Abstract: Graph-based approaches are empirically shown to be very successful for the nearest neighbor search (NNS). However, there has been very little research on their theoretical guarantees. In this work, we fill this gap and rigorously analyze the performance of graph-based NNS algorithms, specifically focusing on the low-dimensional (d << log n) regime. In addition to the basic greedy algorithm on the nearest neighbor graph, we also analyze the most successful heuristics commonly used in practice: speeding up via adding shortcut edges and improving accuracy via maintaining a dynamic list of candidates. We believe that our theoretical results supported by experimental analysis are an important step towards understanding the limits and benefits of graph-based NNS algorithms.