Efficient Distributed Algorithms for the $K$-Nearest Neighbors Problem

The $K$-nearest neighbors is a basic problem in machine learning with numerous applications. In this problem, given a (training) set of $n$ data points with labels and a query point $p$, we want to assign a label to $p$ based on the labels of the $K$-nearest points to the query. We study this problem in the {\em $k$-machine model}, (Note that parameter $k$ stands for the number of machines in the $k$-machine model and is independent of $K$-nearest points.) a model for distributed large-scale data. In this model, we assume that the $n$ points are distributed (in a balanced fashion) among the $k$ machines and the goal is to quickly compute answer given a query point to a machine. Our main result is a simple randomized algorithm in the $k$-machine model that runs in $O(\log K)$ communication rounds with high probability success (regardless of the number of machines $k$ and the number of points $n$). The message complexity of the algorithm is small taking only $O(k\log K)$ messages. Our bounds are essentially the best possible for comparison-based algorithms (Algorithms that use only comparison operations ($\leq, \geq, =$) between elements to distinguish the ordering among them). This is due to the existence of a lower bound of $\Omega(\log n)$ communication rounds for finding the {\em median} of $2n$ elements distributed evenly among two processors by Rodeh \cite{rodeh}. We also implemented our algorithm and show that it performs well compared to an algorithm (used in practice) that sends $K$ nearest points from each machine to a single machine which then computes the answer.