Tell about the idea of HNSW and how the hierarchical graph is structured.

HNSW (Hierarchical Navigable Small World) is an algorithm for efficient nearest neighbor search in large vector spaces. The idea is based on constructing a hierarchical graph where nodes are objects (vectors), and edges connect similar objects based on a metric.

The hierarchical graph is organized as follows:

• There are multiple levels, starting from the top where the graph is very sparse, and ending at the bottom where the graph is dense.
• Each object is placed on a random level with a probability that decreases exponentially.
• The search begins at the top level, where an approximate neighbor is quickly found, then moves down to the next level, refining the search.
• At each level, the search is conducted on the graph using a greedy algorithm, moving to neighbors closer to the target vector.

This approach allows for rapid reduction of the search space, providing a balance between speed and accuracy. HNSW scales well and is often used in vector search systems, for example, for recommendations or embedding-based search.