Tight upper bounds for SNG and HNSW shortcut costs in the dense regime

Derive a tight upper bound for the shortcut-phase search cost of the Sparse Neighborhood Graph (SNG) and Hierarchical Navigable Small World (HNSW) indexes in the dense regime, where the intrinsic dimensionality grows sublogarithmically with the dataset size.

Background

The paper analyzes graph-based vector-search cost by separating beam search into a shortcut phase, which navigates from an entry point toward the query neighborhood, and an exploration phase, which searches that neighborhood. In the dense regime, the paper establishes bounds for Vamana and lower bounds for SNG and HNSW, but the shortcut behavior of SNG and HNSW is more difficult to characterize.

For SNG, the paper proves that each shortcut edge has limited length with high probability, yielding a lower bound on the number of shortcut steps. It notes that the actual number of steps may be larger because beam search can encounter local minima. Since each HNSW layer behaves similarly to SNG, the same difficulty applies to HNSW. A tight upper bound would complete the theoretical characterization of shortcut costs for these indexes in the dense regime.

References

Since each layer of HNSW behaves similarly to SNG, similar concerns around an upper bound on its search cost persist. We leave deriving a tight upper bound for SNG and HNSW as an interesting open question for future work.

— A Power Law in Logarithm's Clothing: On the Scalability of Graph-Based Vector Search  (2609.02143 - Maghrebi et al., 2 Sep 2026) in Section 6.2, subsection “Dense Regime,” paragraph following Lemma “SNG Maximum Progress”