The Price of Almost Navigability

Navigability is a fundamental property of graph-based search structures and plays an important role in the analysis of nearest-neighbor algorithms. Informally, a graph is navigable if, from any current point and toward any desired target, there is always an outgoing edge that moves strictly closer to that target. While this property provides a strong guarantee for greedy search, it can be inherently expensive: in the worst case, navigable graphs require $Ω(n^{3/2})$ edges, where $n$ is the size of the dataset. Recently, Avi and Musco introduced $(1-ε)$-almost navigability, a natural relaxation in which, from every current point, such a progress-making edge is required for almost all targets, while an $ε$ fraction of targets may fail this condition \cite{avimusco2026almost}. They showed that every dataset admits such a graph with $O(n/ε)$ edges. In this work, we prove a matching lower bound, establishing the optimality of their construction and giving a complete characterization of the sparsity achievable by almost-navigable graphs. For most values of $ε$, our hard instances lie in Euclidean spaces of polylogarithmic dimension, across the full worst-case range of $ε$, dimension $d=O(\sqrt{n}\log^{3/2} n)$ suffices. The same construction also sharpens our understanding of ordinary navigability: in the latter dimension, we exhibit datasets for which every navigable graph has $Ω(n^{3/2})$ edges. Our proof reveals a surprising connection between almost navigability and the classical Zarankiewicz problem of constructing dense graphs with limited pairwise neighborhood overlap. This connection lets us translate extremal graph constructions into hard geometric instances for navigation, linking two seemingly different notions of graph sparsity.