We study approximation algorithms for constructing sparse navigable graphs in general metrics. Given an $n$-point metric $(P,d)$ and a parameter $α\ge1$, a directed graph on $P$ is $α$-navigable if, for every pair of distinct points $s,t\in P$, the source $s$ has an outgoing neighbor $p$ with $d(p,t)<d(s,t)/α$. The objective is to minimize the number of edges. Recent independent works of Khanna, Padaki, and Waingarten (2026) and Conway et al. (2026) give $O(\log n)$-approximation algorithms and show that an $o(\log n)$ approximation is NP-hard in general. Before our work, the best known running time for such an approximation for general $α\ge1$ was $\widetilde O(\min\{n\operatorname{OPT}_α,n^{2.5}\})$, where $\operatorname{OPT}_α$ is the minimum number of edges in an $α$-navigable graph. In this paper, we give a randomized algorithm that, with high probability, runs in $\widetilde O(n^2)$ time and achieves an $O(\log n)$ approximation for every $α\ge1$. In general, these guarantees are nearly optimal up to constant factors in the approximation and polylogarithmic factors in the running time.

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