Generalized relative neighborhood graph (GRNG) for similarity search

Abstract

A critical factor in graph-based similarity search is the choice of graph that represents the underlying space’s structure. Proximity graphs, such as Relative Neighbor Graphs (RNG), are defined by the neighborhood relations between pairs of points. As a result, computing these graphs typically involves a more computationally intensive trinary operation, making them an order of magnitude more expensive than the widely used $k$-Nearest Neighbors ($k$NN) graph, which relies only on pairwise distances. However, the $k$NN graph often suffers from disconnections and requires manual parameter selection, whereas the RNG better captures the geometry of the space. While several methods have attempted to reduce the computational cost of constructing an RNG, these are usually approximate and lack scalability. This paper introduces an incremental, hierarchical method that employs a novel proximity graph called the Generalized Relative Neighborhood Graph (GRNG). The GRNG organizes a pivot layer that efficiently guides the exact construction of the graph for the subsequent layer. This multi-layer, exact approach to RNG construction represents a significant improvement over existing methods that only produce approximate RNGs.