Truly Subquadratic Time Algorithms for Diameter and Related Problems in Graphs of Bounded VC-dimension

We give the first truly subquadratic time algorithm, with ${O^{\ast}}\left( {{n^{2 - 1/18}}} \right)$ running time, for computing the diameter of an n-vertex unit-disk graph, resolving a central open problem in the literature. Our result is obtained as an instance of a general framework, applicable to different graph families and distance problems. Surprisingly, our framework completely bypasses sublinear separators (or r-divisions) which were used in all previous algorithms. Instead, we use low-diameter decompositions in their most elementary form. We also exploit bounded VC-dimension of set systems associated with the input graph, as well as new ideas on geometric data structures. Among the numerous applications of the general framework, we obtain:1)An $\tilde O\left( {m{n^{1 - 1/(2d)}}} \right)$ time algorithm for computing the diameter of m-edge sparse unweighted graphs with constant VC-dimension d. The previously known algorithms by Ducoffe, Habib, and Viennot [SODA 2019] and Duraj, Konieczny, and Potępa [ESA 2024] are truly subquadratic only when the diameter is a small polynomial. Our result thus generalizes truly subquadratic time algorithms known for planar and minor-free graphs (in fact, it slightly improves the previous time bound for minor-free graphs).2)An $\tilde O\left( {{n^{2 - 1/12}}} \right)$ time algorithm for computing the diameter of intersection graphs of axis-aligned squares with arbitrary size. The best-known algorithm by Duraj, Konieczny, and Potępa [ESA 2024] only works for unit squares and is only truly subquadratic in the low-diameter regime.3)The first algorithms with truly subquadratic complexity for other distance-related problems, including all-vertex eccentricities, Wiener index, and exact distance oracles. In particular, we obtain the first exact distance oracle with truly subquadratic space and $\tilde O(1)$ query time for any sparse graph with bounded VC-dimension, again generalizing previous results for planar and minor-free graphs.