A lowcomplexity cell clustering algorithm in dense small cell networks
 Ryuma Seno^{1}Email authorView ORCID ID profile,
 Tomoaki Ohtsuki^{1},
 Wenjie Jiang^{2} and
 Yasushi Takatori^{2}
https://doi.org/10.1186/s1363801607653
© The Author(s) 2016
Received: 27 April 2016
Accepted: 31 October 2016
Published: 14 November 2016
Abstract
Clustering plays an important role in constructing practical network systems. In this paper, we propose a novel clustering algorithm with low complexity for dense small cell networks, which is a promising deployment in nextgeneration wireless networking. Our algorithm is a matrixbased algorithm where metrics for the clustering process are represented as a matrix on which the clustering problem is represented as the maximization of elements. The proposed algorithm simplifies the exhaustive search for all possible clustering formations to the sequential selection of small cells, which significantly reduces the clustering process complexity. We evaluate the complexity and the achievable rate with the proposed algorithm and show that our algorithm achieves almost optimal performance, i.e., almost the same performance achieved by exhaustive search, while substantially reducing the clustering process complexity.
Keywords
1 Introduction
In the past few years, fifth generation (5G) wireless communication has become the center of discussion for researchers in this area [1, 2]. 5G is entirely different from conventional standards in that additional improvements cannot meet their requirements. In addition to the exponential increase in the data amount, the number of wireless devices connected to wireless networks should be considered. As represented by the term “Internet of Things” (IoT), it is predicted that many kinds of devices will operate on wireless networks, which will produce demand for much more network capacity [3].
A simple way to increase the network capacity is narrowing the range of cells and increasing the number of cells. Although this method is effective, intercell interference becomes more critical when densifying cells with small coverage [4], where radio resource management (RRM) would be of importance for interference mitigation. RRM schemes are mainly classified into two categories: frequency partitioning schemes and universal frequency reuse schemes. In the former, frequency resources that are orthogonal with each other are assigned to each cell [5]. On the other hand, in the latter, many cells in the network share the same bandwidth [6]. These schemes may be combined into a hybrid scheme where the network is partitioned into a number of groups of cells and the cells in the same group share the same bandwidth. In [7], the network is classified into two areas, interferencesensitive area (ISA) and notinterferencesensitive area (NISA), and the entire bandwidth is partitioned into two parts, one where interference is tolerated and one where interference is prohibited, which results in the improved frequency utilization efficiency. In this scheme, how to partition the network greatly affects the performance achieved.
Ways for interference mitigation are also classified into intercell interference coordination (ICIC) and base station cooperation (BSC). In ICIC, neighboring cells use the orthogonal bandwidth and avoid the performance degradation of celledge users in each cell [8]. In [9], the dynamical management of bandwidth and cluster size results in improved frequency utilization efficiency. On the other hand, in BSC, the neighboring cells share the same bandwidth and cooperate with each other to make the interference beneficial [10]. Since a number of cells coordinate in BSC, the construction of groups in which cells coordinate greatly affects the performance. In both schemes, grouping cells that use the same bandwidth is essential to better performance.
A particular example of a network deployment where intercell interference is notably critical is heterogeneous networks (small cell networks) [11]. Network operation by using only macro cells has a capacity hole problem where the achievable rate is extremely low. Small cell networks are attractive for addressing this problem because small cells can fill the capacity holes. Although small cells can enhance the network capacity, the dense deployment of cells results in severe intercell interference. The intercell interference problem in small cell networks has been addressed in many studies [12–15, 17, 19]. In [13], semidistributed interference management in which cell clustering and resource allocation are jointly conducted was proposed for an orthogonal frequency division multiple access (OFDMA)based twotier cellular network. In [14], the authors proposed the joint optimization of small cell clustering and a beamforming vector at each base station in dense heterogeneous networks. In [15], interference alignment [16] was utilized as the interference management technique in heterogeneous networks where intertier and intratier interferences are eliminated by adequate beamforming and filtering. In [17], the authors proposed a scheme where some available degrees of freedom (DoF) of macro cells (primary users) were left and utilized for small cells (secondary users) in which interference alignment was achieved. This enabled the achievable DoF of small cells to be maximized while satisfying the required DoF of macro cells. In [18], clustering was performed on the basis of the rate loss caused by cells interfering with each other, where the sum of rate loss between cells belonging to different clusters is minimized, which means the rate loss caused by intercluster interference is minimized. In [19], the authors proposed clustering small cells in heterogeneous networks where interference alignment is utilized and found that clustering small cells is effective for improving spectral efficiency normalized with the number of antennas at a small cell base station.
The above discussion indicates that “clustering” plays an important role when operating a practical network system, and a clustering algorithm with lower complexity is desirable. Many kinds of clustering algorithms have been derived in the literature. A most simple clustering algorithm is an exhaustive search of all possible clustering patterns (exhaustive algorithm) [20]. Although an exhaustive algorithm can achieve global optimal performance, its complexity is significantly high, which is impractical. To achieve a practical clustering algorithm, many studies [21–23] have proposed schemes where each cluster is determined sequentially, i.e., a cluster is determined from cells that have not been selected yet. Papadogiannis et al. [21], Ng and Huang [22], and Qin and Tian [23] assume wireless networks adopting the coordination between BSs and consider the selection of coordinating BSs. Papadogiannis et al. [21] proposes changing coordination sets dynamically to fully exploit the macro diversity, where each set is determined from BSs which are not selected yet so as to maximize the achievable rate in the set. Ng and Huang [22] consider wireless networks where each user receives the signal from several BSs and propose the cooperative precoding weight design. The set of coordinating BSs is determined so that the BSs within the set interfere with each other most strongly, i.e., each UE selects, at first, the geographically nearest BS and picks up some BSs to which the BS interferes most strongly, where the selected BSs are unified into a coordinating group. Qin and Tian [23] consider, as in [22], to group BSs interfering each other most strongly into a coordination set. In particular, the connectivity between BSs is modeled as a graph whose node corresponds to a BS and the weight of whose edge represents the level of the interference between two BSs and based on the graph the grouping is conducted. A partial graph holding the largest sum of weights among all possible sets, which becomes a coordination set, is separated from the whole graph, and this procedure is sequentially conducted until the grouping is completed. Note that all of these schemes have the feature that each BS cluster is determined sequentially one after another from BSs which are not selected yet. Such schemes are called “greedy algorithms” in general. Although such algorithms significantly reduce clustering process complexity, they necessitate sacrificing solution optimality.
In this paper, we propose a novel algorithm for cell clustering. Our algorithm is a matrixbased algorithm where metrics for the clustering process are represented as a matrix on which the clustering problem can be represented as the maximization of elements. In the proposed algorithm, we simplify the exhaustive algorithm into sequential selection of small cells through twostep transformations. The clustering problem is divided into some subproblems each of whose objective function is represented as a sum of elements in the matrix, where each subproblem corresponds to selecting a small cell. The transformations are conducted to narrow the search range in the same way as the transformation from an exhaustive algorithm to a greedy algorithm where solution optimality is replaced with complexity reduction. We evaluate the complexity and the achievable rate per small cell with the proposed algorithm and show that our algorithm achieves almost optimal performance, i.e., almost the same performance achieved in an exhaustive search, while significantly reducing the clustering process complexity.
The rest of the paper is organized as follows. Section 2 describes the system model and Section 3 describes the proposed algorithm. In Section 4, we present a performance evaluation in which we evaluated the clustering process complexity and the achievable rate per small cell. Finally, Section 5 concludes the paper with a summary of key points.
2 System model
In the macro cell, there are K small cells and K _{mue} macro user equipments (UEs), and in each small cell, there are K _{sue} small cell UEs. The macro base station (BS) and each small cell BS are respectively equipped with N _{mbs} and N _{sbs} antennas, and each UE is equipped with N _{ue} antennas. Each BS transmits d data streams to corresponding UE/UEs with spatial multiplexing. Note that in this paper, we assume there is only one macro cell because we focus on the clustering problem of small cells.
where \({k_{l}^{n}}\) denotes the index of the lth small cell in the nth cluster, C _{ n } denotes the number of small cells in the nth cluster, and \(\mathcal {K}\) denotes the set of small cells. Note that the clustering formation is determined at a central unit to whom macro BS and each small cell BS connect via wired backhaul link. The required information (the selected clustering pattern, the channel information, etc.) are assumed to be exchanged via the backhaul link without delaying.
In general, clustering problems are formalized as the maximization or minimization of the sum of metrics within each cluster. In this paper, without loss of generality, we formulate the clustering problem as follows.
where \(\mathbb {C}^{*}\) denotes the optimal clustering formation.
Although we focus on the clustering problem of small cells in this paper, the proposed algorithm can be easily applied to more general environments, e.g., multicell cellular networks or ad hoc sensor networks. In these cases, we only require that appropriate metrics are defined between cells or sensors, like w _{ i,j } which is defined between small cells in this paper.
Notation
Symbol  Description 

K  Number of small cells in a macro cell 
N  Number of small cell clusters in a macro cell 
\(\mathbb {C}\)  A clustering formation ∈Ω 
Ω  Set of clustering formations 
C _{ n }  nth cluster 
\({k_{l}^{n}}\)  lth small cell in nth cluster 
\(\mathcal {K}\)  Set of small cells 
w _{ i,j }  Metric between small cell i and small cell j 
Δ _{ i,j }  Effect from small cell j to small cell i 
\(\mathcal {W} \left (C_{n} \right)\)  Sum of metrics in C _{ n } 
\(\mathcal {U} \left (\mathbb {C} \right)\)  Sum of metrics within each cluster under \(\mathbb {C}\) 
Δ  Matrix representing metrics 
Δ(C _{ i })  Submatrix corresponding to C _{ i } in Δ 
Θ  Label based on which elements in Δ are arranged 
\(\mathcal {X}_{l}^{n}\)  Set of small cells that have been selected 
By the (l−1)th cell in C _{ n }  
\(\mathcal {Y}^{n}\)  Set of small cells that are included in from C _{1} to C _{ n } 
\(\mathcal {Z}_{l}^{n}\)  Set of small cells in C _{ n } up to lth cell 
T ^{ n }  Number of small cells that have been 
Selected from C _{1} to C _{ n } 
3 Proposed method
In the proposed algorithm, metrics between small cells are represented as a matrix on the basis of which we transform the clustering problem. The clustering problem is reformulated as the maximization of elements in the matrix, and we transform the exhaustive search of the optimal clustering pattern to the sequential selection of small cells.
3.1 Transformation of clustering process
where Δ _{ i,j } denotes the effect of small cell j against small cell i, the SINR or the achievable rate loss, for example. For SINR, Δ _{ i,j } indicates SINR at small cell i when interfered with by small cell j, and for the rate loss, Δ _{ i,j } indicates the rate loss at small cell i when interfered with by small cell j. In the following, these effects are contained in Δ _{ i,j }. Note that we assume Δ _{ i,i }=0, \(\forall \ i \in \mathcal {K}\). Although Δ _{ i,j } should contain the effect of “all other small cells” against the small cell i, we emphasize that for the simplicity Δ _{ i,j } contains the effect of “just the small cell j” against the small cell i.
Note that the sum of elements being symmetrical with respect to the diagonal line is the metric between small cells, i.e., Δ _{ i,j }+Δ _{ j,i }=w _{ i,j }.
which means that the ith row (and the ith column) is labeled by small cell i.
Note that \(\mathbf {\Theta } \left (\mathbb {C} \right)\) represents the clustering formation \(\mathbb {C}\). In the proposed algorithm, we select small cells sequentially from the first element of Θ to the last one, which is equivalent to determining the clustering formation.

STEP 1 : exhaustive search for all possible clustering formations is transformed to the sequential determination of each cluster in the same way as in a greedy algorithm

STEP 2 : sequential determination of each cluster is transformed to the sequential selection of small cells
These transformations are conducted to narrow the search range, which results in reduced complexity. In particular, after the step 1 transformation, we determine each cluster sequentially instead of determining all clusters together, which means we search for small cells that have not been selected yet when determining a cluster. In addition, after the step 2 transformation, we sequentially determine each small cell instead of each cluster, which further reduces the complexity. By the sequential selection of small cells, we finally derive the clustering formation as represented in Eq. (9).
3.1.1 Step 1
where \(\sum _{i, j \in C_{n}} \Delta _{i, j}\) represents the sum of metrics in C _{ n }.
where \(\mathcal {Y}^{n1}\) denotes the set of small cells included in from C _{1} to C _{ n−1}.
Equation (17) is called greedy algorithm in general.
3.1.2 Step 2
where \(\mathcal {Z}_{l1}^{n}\) denotes the set of small cells included in C _{ n } up to the (l−1)th cell.
Equation (25) means that for the first small cell in each cluster, we select the one that maximizes the sum of metrics between all small cells that have not been selected yet. Since we do not know which small cells are to be included in the same cluster when selecting the first small cell, it is better to select the one that maximizes the sum of metrics even if any small cells are selected as the same cluster, which corresponds to Eq. (25).
3.2 Proposed algorithm
The proposed algorithm is summarized in Algorithm 1.
4 Performance evaluation
We evaluate the proposed algorithm in terms of two aspects: the complexity of clustering process and the achievable rate per small cell. As mentioned above, there is a tradeoff between the complexity and the achievable rate, which means we have to sacrifice the optimality of the solution when reducing the complexity. However, we show that our algorithm is capable of achieving almost optimal performance, i.e., the same performance achieved by an exhaustive algorithm. Therefore, our algorithm can achieve almost optimal performance while reducing the complexity significantly.
where L denotes a constant value that holds LN=K. Note that although there are some specific schemes in the literature [21–23], we exploit the essence of those schemes and refine them into the schemes described in the previous section, i.e., exhaustive algorithm and greedy algorithm, when comparing the performance of the proposed scheme, which is more concise.
4.1 Complexity of clustering process
All algorithms in this paper, i.e., exhaustive algorithm, greedy algorithm, and proposed algorithm have the objective function comprising only additions of metrics. Therefore, we use the total number of additions in the clustering process as the complexity. Note that the complexity to calculate the metrics themselves is independent of the clustering algorithm used, which allows us to ignore the complexity for the calculation of the metrics themselves. In the following, we derive the complexity of each algorithm quantitatively.
4.1.1 Exhaustive algorithm
where the second term in Eq. (27) denotes the number of combinations of two small cells from C _{ n }.
where the derivation is given in Appendix 1.
4.1.2 Greedy algorithm
4.1.3 Proposal
where T ^{(n−1)} denotes the number of small cells selected from C _{1} to C _{ n−1}, i.e., T ^{ n−1}=L(n−1).
where K=NL.
4.1.4 Comparison
Complexity of algorithms
Algorithm  Quantitative  Order 

representation  representation  
Exhaustive  Eq. (29)  \(\mathcal {O} \left (L^{2} N^{\left (L \left (N  1 \right) + 1 \right)} \right)\) 
algorithm  
Greedy  Eq. (33)  \(\mathcal {O} \left (L^{2} N^{L} \right)\) 
algorithm  
Proposal  Eq. (38)  \(\mathcal {O} \left (N^{3} L^{2} + N^{2} L^{3} \right)\) 
As shown in Table 2, the complexity of exhaustive algorithm increases exponentially with respect to both N and L, which can be verified in Figs. 2 and 3. In terms of greedy algorithm, as shown in Table 2, we found that the complexity increases exponentially with respect to L while polynomially with respect to N, which can be verified in Figs. 2 and 3.
Compared to the aforementioned two algorithms, i.e., exhaustive algorithm and greedy algorithm, the complexity of the proposed algorithm increases polynomially with respect to both N and L as shown in Table 2. Therefore, our algorithm reduces the complexity significantly compared to the conventional algorithms, particularly when N or L increases.
4.2 Achievable rate
Simulation parameters
Macro cell  1  
Macro UE (K _{mue})  12  
Small cell (K)  12  
Small cell UE (K _{sue})  1 per small cell  
Macrocell radius  500 m  
Small cell radius  50 m  
Cluster size (L)  2  4  6  
Transmit stream (d)  2  
Rx antenna (N _{ue})  4  
Tx antenna  macro BS (N _{mbs})  48  
small BS (N _{sbs})  4  12  20  
Channel coefficient  i.i.d. Gaussian  
Path loss model  −38.46−log10(distance (km)) 
Design of weights
Type of weight  How to design  

Precoding weight  Macro BS  Direct the null space to macro UEs except for the desired UE and eliminate interstream interference 
Small BS  Align the intracluster interference to the strongest interference from macro BS  
Postcoding weight  Macro UE  Eliminate interstream interference 
Small UE  ZF weight nulls out the aligned interference and MMSE weight mitigate the residual interference 
where \(\sqrt {\rho _{ij}}\) and P _{ i } respectively denote the path loss effect between small BS j and small UE i and the transmit power at small BS i. Note that, as in [19], we assume each small cell BS has the same transmit power, i.e., P _{ i } has the same value for all i, and we omit the effect of multipath fading in Eq. (41) for the simplicity.
Although there are some parameters in Table 3, only a few (“macro UE,” “small cell,” and “transmit stream”) can be set arbitrarily and the others should be set to a certain value(s) compulsorily due to the constraint regarding, mainly, the calculation of precoding or postcoding weight [19]. In the following, we investigate the performance when changing these parameters. Note that the cluster size is fixed to 4 unless otherwise specified.
5 Conclusions
In this paper, we proposed a novel cell clustering algorithm that achieves low complexity and high performance. Our algorithm is a matrixbased algorithm where the clustering problem is represented as the maximization of elements in a matrix representing the metric for the clustering process. The algorithm transforms an exhaustive search for all possible clustering formations to a sequential selection of small cells, which resulted in significantly reduced complexity. We evaluated the complexity and the achievable rate per small cell and showed that our algorithm achieved almost optimal performance, i.e., almost the same performance achieved by exhaustive search, while significantly reducing the clustering process complexity.
6 Appendices
6.1 Appendix 1: the derivation of the complexity order for exhaustive algorithm
6.2 Appendix 2: the derivation of the complexity order for greedy algorithm
where we used the relationship shown in Eq. (53).
Declarations
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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