# A Comprehensive Survey of Clustering Algorithms

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## Abstract

Data analysis is used as a common method in modern science research, which is across communication science, computer science and biology science. Clustering, as the basic composition of data analysis, plays a significant role. On one hand, many tools for cluster analysis have been created, along with the information increase and subject intersection. On the other hand, each clustering algorithm has its own strengths and weaknesses, due to the complexity of information. In this review paper, we begin at the definition of clustering, take the basic elements involved in the clustering process, such as the distance or similarity measurement and evaluation indicators, into consideration, and analyze the clustering algorithms from two perspectives, the traditional ones and the modern ones. All the discussed clustering algorithms will be compared in detail and comprehensively shown in Appendix Table 22.

## Keywords

Clustering Clustering algorithm Clustering analysis Survey Unsupervised learning## 1 Introduction

- (1)
Instances, in the same cluster, must be similar as much as possible;

- (2)
Instances, in the different clusters, must be different as much as possible;

- (3)
Measurement for similarity and dissimilarity must be clear and have the practical meaning;

- (1)
Feature extraction and selection: extract and select the most representative features from the original data set;

- (2)
Clustering algorithm design: design the clustering algorithm according to the characteristics of the problem;

- (3)
Result evaluation: evaluate the clustering result and judge the validity of algorithm;

- (4)
Result explanation: give a practical explanation for the clustering result;

## 2 Distance and Similarity

Distance (dissimilarity) and similarity are the basis for constructing clustering algorithms. As for quantitative data features, distance is preferred to recognize the relationship among data. And similarity is preferred when dealing with qualitative data features [2].

Distance functions

Name | Formula | Explanation |
---|---|---|

Minkowski distance | \(\left( {\sum \limits _{l=1}^d {\left| {x_{il} -x_{jl} } \right| ^{n}} } \right) ^{1/n}\) | A set of definitions for distance: |

1. City-block distance when n = 1 | ||

2. Euclidean distance when n = 2 | ||

3. Chebyshev distance when n \(\rightarrow \quad \infty \) | ||

Standardized Euclidean distance | \(\left( {\sum \limits _{l=1}^d {\left| {\frac{x_{il} -x_{jl} }{s_l }} \right| ^{2}} } \right) ^{1/2}\) | 1. S stands for the standard deviation |

2. A weighted Euclidean distance based on the deviation | ||

Cosine distance | \(1-\cos \alpha =\frac{x_i ^{T}x_j }{\left\| {x_i } \right\| \left\| {x_j } \right\| }\) | 1. Stay the same in face of the rotation change of data |

2. The most commonly used distance in document area | ||

Pearson correlation distance | \(1-\frac{Cov\left( {x_i ,x_j } \right) }{\sqrt{D\left( {x_i } \right) }\sqrt{D\left( {x_j } \right) }}\) | 1. Cov stands for the covariance for and D stands for the variance |

2. Measure the distance based on linear correlation | ||

Mahalanobis distance | \(\sqrt{\left( {x_i -x_j } \right) ^{T}S^{-1}\left( {x_i -x_j } \right) }\) | 1. S is the covariance matrix inside the cluster |

2. With high computation complexity |

Similarity functions

Name | Function formula or measure method | Explanation |
---|---|---|

Jaccard similarity | \(J\left( {A,B} \right) =\frac{\left| {A\cap B} \right| }{\left| {A\cup B} \right| }\) | 1. Measure the similarity of two sets |

2. \(|\hbox {X}|\) Stands for the number of elements of set X | ||

3. Jaccard distance = 1 \(-\) Jaccard similarity | ||

Hamming similarity | The minimum number of substitutions needed to change one data point into the other | The number is smaller, the similarity is more |

Hamming distance is the opposite of Hamming similarity | ||

Especially for the data of string | ||

For data of mixed type | Map the feature into (0, 1) | |

Transform the feature into dichotomous one | ||

\(S_{ij} =\frac{1}{d}\sum \limits _{l=1}^d {S_{ijl} } \) | ||

\(S_{ij} ={\left( {\sum _{l=1}^d {\eta _{ijl} S_{ijl} } } \right) }/{\left( {\sum _{l=1}^d {\eta _{ijl} } } \right) }\) |

## 3 Evaluation Indicator

The main purpose of evaluation indicator is to test the validity of algorithm. Evaluation indicators can be divided into two categories, the internal evaluation indicators and the external evaluation indicators, in terms of the test data whether in the process of constructing the clustering algorithm.

Evaluation indicators

Name | Formula or measure method | Explanation |
---|---|---|

Davies–Bouldin indicator | \(DB=\frac{1}{k}\sum \limits _{i=1}^k {\max \limits _{i\ne j} \left( {\frac{\sigma _i +\sigma _j }{d\left( {c_i ,c_j } \right) }} \right) } \) | K stands for the number of clusters, \(C_x \) is the center of cluster |

Dunn indicator | \(D=\min \limits _{1\le i\le n} \left\{ \min \limits _{1\le j\le n,i\ne j} \left\{ {\frac{d\left( {i,j} \right) }{\max _{1\le k\le n} {d}'\left( k \right) }} \right\} \right\} \) | 1. Mainly for the data that has even density and distribution |

2. \(d\left( {c_i ,c_j } \right) \)is the distance between \(c_i \) and \(c_j \), \({d}'\left( k \right) \) stands for the distance in cluster | ||

Silhouette coefficient | Evaluate the clustering result based on the average distance between a data point and other data points in the same cluster and average distance among different clusters |

Evaluation indicators

Name | Formula or measure method | Explanation |
---|---|---|

Rand indicator | \(RI=\frac{TP+TN}{TP+FP+FN+TN}\) | 1. TP is the number of true positives |

2. TN is the number of true negatives | ||

3. FP is the number of false positives | ||

4. FN is the number of false negatives | ||

F indicator | \(F_\beta =\frac{\left( {\beta ^{2}+1} \right) \cdot P\cdot R}{\beta ^{2}\cdot P+R}\) | 1. \(P=\frac{TP}{TP+FP}\) stands for the accuracy, \(R=\frac{TP}{TP+FN}\) stands for the recall rate |

2. TP, TN, FP, and FN are defined as before | ||

Jaccard indicator | \(J\left( {A,B} \right) =\frac{\left| {A\cap B} \right| }{\left| {A\cup B} \right| }=\frac{TP}{TP+FP+FN}\) | 1. Measure the similarity of two sets |

2. \(|\hbox {X}|\) Stands for the number of elements of set X | ||

3. TP, TN, FP, and FN are defined as before | ||

Fowlkes–Mallows indicator | \(FM=\sqrt{\frac{TP}{TP+FP}\cdot \frac{TP}{TP+FN}}\) | TP, TN, FP, and FN are defined as before |

Mutual information | To measure, based on information theory, how much information is shared by two clusters, between which the nonlinear correlation can be detected | |

Confusion matrix | To figure out the difference between a cluster and a gold-standard cluster |

In the following sections, especially in the analysis of time complexity, n stands for the number of total objects/data points, k stands for the number of clusters, s stands for the number of sample objects/data points, and t stands for the number of iterations.

## 4 Traditional Clustering Algorithms

Traditional algorithms

Category | Typical algorithm |
---|---|

Clustering algorithm based on partition | K-means, K-medoids, PAM, CLARA, CLARANS |

Clustering algorithm based on hierarchy | BIRCH, CURE, ROCK, Chameleon |

Clustering algorithm based on fuzzy theory | FCM, FCS, MM |

Clustering algorithm based on distribution | DBCLASD, GMM |

Clustering algorithm based on density | DBSCAN, OPTICS, Mean-shift |

Clustering algorithm based on graph theory | CLICK, MST |

Clustering algorithm based on grid | STING, CLIQUE |

Clustering algorithm based on fractal theory | FC |

Clustering algorithm based on model | COBWEB, GMM, SOM, ART |

### 4.1 Clustering Algorithm Based on Partition

The basic idea of this kind of clustering algorithms is to regard the center of data points as the center of the corresponding cluster. K-means [7] and K-medoids [8] are the two most famous ones of this kind of clustering algorithms. The core idea of K-means is to update the center of cluster which is represented by the center of data points, by iterative computation and the iterative process will be continued until some criteria for convergence is met. K-mediods is an improvement of K-means to deal with discrete data, which takes the data point, most near the center of data points, as the representative of the corresponding cluster. The typical clustering algorithms based on partition also include PAM [9], CLARA [10], CLARANS [11].

For more information about this kind of clustering algorithms, you can refer to [12, 13, 14].

- (1)
Time complexity (Table 6):

- (2)
Advantages: relatively low time complexity and high computing efficiency in general;

- (3)
Disadvantages: not suitable for non-convex data, relatively sensitive to the outliers, easily drawn into local optimal, the number of clusters needed to be preset, and the clustering result sensitive to the number of clusters.;

- (4)
AP algorithm [15], which will be discussed in the section Clustering algorithm based on affinity propagation, can also be considered as one of this kind of clustering algorithm.

Time complexity

K-means | K-medoids | PAM | CLARA | CLARANS |
---|---|---|---|---|

O(knt) | O(k(n-k)\(\hat{\,\,}\)2) | O(k\(\hat{\,\,}\)3*n\(\hat{\,\,}\)2) | O(ks\(\hat{\,\,}\)2+k(n-k)) | O(n\(\hat{\,\,}\)2) |

Low | High | High | Middle | High |

### 4.2 Clustering Algorithm Based on Hierarchy

The basic idea of this kind of clustering algorithms is to construct the hierarchical relationship among data in order to cluster [16]. Suppose that each data point stands for an individual cluster in the beginning, and then, the most neighboring two clusters are merged into a new cluster until there is only one cluster left. Or, a reverse process. Typical algorithms of this kind of clustering include BIRCH [17], CURE [18], ROCK [19], Chameleon [20]. BIRCH realizes the clustering result by constructing the feature tree of clustering, CF tree, of which one node stands for a subcluster. CF tree will dynamically grow when a new data point comes. CURE, suitable for large-scale clustering, takes random sampling technique to cluster sample separately and integrates the results finally. ROCK is an improvement of CURE for dealing with data of enumeration type, which takes the effect on the similarity from the data around the cluster into consideration. Chameleon, at first, divides the original data into clusters with smaller size based on the nearest neighbor graph, and then the clusters with small size are merged into a cluster with bigger size, based on agglomerative algorithm, until satisfied.

For more information about this kind of clustering algorithms, you can refer to [21, 22].

- (1)
Time complexity (Table 7):

- (2)
Advantages: suitable for the data set with arbitrary shape and attribute of arbitrary type, the hierarchical relationship among clusters easily detected, and relatively high scalability in general;

- (3)
Disadvantages: relatively high in time complexity in general, the number of clusters needed to be preset.

Time complexity

BIRCH | CURE | ROCK | Chameleon |
---|---|---|---|

O(n) | O(s\(\hat{\,\,}\)2*s), | O(n\(\hat{\,\,}\)2*logn) | O(n\(\hat{\,\,}\)2) |

Low | Low | High | High |

### 4.3 Clustering Algorithm Based on Fuzzy Theory

The basic idea of this kind of clustering algorithms is that the discrete value of belonging label, {0, 1}, is changed into the continuous interval [0, 1], in order to describe the belonging relationship among objects more reasonably. Typical algorithms of this kind of clustering include FCM [23, 24, 25], FCS [26] and MM [27]. The core idea of FCM is to get membership of each data point to every cluster by optimizing the object function. FCS, different from the traditional fuzzy clustering algorithms, takes the multidimensional hypersphere as the prototype of each cluster, so as to cluster with the distance function based on the hypersphere. MM, based on the Mountain Function, is used to find the center of cluster.

For more information about this kind of clustering algorithms, you can refer to [28, 29, 30].

- 1)
Time complexity (Table 8):

- 2)
The time complexity of FCS is high for the kernel involved in the algorithm;

- 3)
Advantages: more realistic to give the probability of belonging, relatively high accuracy of clustering;

- 4)
Disadvantages: relatively low scalability in general, easily drawn into local optimal, the clustering result sensitive to the initial parameter values, and the number of clusters needed to be preset.

Time complexity

FCM | FCS | MM |
---|---|---|

O(n) | (kernel) | O(v\(\hat{\,\,}\)2*n) |

Low | High | Middle |

### 4.4 Clustering Algorithm Based on Distribution

The basic idea is that the data, generated from the same distribution, belongs to the same cluster if there exists several distributions in the original data. The typical algorithms are DBCLASD [31] and GMM [32]. The core idea of DBCLASD, a dynamic incremental algorithm, is that if the distance between a cluster and its nearest data point satisfies the distribution of expected distance which is generated from the existing data points of that cluster, the nearest data point should belong to this cluster. The core idea of GMM is that GMM consists of several Gaussian distributions from which the original data is generated and the data, obeying the same independent Gaussian distribution, is considered to belong to the same cluster.

For more information about this kind of clustering algorithms, you can refer to [33, 34].

- (1)
Time complexity (Table 9):

- (2)
Advantages: more realistic to give the probability of belonging, relatively high scalability by changing the distribution, number of clusters and so on, and supported by the well developed statistical science;

- (3)
Disadvantages: the premise not completely correct, involved in many parameters which have a strong influence on the clustering result and relatively high time complexity.

Time complexity

DBCLASD | GMM |
---|---|

O(n*logn) | O(n\(\hat{\,\,}\)2*kt) |

Middle | High |

### 4.5 Clustering Algorithm Based on Density

The basic idea of this kind of clustering algorithms is that the data which is in the region with high density of the data space is considered to belong to the same cluster [35]. The typical ones include DBSCAN [36], OPTICS [37] and Mean-shift [38]. DBSCAN is the most well known density-based clustering algorithm, which is generated from the basic idea of this kind of clustering algorithms directly. OPTICS is an improvement of DBSCAN and it overcomes the shortcoming of DBSCAN that being sensitive to two parameters, the radius of the neighborhood and the minimum number of points in a neighborhood. In the process of Mean-shift, the mean of offset of current data point is calculated at first, the next data point is figured out based on the current data point and the offset then, and last, the iteration will be continued until some criteria are met.

For more information about this kind of clustering algorithms, you can refer to [39, 40, 41, 42].

- (1)
Time complexity (Table 10):

- (2)
The time complexity of Mean-shift is high for the kernel involved in the algorithm;

- (3)
Advantages: clustering in high efficiency and suitable for data with arbitrary shape;

- (4)
Disadvantages: resulting in a clustering result with low quality when the density of data space isn’t even, a memory with big size needed when the data volume is big, and the clustering result highly sensitive to the parameters;

- (5)
DENCLUE algorithm [43], which will be discussed in the section Clustering algorithm for large-scale data, can also be considered as one of this kind of clustering algorithms.

Time complexity

DBSCAN | OPTICS | Mean-shift |
---|---|---|

O(n*logn) | O(n*logn) | (kernel) |

Middle | Middle | High |

### 4.6 Clustering Algorithm Based on Graph Theory

According to this kind of clustering algorithms, clustering is realized on the graph where the node is regarded as the data point and the edge is regarded as the relationship among data points. Typical algorithms of this kind of clustering are CLICK [44] and MST-based clustering [45]. The core idea of CLICK is to carry out the minimum weight division of the graph with iteration in order to generate the clusters. Generating the minimum spanning tree from the data graph is the key step to do the cluster analysis for the MST-based clustering algorithm.

For more detailed information about this kind of clustering algorithms, you can refer to [1, 20, 46, 47, 48, 49].

- (1)
Time complexity (Table 11):

where v stands for the number of vertices, e stands for the number of edges, and f(v, e) stands for the time complexity of computing a minimum cut;

- (2)
Advantages: clustering in high efficiency, the clustering result with high accuracy;

- (3)
Disadvantages: the time complexity increasing dramatically with the increasing of graph complexity;

- (4)
SM algorithm [50] and NJW algorithm [51], which will be discussed in the section Clustering algorithm based on spectral graph theory, can also be considered as ones of this kind of clustering algorithms.

Time complexity

CLICK | MST |
---|---|

O(k*f(v, e)) | O(e*logv) |

Low | Middle |

### 4.7 Clustering Algorithm Based on Grid

The basic idea of this kind of clustering algorithms is that the original data space is changed into a grid structure with definite size for clustering. The typical algorithms of this kind of clustering are STING [52] and CLIQUE [53]. The core idea of STING which can be used for parallel processing is that the data space is divided into many rectangular units by constructing the hierarchical structure and the data within different structure levels is clustered respectively. CLIQUE takes advantage of the grid-based clustering algorithms and the density-based clustering algorithms.

For more detailed information about this kind of clustering algorithms, you can refer to [41, 54, 55, 56, 57].

- (1)
Time complexity (Table 12):

- (2)
Advantages: low time complexity, high scalability and suitable for parallel processing and increment updating;

- (3)
Disadvantages: the clustering result sensitive to the granularity (the mesh size), the high calculation efficiency at the cost of reducing the quality of clusters and reducing the clustering accuracy;

- 4)
Wavecluster algorithm [54], which will be discussed in the section Clustering algorithm for spatial data, can also be considered as ones of this kind of clustering algorithms.

Time complexity

STING | CLIQUE |
---|---|

O(n) | O(n+k\(\hat{\,\,}\)2) |

Low | Low |

### 4.8 Clustering Algorithm Based on Fractal Theory

Fractal stands for the geometry that can be divided into several parts which share some common characters with the whole [58]. The typical algorithm of this kind of clustering is FC [59] of which the core idea is that the change of any inner data of a cluster does not have any influence on the intrinsic quality of the fractal dimension.

For more detailed information about this kind of clustering algorithms, you can refer to [60, 61, 62, 63].

- (1)
The time complexity of FC is O(n);

- (2)
Advantages: clustering in high efficiency, high scalability, dealing with outliers effectively and suitable for data with arbitrary shape and high dimension;

- (3)
Disadvantages: the premise not completely correct, the clustering result sensitive to the parameters.

### 4.9 Clustering Algorithm Based on Model

The basic idea is to select a particular model for each cluster and find the best fitting for that model. There are mainly two kinds of model-based clustering algorithms, one based on statistical learning method and the other based on neural network learning method.

The typical algorithms, based on statistical learning method, are COBWEB [64] and GMM [32]. The core idea of COBWEB is to build a classification tree, based on some heuristic criteria, in order to realize hierarchical clustering on the assumption that the probability distribution of each attribute is independent. The typical algorithms, based on neural network learning method, are SOM [65] and ART [66, 67, 68, 69]. The core idea of SOM is to build a mapping of dimension reduction from the input space of high dimension to output space of low dimension on the assumption that there exists topology in the input data. The core idea of ART, an incremental algorithm, is to generate a new neuron dynamically to match a new pattern to create a new cluster when the current neurons are not enough. GMM has been discussed in the section Clustering algorithm based on distribution.

For more detailed information about this kind of clustering algorithms, you can refer to [70, 71, 72, 73, 74, 75].

- (1)
Time complexity (Table 13):

- (2)
The time complexity of COBWEB is generally low, which depends on the distribution involved in the algorithm;

- (3)
The time complexity of SOM is generally high, which depends on the layer construction involved in the algorithm;

- (4)
The time complexity of ART is generally middle, which depends on the type of ART and the layer construction involved in the algorithm;

- (5)
Advantages: diverse and well developed models providing means to describe data adequately and each model having its own special characters that may bring about some significant advantages in some specific areas;

- (6)
Disadvantages: relatively high time complexity in general, the premise not completely correct, and the clustering result sensitive to the parameters of selected models.

Time complexity

COBWEB | GMM | SOM | ART |
---|---|---|---|

(distribution) | O(n\(\hat{\,\,}\)2*kt) | (layer) | (type+layer) |

Low | High | High | Middle |

## 5 Modern Clustering Algorithms

Modern algorithms

Category | Typical algorithm |
---|---|

Clustering algorithm based on kernel | kernel K-means, kernel SOM, kernel FCM, SVC, MMC, MKC |

Clustering algorithm based on ensemble | Methods for generating the set of clusters: 4 types Consensus function: CSPA, HGPA, MCLA, VM, HCE, LAC, WPCK, sCSPA, sMCLA, sHBGPA |

Clustering algorithm based on swarm intelligence | ACO_based(LF), PSO_based, SFLA_based, ABC_based |

Clustering algorithm based on quantum theory | QC, DQC |

Clustering algorithm based on spectral graph theory | SM, NJW |

Clustering algorithm based on affinity propagation | AP |

Clustering algorithm based on density and distance | DD |

Clustering algorithm for spatial data | DBSCAN, STING, Wavecluster, CLARANS |

Clustering algorithm for data stream | STREAM, CluStream, HPStream, DenStream |

Clustering algorithm for large-scale data | K-means, BIRCH, CLARA, CURE, DBSCAN, DENCLUE, Wavecluster, FC |

### 5.1 Clustering Algorithm Based on Kernel

The basic idea of this kind of clustering algorithms is that data in the input space is transformed into the feature space of high dimension by the nonlinear mapping for the cluster analysis. The typical algorithms of this kind of clustering include kernel K-means [76], kernel SOM [77], kernel FCM [78], SVC [79], MMC [80] and MKC [81]. The basic idea of kernel K-means, kernel SOM and kernel FCM is to take advantage of the kernel method and the original clustering algorithm, transforming the original data into a high dimensional feature space by nonlinear kernel function in order to carry out the original clustering algorithm. The core idea of SVC is to find the sphere with the minimum radius that can cover all the data point in the high dimensional feature space, then map the sphere back into the original data space to form the isoline, namely the border of clusters, covering the data, and the data in the closed isoline should belong to the same cluster. MMC tries to find the hyperplane with the maximum margin to cluster and it can be promoted for the multi-label clustering problem. MKC, an improvement of MMC, tries to find the best hyperplane based on several kernels to cluster. MMC and MKC share the limitation of computation to a degree.

For more detailed information about this kind of clustering algorithms, you can refer to [82, 83, 84].

- (1)
Time complexity (Table 15):

- (2)
The time complexity of this kind of clustering algorithms is generally high for the kernel involved in the algorithm;

- (3)
Advantages: more easy to cluster in the high dimensional feature space, suitable for data with arbitrary shape, able to analyze the noise and separate the overlapping clusters, and not needed to have the preliminary knowledge about the topology of data;

- (4)
Disadvantages: the clustering result sensitive to the type of kernel and its parameters, time complexity being high, and not suitable for large-scale data.

Time complexity

kernel K-means | kernel SOM | kernel FCM | SVC | MMC | MKC |
---|---|---|---|---|---|

(kernel) | (kernel) | (kernel) | (kernel) | (kernel) | (kernel) |

High | High | High | High | High | High |

### 5.2 Clustering Algorithm Based on Ensemble

- (1)
For the same data set, employ the same algorithm with the different parameters or the different initial conditions [85];

- (2)
For the same data set, employ the different algorithms [86];

- (3)
For the subsets, carry out the clustering respectively [86];

- (4)
For the same data set, carry out the clustering in different feature spaces based on different kernels [87].

Consensus functions

Name | Typical algorithm and application |
---|---|

Based on co-association matrix | [85] |

Based on graph partition | CSPA, HGPA and MCLA [86] |

Based on relabeling and voting | VM [88] |

Based on the hybrid model | [89] |

Based on information theory | [90] |

Based on genetic algorithm | HCE [91] |

Based on local adaptation | LAC [92] |

Based on kernel method | WPCK [93] |

Based on fuzzy theory | sCSPA, sMCLA and sHBGPA [94] |

For more detailed information about this kind of clustering algorithms, you can refer to [95].

- (1)
The time complexity of this kind of algorithm is based on the specific method and algorithms involved in the algorithm;

- (2)
Advantages: robust, scalable, able to be parallel and taking advantage of the strengths of the involved algorithms;

- (3)
Disadvantages: inadequate understanding about the difference among the initial clustering results, existing deficiencies of the design of the consensus function.

### 5.3 Clustering Algorithm Based on Swarm Intelligence

The basic idea of this kind of clustering algorithms is to simulate the changing process of the biological population. Typical algorithms include the 4 main categories: ACO_based [96, 97], PSO_based [97, 98], SFLA_based [99] and ABC_based [100]. The core idea of LF [101], the typical algprithm of the ACO_based, is that data is distributed randomly on the grid of two dimensions first, then the data is selected or not for further operation based on the decision of an ant and this process is iterated until a satisfactory clustering result is got. The PSO_based algorithms regard the data point as a particle. The initial clusters of particles is got by the other clustering algorithm first, then the clusters of particles is updated continuously based on the center of clusters and the location and speed of each particle, until a satisfactory clustering result is got. The core idea of the SFLA_based algorithms is to simulate the information interaction of frogs and taking advantage of the local search and the global information interaction. The core idea of the ABC_based algorithms is to simulate the foraging behavior of three types of bee, of which the duty is to determine the food source, in a bee population and making use of the exchange of local information and global information for clustering.

For more detailed information about this kind of clustering algorithms, you can refer to [102, 103, 104].

- (1)
Time complexity (Table 17):

- (2)
The time complexity of this kind of algorithm is high, mainly for the large number of iterations;

- (3)
Advantages: algorithm with the character of overcoming being easily drawn into local optimal and getting the global optimal, easy to understand the algorithm;

- (4)
Disadvantages: low scalability, low operating efficiency and not suitable for high dimensional or large-scale data.

Time complexity

ACO_based (LF) | PSO_based | SFLA_based | ABC_based |
---|---|---|---|

High | High | High | High |

### 5.4 Clustering Algorithm Based on Quantum Theory

The clustering algorithm based on quantum theory is called quantum clustering, of which the basic idea is to study the distribution law of sample data in the scale space by studying the distribution law of particles in the energy field. The typical algorithms of this kind include QC [105, 106] and DQC [107]. The core idea of QC (quantum clustering), suitable for high dimensional data, is to get the potential energy of each object by Schrodinger Equation using the iterative gradient descent algorithm, regard the object with low potential energy as the center of the cluster, and put the objects into different clusters by the defined distance function. DQC, an improvement of QC, adopts the time-based Schrodinger Equation in order to study the change of the original data set and the structure of the quantum potential energy function dynamically.

For more detailed information about this kind of clustering algorithms, you can refer to [108, 109, 110].

- (1)
Time complexity (Table 18):

- (2)
The time complexity of QC is high, for the process of solving the Schrodinger Equation and the large number of iterations;

- (3)
The time complexity of DQC which is more practical compared with DQ, is middle for the process of solving the Schrodinger Equation;

- (4)
Advantages: the number of parameters involved in this kind of algorithm being small, the determination of the center of a cluster based on the potential information of sample data;

- (5)
Disadvantages: the clustering result sensitive to the parameters of the algorithm, the algorithm model not able to describe the change law of data completely.

Time complexity

QC | DQC |
---|---|

(Schrodinger Equation + a large number of iterations) | (Schrodinger Equation) |

High | Middle |

### 5.5 Clustering Algorithm Based on Spectral Graph Theory

The basic idea of this kind of clustering algorithms is to regard the object as the vertex and the similarity among objects as the weighted edge in order to transform the clustering problem into a graph partition problem. And the key is to find a method of graph partition making the weight of connection between different groups small as much as possible and the total weight of connection among the edges within the same group high as much as possible [111]. The typical algorithms of this kind of clustering can be mainly divided into two categories, recursive spectral and multiway spectral and the typical algorithms of this two categories are SM [50] and NJW [51] respectively. The core idea of SM which is usually used for image segmentation is to minimize Normalized Cut by heuristic method, based on the eigenvector. And NJW carries out the clustering analysis in the feature space constructed by the eigenvectors corresponding to the k largest eigenvalues of the Laplacian matrix.

For more detailed information about this kind of clustering algorithms, you can refer to [51, 84, 112, 113, 114].

- (1)
Time complexity (Table 19):

- (2)
The time complexity of SM is high, for the process of figuring out the eigenvectors and the heuristic method involved in the algorithm;

- (3)
The time complexity of NJW is high, for the process of figuring out the eigenvectors;

- (4)
Advantages: suitable for the data set with arbitrary shape and high dimension, converged to the global optimal, only the similarity matrix needed as the input, and not sensitive to the outliers;

- (5)
Disadvantages: the clustering result sensitive to the scaling parameter, time complexity relatively high, unclear about the construction of similarity matrix, the selection of eigenvector not optimized and the number of clusters needed to be preset.

Time complexity

SM | NJW |
---|---|

(Eigenvector + heuristic method) | (Eigenvector) |

High | High |

### 5.6 Clustering Algorithm Based on Affinity Propagation

AP (affinity propagation clustering) is a significant algorithm, which was proposed in Science in 2007. The core idea of AP is to regard all the data points as the potential cluster centers and the negative value of the Euclidean distance between two data points as the affinity. So, the sum of the affinity of one data point for other data points is bigger, the probability of this data point to be the cluster center is higher. AP algorithm takes the greedy strategy which maximizes the value of the global function of the clustering network during every iteration [15].

For more detailed information about this kind of clustering algorithms, you can refer to [115, 116, 117].

- (1)
The time complexity of AP is O(n\(\hat{\,\,}\)2*logn);

- (2)
Advantages: simply and clear algorithm idea, insensitive to the outliers and the number of clusters not needed to be preset;

- (3)
Disadvantages: high time complexity, not suitable for very large data set, and the clustering result sensitive to the parameters involved in AP algorithm.

### 5.7 Clustering Algorithm Based on Density and Distance

- (1)
with high local density: the number of data points near the cluster center within a certain scope must be big enough;

- (2)
away from other data points with high local density: cluster center must be away from other data points that could be the center of a cluster.

- (1)
The time complexity of DD is O(n\(\hat{\,\,}\)2);

- (2)
Advantages: simply and clear algorithm idea, suitable for the data set with arbitrary shape and insensitive to the outliers;

- (3)
Disadvantages: relatively high time complexity, relatively strong subjectivity for the selection of the cluster center based on the decision graph and the clustering result sensitive to the parameters involved in DD algorithm.

### 5.8 Clustering Algorithm for Spatial Data

Spatial data refers to the data with the two dimensions, time and space, at the same time, sharing the characteristics of large in scale, high in speed and complex in information. The typical algorithms of this kind of clustering include DBSCAN [36], STING [52], Wavecluster [54] and CLARANS [11]. The core idea of Wavecluster which can be used for parallel processing is to carry out the clustering in the new feature space by applying the Wavelet Transform to the original data. And the core idea of CLARANS is to sample based on CLARA [10] and carry out clustering by PAM [9]. DBSCAN has been discussed in the section Clustering algorithm based on density and STING has been discussed in the section Clustering algorithm based on grid.

For more detailed information about this kind of clustering algorithms, you can refer to [119, 120, 121, 122], ST-DBSCAN [123].

Time complexity

DBSCAN | STING | Wavecluster | CLARANS |
---|---|---|---|

O(n*logn) | O(n) | O(n) | O(n\(\hat{\,\,}\)2) |

Middle | Low | Low | High |

### 5.9 Clustering Algorithm for Data Stream

Data stream shares the characteristics of arriving based on sequence, large in scale and limited frequency of reading. The typical algorithms of this kind of clustering include STREAM [124], CluStream [125], HPStream [126], DenStream [127] and the latter three are incremental algorithms. STREAM, based on the idea of divide and conquer, deals with the data successively according to the sequence of data arriving in order to construct the hierarchical clustering structure. CluStream, which mainly deals with the shortcoming of STREAM that only describing the original data statically, regards data as a dynamic changing process. So CluStream can not only give the timely response for a request, but it also gives the clustering result in terms of different time granularities by figuring out the Micro-clusters online and offline. HPStream, an improvement of CluStream, takes the attenuation of data’s influence over time into consideration and is more suitable for clustering data with high dimension. DenStream, which takes the core idea of the clustering algorithm based on density, is suitable for the nonconvex data set and can deal with outliers efficiently, compared with the algorithms mentioned above in this section.

For more detailed information about this kind of clustering algorithms, you can refer to [128, 129, 130, 131], D-Stream [41, 132].

Time complexity

STREAM | CluStream | HPStream | DenStream |
---|---|---|---|

O(kn) | (online and offline processes) | ||

Low | Low |

The time complexity of CluStream, HPStream and DenStream is involved in the online and offline processes.

### 5.10 Clustering Algorithm for Large-Scale Data

- (1)
- (2)
- (3)
- (4)

For more detailed information about this kind of clustering algorithms, you can refer to [2, 13, 140, 141].

The time complexity of DENCLUE is O(nlogn) and the complexities of K-means, BIRCH, CLARA, CURE, DBSCAN, Wavecluster and FC have been described before in other sections.

## 6 Conclusions

This paper starts at the basic definitions of clustering and the typical procedure, lists the commonly used distance (dissimilarity) functions, similarity functions, and evaluation indicators that lay the foundation of clustering, and analyzes the clustering algorithms from two perspectives, the traditional ones that contain 9 categories including 26 algorithms and the modern ones that contain 10 categories including 45 algorithms. The detailed and comprehensive comparisons of all the discussed clustering algorithms are summarized in Appendix Table 22.

The main purpose of the paper is to introduce the basic and core idea of each commonly used clustering algorithm, specify the source of each one, and analyze the advantages and disadvantages of each one. It is hard to present a complete list of all the clustering algorithms due to the diversity of information, the intersection of research fields and the development of modern computer technology. So 19 categories of the commonly used clustering algorithms, with high practical value and well studied, are selected and one or several typical algorithm(s) of each category is(are) discussed in detail so as to give readers a systematical and clear view of the important data analysis method, clustering.

## Notes

### Acknowledgments

This work has been partially supported by grants form National Natural Science Foundations of China (Nos. 61472390, 11271361, 71331005, and 11226089), Major International (Regional) Joint Research Project (No. 71110107026).

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