发表机构
Taub Faculty of Computer Science, Technion – Israel Institute of Technology（泰伯计算机科学与工程学院，以色列理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对几乎可导航性，证明了与已有上界匹配的下界，确立其构造最优性，还揭示了其与Zarankiewicz问题的关联，同时加深了对普通可导航性的理解。
AI 中文摘要
可导航性是基于图的搜索结构的基本性质，在最近邻算法分析中发挥重要作用。非正式地，若从任意当前点到任意期望目标，总存在一条出边能严格靠近该目标，则该图是可导航的。尽管该性质为贪心搜索提供强保证，但它的代价可能极高：最坏情况下，可导航图需要Ω(n^(3/2))条边，其中n为数据集规模。近期，Avi和Musco提出了(1-ε)-几乎可导航性，这是一种自然的松弛：从每个当前点出发，对几乎所有目标都存在这样的推进边，而ε比例的目标可不满足该条件。他们证明每个数据集都存在此类图，边数为O(n/ε)。本研究证明了匹配的下界，确立了该构造的最优性，并完整刻画了几乎可导航图可达到的稀疏性。对大多数ε值，我们的困难实例位于多对数维的欧氏空间中；在ε的整个最坏情况范围内，维度d=O(√n log^(3/2) n)即足够。该构造也加深了我们对普通可导航性的理解：在上述维度下，我们展示了部分数据集，其对应的每一个可导航图都需要Ω(n^(3/2))条边。我们的证明揭示了几乎可导航性与经典Zarankiewicz问题（构造成对邻域重叠受限的稠密图）之间的意外联系，该联系使我们能将极值图构造转化为导航问题的困难几何实例，关联了两种看似不同的图稀疏性概念。
英文摘要
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.
