**1. Introduction**

Clustering can be considered as a key problem in data analysis and data mining [1]. It tries to group similar data objects into the sets of disjoint classes or clusters [2,3]. Symmetry can be considered as a pre-attentive feature that can improve the shapes and objects as well as reconstruction and recognition. Furthermore, the symmetry-based clustering methods search for clusters that are symmetric with respect to their centers. As each cluster has the point symmetry property, an efficient point symmetry distance (PSD) was proposed by Su and Chou [4] to support partitioning the dataset into the clusters. They proposed a symmetry-based K-means (SBKM) algorithm that can give symmetry with minimum movement [5].

The literature shows that there are many techniques to solve clustering problems. They are basically categorized as the hierarchical clustering algorithm and the partitional clustering algorithm [6,7].

The hierarchical clustering algorithms mainly generate clusters hierarchies signified in a tree structure called a dendrogram. It can be successively proceeded throughout either splitting larger clusters or merging smaller clusters into larger ones. The clusters identify as the connected components in a tree. Hence, the output of hierarchical clustering algorithms is a tree of clusters [8–10].

However, partitional clustering algorithms attempt to treat a dataset as one cluster and try to decompose it into disjoint clusters. As partitional clustering does not need to make a

**Citation:** Alotaibi, Y. A New Meta-Heuristics Data Clustering Algorithm Based on Tabu Search and Adaptive Search Memory. *Symmetry* **2022**, *14*, 623. https://doi.org/ 10.3390/sym14030623

Academic Editor: Kuo-Hui Yeh

Received: 3 March 2022 Accepted: 18 March 2022 Published: 20 March 2022

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**Copyright:** © 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

tree structure, it can be considered to be easier than hierarchy clustering. The partitional algorithms are restricted with a criterion function while decomposing the dataset, and thus this function leads to minimizing dissimilarity in each cluster and maximizing similarity in data objects of each cluster [11].

The K-means (K-M) algorithm is considered to be one of the most common algorithms of partitional clustering [12–14]. However, it has some disadvantages, such as specifying the number of k clusters, as it is based on practical experience and knowledge. The main K-M problems are its sensitivity to initialization and becoming trapped in local optima [15,16]. Therefore, meta-heuristics algorithms are used to escape from these problems.

Meta-heuristics is an iterative generation processes that can guide a heuristic algorithm by different responsive concepts for exploring and exploiting search space [17]. It is a class of approximate methods that can attack hard combinatorial problems when the classical optimization methods fail. It allows for the creation of new hybrid methods by combining different concepts [18,19].

Tabu Search (TS) is considered to be one of the meta-heuristics algorithms based on memory objects to make economic exploitation and exploration for search space. Tabu list (TL) is one of the main memory elements of TS. It is a list that contains recently visited solutions by TS. It is used by TS in order to avoid re-entering pre-visited regions [20,21].

This paper aims to propose a new meta-heuristics Tabu Search with an Adaptive Search Memory algorithm (MHTSASM) for data clustering, which is based on TS and K-M. MHTSASM is an advanced TS that uses responsive and intelligent memory. The proposed MHTSASM algorithm uses multiple memory elements. For example, it uses the elite list (EL) memory element of TS to store the best solutions and Adaptive Search Memory (ASM) to store information about features of each solution. The ASM partitions each feature range into p partitions and stores information about these partitions in each center. There are two types of ASM used in the proposed algorithm: (1) ASMv stores number of visits; and (2) ASMu stores the number of improvement updates. In addition, there are two TS strategies used in the proposed algorithm: (1) the intensification strategy, which allows MHTSASM to focus on the best pre-visited regions; and (2) the diversification strategy, which allows MHTSASM to explore un-visited regions to ensure good exploration for search space.

The results of MHTSASM are specifically compared with other optimization and metaheuristics algorithms. The performance of MHTSASM is compared with several variable neighborhood searches, such as Variable Neighborhood Search (VNS), VNS+, and VNS-1 algorithms [22], Simulated Annealing (SA) [23], and global K-means (G1) [24]. Additionally, it is compared with hybrid algorithms, such as a new hybrid algorithm based on particle swarm optimization with K-harmonic means (KHM) [25], and an ant colony optimization clustering algorithm (ACO) with KHM [26,27].

The remainder of this paper is organized as follows: Section 2 presents the background of the proposed MHTSASM clustering algorithm. Section 3 presents the proposed MHTSASM clustering algorithm. Section 4 illustrates experimental results. Finally, the conclusion is presented in Section 5.
