The Fast $k-\text{NN}$ Algorithm Based on a Fixed Grid Method

$k-\text{Nearest}$ Neighbor $(k-\text{NN})$ is a well-known instance-based learning algorithm; widely used in pattern recognition. A $k-\text{NN}$ classifier can generate highly accurate predictions if provided with sufficient training instances. Thus, it plays a pivotal role in many fields. However, though its accuracy can improve with more data, the need for computational resources increases as well. In this paper, we propose a novel approach which pre-partitions instance space into smaller cells in order to reduce computational cost and greatly reduce the time complexity. Assume every instance is mapped to a point in the $d-\text{dimensional}$ Euclidean space and a training set $D$ contains $n$ training instances. Given a query instance, the brute force $k-\text{NN}$ algorithm has the $O(nd)$ time complexity for predicting the class of a query instance. This algorithm can improve the time complexity.