Contributions to the Field of Sorting and Priority Queue Applications
- 10.3 Final Thoughts on Choosing and Optimizing Sorting Algorithms
- 10.4 Closing Remarks on Future Developments in Sorting with Priority Queues
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