The Hierarchical Navigable Small World (HNSW) algorithm is a graph-based approximate nearest neighbor search technique that efficiently searches vector databases using a multi-layered graph structure. HNSW creates hierarchical layers where upper layers have fewer nodes with longer connections, while lower layers contain all nodes with shorter connections. The algorithm works by probabilistically assigning nodes to different layers (similar to skip lists), enabling logarithmic time complexity for approximate nearest neighbor search. This makes HNSW particularly effective for high-dimensional vector similarity search, overcoming the curse of dimensionality that affects traditional search methods.