Fractal Dimension and Lower Bounds for Geometric Problems

We study the complexity of geometric problems on spaces of low fractal dimension . It was recently shown in Sidiropoulos and Sridhar (33rd International Symposium on Computational Geometry (Brisbane 2017). Leibniz Int. Proc. Inform., vol. 77, # 58. Leibniz-Zent. Inform., Wadern, 2017) that several problems admit improved solutions when the input is a pointset in Euclidean space with fractal dimension smaller than the ambient dimension. In this paper we prove nearly-matching lower bounds, thus establishing nearly-optimal bounds for various problems as a function of the fractal dimension. More specifically, we show that for any integer d > 1 , any δ∈ (1,d) , and any n ∈ℕ , there exists a set X of n points in ℝ^d , with fractal dimension δ such that for any ε > 0 and c ≥ 1 , any c -spanner of X has treewidth Ω ( n^1-1/(δ - ϵ )/c^d-1 ) . This lower bound matches the previous upper bound. The construction used to prove this lower bound on the treewidth of spanners, can also be used to derive lower bounds on the running time of algorithms for various problems, assuming the Exponential Time Hypothesis. We provide two prototypical results of this type: For any δ∈ (1,d) and any ε >0 , d -dimensional Euclidean TSP on n points with fractal dimension at most δ cannot be solved in time 2^O(n^1-1/(δ - ε ) ) . The best-known upper bound is 2^O(n^1-1/δlog n) . For any δ∈ (1,d) and any ε >0 , the problem of finding k -pairwise non-intersecting d -dimensional unit balls/axis-parallel unit cubes with centers having fractal dimension at most δ cannot be solved in time f(k) n^O (k^1-1/(δ - ε )) for any computable function f . The best-known upper bound is n^O(k^1-1/δlog n) . The above results nearly match previously known upper bounds from [op. cit.], and generalize analogous lower bounds for the case of ambient dimension due to Marx and Sidiropoulos (30th Annual Symposium on Computational Geometry (Kyoto 2014), pp. 67–76. ACM, New York, 2014).