Efficient Selection Algorithm for Nondistinct Elements: Difference between revisions

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Created page with "Title: Efficient Selection Algorithm for Nondistinct Elements Abstract: This research article presents an improved version of the selection algorithm known as Select, originally proposed by Floyd and Rivest in 1975. The algorithm is used to find the kth smallest element in a set of n elements, with nondistinct elements. The main contribution of this research is the analysis of the algorithm, showing that it requires at most n + min {k, n - k} + O(n^2/3 ln 1/3 n) compari..."
 
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Title: Efficient Selection Algorithm for Nondistinct Elements
Title: Efficient Selection Algorithm for Nondistinct Elements


Abstract: This research article presents an improved version of the selection algorithm known as Select, originally proposed by Floyd and Rivest in 1975. The algorithm is used to find the kth smallest element in a set of n elements, with nondistinct elements. The main contribution of this research is the analysis of the algorithm, showing that it requires at most n + min {k, n - k} + O(n^2/3 ln 1/3 n) comparisons on average. This result is significant because it improves upon the previously assumed upper bound and provides a more accurate estimate of the algorithm's performance. The research also proves that non-recursive versions of the algorithm, which use other selection or sorting algorithms for smaller subproblems, require at most n + min {k, n - k} + o(n) comparisons with high probability. These results make Select a more attractive choice for finding the kth smallest element in large sets of nondistinct elements.
Abstract: This research article presents an improved version of the Select algorithm for finding the kth smallest element in a set of n elements, even when those elements are not distinct. The algorithm uses a randomized selection method with quintary partitions, which significantly reduces the average number of comparisons needed to find the kth smallest element. The results show that the Select algorithm can be more efficient than other existing methods, making it a potentially better choice for practical applications.


Main Research Question: Can the performance of the Select algorithm be improved for nondistinct elements, and if so, how?
Main Research Question: Can the Select algorithm be improved to work more efficiently with nondistinct elements?


Methodology: The research uses the comparison model to analyze the performance of the Select algorithm. The algorithm is modified slightly to simplify the analysis, and the average and high-probability bounds for the number of comparisons needed to find the kth smallest element are derived. The results are based on the randomized selection method, which involves taking a random sample of the elements and comparing them with the other elements to find the kth smallest one.
Methodology: The study uses a randomized selection method with quintary partitions, which involves dividing the set of elements into five subsets based on their values. This method is applied to the Select algorithm to reduce the number of comparisons needed to find the kth smallest element.


Results: The main result is that the Select algorithm requires at most n + min {k, n - k} + O(n^2/3 ln 1/3 n) comparisons on average to find the kth smallest element. This is an improvement over the previously assumed upper bound and provides a more accurate estimate of the algorithm's performance. Additionally, the research shows that non-recursive versions of the algorithm require at most n + min {k, n - k} + o(n) comparisons with high probability.
Results: The research shows that the improved Select algorithm requires at most n + min {k, n - k} + O(n^2/3 ln^1/3 n) comparisons on average. This result agrees with the previous analysis of the Select algorithm, which was based on an unproven assumption. The study also proves that non-recursive versions of the Select algorithm, which use other selection or sorting algorithms for smaller subproblems, require at most n + min {k, n - k} + o(n) comparisons with high probability.


Implications: The improved performance of the Select algorithm makes it a more attractive choice for finding the kth smallest element in large sets of nondistinct elements. The results also provide a better understanding of the algorithm's performance and may lead to further improvements in the design and analysis of selection algorithms.
Implications: The improved Select algorithm presented in this study has significant implications for practical applications. It offers a more efficient method for finding the kth smallest element in a set of n elements, even when those elements are not distinct. This could lead to faster implementation of the algorithm in various applications, such as sorting and finding convex hulls.


In conclusion, this research presents an improved version of the Select algorithm for nondistinct elements, showing that it requires at most n + min {k, n - k} + O(n^2/3 ln 1/3 n) comparisons on average and at most n + min {k, n - k} + o(n) comparisons with high probability. These results make Select a more efficient choice for finding the kth smallest element in large sets of nondistinct elements.
Keywords: Selection algorithm, nondistinct elements, quintary partitions, computational complexity


Link to Article: https://arxiv.org/abs/0204033v1
Link to Article: https://arxiv.org/abs/0312055v1
Authors:  
Authors:  
arXiv ID: 0204033v1
arXiv ID: 0312055v1


[[Category:Computer Science]]
[[Category:Computer Science]]
[[Category:Algorithm]]
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[[Category:N]]
[[Category:Algorithm]]
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[[Category:Elements]]
[[Category:Elements]]
[[Category:Select]]
[[Category:Select]]
[[Category:Selection]]

Latest revision as of 15:09, 24 December 2023

Title: Efficient Selection Algorithm for Nondistinct Elements

Abstract: This research article presents an improved version of the Select algorithm for finding the kth smallest element in a set of n elements, even when those elements are not distinct. The algorithm uses a randomized selection method with quintary partitions, which significantly reduces the average number of comparisons needed to find the kth smallest element. The results show that the Select algorithm can be more efficient than other existing methods, making it a potentially better choice for practical applications.

Main Research Question: Can the Select algorithm be improved to work more efficiently with nondistinct elements?

Methodology: The study uses a randomized selection method with quintary partitions, which involves dividing the set of elements into five subsets based on their values. This method is applied to the Select algorithm to reduce the number of comparisons needed to find the kth smallest element.

Results: The research shows that the improved Select algorithm requires at most n + min {k, n - k} + O(n^2/3 ln^1/3 n) comparisons on average. This result agrees with the previous analysis of the Select algorithm, which was based on an unproven assumption. The study also proves that non-recursive versions of the Select algorithm, which use other selection or sorting algorithms for smaller subproblems, require at most n + min {k, n - k} + o(n) comparisons with high probability.

Implications: The improved Select algorithm presented in this study has significant implications for practical applications. It offers a more efficient method for finding the kth smallest element in a set of n elements, even when those elements are not distinct. This could lead to faster implementation of the algorithm in various applications, such as sorting and finding convex hulls.

Keywords: Selection algorithm, nondistinct elements, quintary partitions, computational complexity

Link to Article: https://arxiv.org/abs/0312055v1 Authors: arXiv ID: 0312055v1