Abstract
The k-means problem has been paid much attention for many applications. In this paper, we define the uncertain constrained k-means problem and propose a (1+ϵ)-approximate algorithm for the problem. First, a general mathematical model of the uncertain constrained k-means problem is proposed. Second, the random sampling properties of the uncertain constrained k-means problem are studied. This paper mainly studies the gap between the center of random sampling and the real center, which should be controlled within a given range with a large probability, so as to obtain the important sampling properties to solve this kind of problem. Finally, using mathematical induction, we assume that the first j−1 cluster centers are obtained, so we only need to solve the j-th center. The algorithm has the elapsed time O((1891ekϵ2)8k/ϵnd), and outputs a collection of size O((1891ekϵ2)8k/ϵn) of candidate sets including approximation centers.
Highlights
Mathematics 2022, 10, 144. https://The k-means problem has received much attention in the past several decades
We prove the random sampling properties of the uncertain constrained k-means problem, which are fundamental for our proposed algorithm
We proposed a general mathematical model of the uncertain constrained k-means problem, and studied the random sampling properties, which are very important to deal with the uncertain constrained k-means problem
Summary
The k-means problem has received much attention in the past several decades. The k-means problems consists of partitioning a set P of points in d-dimensional space Rd into k subsets P1 , . . . , Pk such that ∑ik=1 ∑ p∈ Pi || p − ci ||2 is minimized, where ci is the center of Pi , and || p − q|| is the distance between two points of p and q. Given a point set P ⊆ Rd , and a positive integer k, a list of constraints L, the constrained k-means problem is to partition P into k clusters P = { P1 , . The existing fastest approximation schemes for the constrained k-means problem takes O(2O(k/e) nd) time [12,13], which was first shown by Bhattacharya, Jaiswai, and Kumar [12]. Their algorithm gives a collection of size O(2O(k/e) ) of candidate approximate centers. By applying random sampling and mathematical induction, we propose a stochastic approximate algorithm with lower complexity for the uncertain constrained k-means problem.
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