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.
Main Research Question: Can the performance of the Select algorithm be improved for nondistinct elements, and if so, how?
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.
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.
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.
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.
Link to Article: https://arxiv.org/abs/0204033v1 Authors: arXiv ID: 0204033v1