Recent research on computing the diameter of geometric intersection graphs has made significant strides, primarily focusing on the 2D case where truly subquadratic-time algorithms were given for simple objects such as unit-disks and (axis-aligned) squares. However, in three or higher dimensions, there is no known truly subquadratic-time algorithm for any intersection graph of non-trivial objects, even basic ones such as unit balls or (axis-aligned) unit cubes. This was partially explained by the pioneering work of Bringmann et al. [SoCG '22] which gave several truly subquadratic lower bounds, notably for unit balls or unit cubes in 3D when the graph diameter $Δ$ is at least $Ω(\log n)$, hinting at a pessimistic outlook for the complexity of the diameter problem in higher dimensions. In this paper, we substantially extend the landscape of diameter computation for objects in three and higher dimensions, giving a few positive results. Our highlighted findings include: - A truly subquadratic-time algorithm for deciding if the diameter of unit cubes in 3D is at most 3 (Diameter-3 hereafter), the first algorithm of its kind for objects in 3D or higher dimensions. Our algorithm is based on a novel connection to pseudolines, which is of independent interest. - A truly subquadratic time lower bound for \Diameter-3 of unit balls in 3D under the Orthogonal Vector (OV) hypothesis, giving the first separation between unit balls and unit cubes in the small diameter regime. Previously, computing the diameter for both objects was known to be truly subquadratic hard when the diameter is $Ω(\log n)$. - A near-linear-time algorithm for Diameter-2 of unit cubes in 3D, generalizing the previous result for unit squares in 2D. - A truly subquadratic-time algorithm and lower bound for Diameter-2 and Diameter-3 of rectangular boxes (of arbitrary dimension and sizes), respectively.

翻译：近年来，几何交图直径计算研究取得重要进展，主要聚焦于二维情形——针对单位圆盘和（轴对齐）正方形等简单对象，已提出真正次二次时间算法。然而在三维或更高维度中，即使对于单位球体或（轴对齐）单位立方体等基本对象，任何非退化对象的交图目前均无已知的真正次二次时间算法。Bringmann等人[SoCG '22]的开创性工作部分解释了这一现象：他们给出了多个真正次二次下界，特别是当图直径$Δ$至少为$Ω(\log n)$时，三维单位球体或单位立方体的直径计算存在显著困难，预示着高维直径问题的复杂性前景黯淡。本文大幅扩展三维及更高维对象直径计算的研究格局，取得若干正面结果。主要发现包括：- 针对三维单位立方体直径是否不超过3（后文简称Diameter-3）的判定问题，提出首个真正次二次时间算法——这也是三维或更高维对象中首个此类算法。该算法基于与伪直线（pseudolines）的新颖联系，该联系本身具有独立研究价值。- 在正交向量（OV）假设下，给出三维单位球体Diameter-3问题的真正次二次时间下界，首次在小直径范围内区分了单位球体与单位立方体的计算复杂度。此前已知当直径为$Ω(\log n)$时，两种对象的直径计算均存在真正次二次困难性。- 提出三维单位立方体Diameter-2问题的近线性时间算法，将二维单位正方形问题的现有结果推广至三维。- 分别给出矩形盒（任意维度和尺寸）Diameter-2与Diameter-3问题的真正次二次时间算法与下界。