  Approximate nearest neighbor search (ANNS) in high-dimensional spaces is a
pivotal challenge in the field of machine learning. In recent years,
graph-based methods have emerged as the superior approach to ANNS, establishing
a new state of the art. Although various optimizations for graph-based ANNS
have been introduced, they predominantly rely on heuristic methods that lack
formal theoretical backing. This paper aims to enhance routing within
graph-based ANNS by introducing a method that offers a probabilistic guarantee
when exploring a node's neighbors in the graph. We formulate the problem as
probabilistic routing and develop two baseline strategies by incorporating
locality-sensitive techniques. Subsequently, we introduce PEOs, a novel
approach that efficiently identifies which neighbors in the graph should be
considered for exact distance calculation, thus significantly improving
efficiency in practice. Our experiments demonstrate that equipping PEOs can
increase throughput on commonly utilized graph indexes (HNSW and NSSG) by a
factor of 1.6 to 2.5, and its efficiency consistently outperforms the
leading-edge routing technique by 1.1 to 1.4 times.
