Efficient $k$ -Means++ Approximation with MapReduce

$k$
-means is undoubtedly one of the most popular clustering algorithms owing to its simplicity and efficiency. However, this algorithm is highly sensitive to the chosen initial centers and thus a proper initialization is crucial for obtaining an ideal solution. To address this problem,
$k$
-means++ is proposed to sequentially choose the centers so as to achieve a solution that is provably close to the optimal one. However, due to its weak scalability,
$k$
-means++ becomes inefficient as the size of data increases. To improve its scalability and efficiency, this paper presents MapReduce
$k$
-means++ method which can drastically reduce the number of MapReduce jobs by using only one MapReduce job to obtain
$k$
centers. The
$k$
-means++ initialization algorithm is executed in the Mapper phase and the weighted
$k$
-means++ initialization algorithm is run in the Reducer phase. As this new MapReduce
$k$
-means++ method replaces the iterations among multiple machines with a single machine, it can reduce the communication and I/O costs significantly. We also prove that the proposed MapReduce
$k$
-means++ method obtains
$O(\alpha^{2})$
approximation to the optimal solution of
$k$
-means. To reduce the expensive distance computation of the proposed method, we further propose a pruning strategy that can greatly avoid a large number of redundant distance computations. Extensive experiments on real and synthetic data are conducted and the performance results indicate that the proposed MapReduce
$k$
-means++ method is much more efficient and can achieve a good approximation.