kM++kNN : A fast algorithm for the exact search of k-nearest neighbors

The k-NN algorithm - k-nearest neighbor - is widely used in Machine Learning and Statistics for tasks involving classification and regression. Having as inputs an instance x, a set of instances T and an integer $k \geqslant 1$, the k-NN performs an exhaustive search in T of the k instances most similar to instance x (k-nearest neighbors). In applications involving many instances and/or instances with high dimensionality, the search process is time-consuming due to the need to perform many calculations of similarity functions between instances. Several proposals to reduce the k-NN search time have been made, some of them aiming at the exact search of the k most similar instances to x in T and, others, reducing the search time via approximate methods to calculate the most similar instances to x. This work proposes an algorithm called $\mathrm{kM}++\mathrm{kNN}$ for the exact search of the k most similar instances to x in T, which uses the triangular inequality concept to reduce the ${\mathrm {k-N N}}$ search time. The ${\mathrm {k M++k N N}}$ algorithm is compared, in experiments to measure the economy of the number of calculations of similarity functions between instances and search time, with an algorithm currently considered fast, the kMkNN.