Efficient Selection Algorithm for Nondistinct Elements

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