Finding the exact nearest neighbors for a query vector in a high-dimensional space, common when working with embeddings from Large Language Models, scales poorly. Searching through millions or billions of vectors for the closest matches using exact $k$-NN becomes computationally infeasible for many applications. Approximate Nearest Neighbor (ANN) algorithms provide practical alternatives by trading perfect accuracy for significant improvements in search speed.

This chapter focuses on the core mechanisms behind several widely used ANN techniques essential for building efficient vector search systems. We will analyze the construction and search processes of Hierarchical Navigable Small Worlds (HNSW), Inverted File Index (IVF) variations, and Product Quantization (PQ). We will also cover alternative graph-based methods and the key trade-offs involved in selecting an appropriate algorithm based on factors like build time, memory usage, and search performance. The chapter concludes with a practical session on implementing and tuning HNSW.