Before revision:  This leads us to consider what we call the Relational Learning Question: ``Which standard learning algorithms can be efficiently implemented on relational data, and for those that can not, is there an alternative algorithm that can be efficiently implemented on relational data and that has similar performance guarantees to the standard algorithm?'' In this paper, we address the relational learning question for two well-known algorithms for the standard k-means clustering problem. We first show that the  k-means+ + algorithm can be efficiently implemented on relational data .
After revision:  This leads us to consider what we call the Relational Learning Question: ``Which standard learning algorithms can be efficiently implemented on relational data, and for those that can not, is there an alternative algorithm that can be efficiently implemented on relational data and that has similar performance guarantees to the standard algorithm?'' In this paper, we address the relational learning question for two well-known algorithms for the standard k-means approximation algorithm that is efficient in the relational algorithms model. This is an algorithm that operates directly on a relational database without performing a join to convert it to a matrix whose rows represent the data points. The running time is potentially exponentially smaller than N, the k-means+ + algorithm can be efficiently implemented on relational data .