# K-RBBSO Algorithm: A Result-Based Stochastic Search Algorithm in Big Data
## 1 Introduction
The following issues have been reported in methods proposed to improve the clustering efficiency:  
* Problem 1: The silhouette coefficient is not suitable for application to the evaluation value-based stochastic optimum solution search algorithm. * Problem 2: As the evaluation of the silhouette coefficient requires significant computation, high computation time is required when applied to big data. * Problem 3: Stochastic search for an optimum solution corresponding to a random initial solution is very time-consuming. Problem 1 is related to the silhouette coefficient and the stochastic search equation of the optimization algorithm. In general, the optimization algorithm performs a stochastic search based on the initial search result. Most evaluation functions search for the optimal evaluation value by progressing from high to low values. For example, when the Euclidean distance is used as the evaluation function for data with k clusters, if the first and second clustering evaluation values are 80 and 70, respectively, the second clustering is considered to be superior to the first one for k clusters. However, the case of the silhouette coefficient is different--its search begins at \(-1\) and progresses toward \(1\). At this point, the algorithm used to determine the probability of identifying the solution in the next iteration based on the result relies on the accepted evaluation value. By design, the evaluation value progressively decreases in a general stochastic search model. Conversely, the evaluation value of the silhouette coefficient increases over time. As a result, the existing probability search formula cannot be used in this case. Moreover, when the silhouette coefficient is applied to the search probability equation based on the evaluation value, a comparison of the random probability in (0, 1) with the search probability yields a value that is unconditionally larger than the former. Consequently, the algorithm does not function properly. Two representative algorithms are associated with Problem 1--simulated annealing (SA) and artificial bee colony (ABC). SA [13] mimics the manufacture of steel products. By design, the algorithm identifies stochastically better solutions by accepting an evaluation value that is stochastically worse than the current evaluation value based on Equation (1):  
\[e^{-(f(n)-f(e))/\mathcal{T}}, \tag{1}\]  
where f(n) denotes the previous evaluation value, f(e) denotes the current evaluation value, and T denotes the current temperature. In the SA algorithm, if the value of the expression in Equation (1) is smaller than the randomly produced value between 0 and 1, the stochastically poorer evaluation value is adopted; otherwise, the current evaluation value is used until the termination condition is met. Next, let us consider the application of the silhouette coefficient to this equation. Assume that the previous evaluation value is \(-0.7\), the current evaluation value is \(-0.1\), and T = 0.8. Then, Equation (1) exhibits a value exceeding 1, and thus, the worse solution is unconditionally accepted. As a result, the corresponding formula must be modified to make it suitable for the application of the silhouette coefficient. The ABC [14] algorithm mimics the honey-gathering process of honeybees. It identifies optimal solutions stochastically. By design, several on-looker bees (OBs) may be selected during the solution search, and one of them is stochastically searched for. The following probability equation is used for the stochastic search:  
\[P_{i}=\frac{f(i)}{\sum_{j=1}^{OB}f(j)}, \tag{2}\]  
where P denotes the search probability for each OB and f(i) denotes the evaluation value corresponding to the solution identified using the ith OB. The denominator is defined to be the sum of the evaluation values corresponding to the solutions identified based on all previous OBs. A stochastic search is performed using this probability. However, the silhouette coefficient cannot be applied without modifying the formula.