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 \(\delta \in (1,d)\), and any \(n \in {\mathbb {N}}\), there exists a set X of n points in \({\mathbb {R}}^{d}\), with fractal dimension \(\delta \) such that for any \(\varepsilon > 0\) and \(c \ge 1\), any c-spanner of X has treewidth \(\Omega ( n^{1-1/(\delta - \epsilon )}/c^{d-1} )\). 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 \(\delta \in (1,d)\) and any \(\varepsilon >0\), d-dimensional Euclidean TSP on n points with fractal dimension at most \(\delta \) cannot be solved in time \(2^{O(n^{1-1/(\delta - \varepsilon )} )}\). The best-known upper bound is \(2^{O(n^{1-1/\delta } \log n)}\).

• For any \(\delta \in (1,d)\) and any \(\varepsilon >0\), the problem of finding k-pairwise non-intersecting d-dimensional unit balls/axis-parallel unit cubes with centers having fractal dimension at most \(\delta \) cannot be solved in time \(f(k)\ n^{O (k^{1-1/(\delta - \varepsilon )})}\) for any computable function f. The best-known upper bound is \(n^{O(k^{1-1/\delta } \log n)}\).

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).

我们研究了低分形维数空间上几何问题的复杂性。最近在 Sidiropoulos 和 Sridhar（第 33 届计算几何国际研讨会（布里斯班，2017 年）。Leibniz Int. Proc. Inform., vol. 77, # 58. Leibniz-Zent. Inform., Wadern, 2017）中展示了几个问题承认当输入是欧几里得空间中分形维数小于环境维数的点集时的改进解决方案。在本文中，我们证明了近似匹配的下界，从而为各种问题建立了近似最优的边界，作为分形维数的函数。更具体地说，我们证明对于任何整数\(d > 1\)、任何\(\delta \in (1,d)\)和任何\(n \in {\mathbb {N}}\)，有存在一个集合\({\mathbb {R}}^{d}\) 中的n个点中的X点 ，具有分形维数\(\delta \)使得对于任何\(\varepsilon > 0\)和\(c \ge 1\ )，X 的任何c -spanner都有树宽\(\Omega ( n^{1-1/(\delta - \epsilon )}/c^{d-1} )\)。该下限与之前的上限相匹配。假设指数时间假设，用于证明生成器树宽的下界的构造也可用于推导出各种问题的算法运行时间的下界。我们提供了这种类型的两个原型结果：

• 对于任何\（\ delta \ in（1，d）\）和任何\（\ varepsilon> 0 \），在n个点上最多具有分形维数\（\ delta \）的d维欧几里得TSP不能及时求解\(2^{O(n^{1-1/(\delta - \varepsilon )} )}\)。最著名的上限是\(2^{O(n^{1-1/\delta } \log n)}\)。

• 对于任何\（\三角洲\在（1，d）\）和任何\（\ varepsilon> 0 \） ，发现的问题ķ -pairwise非相交d维单位球/轴平行的单位立方体与具有中心分形维数至多\(\delta \)无法及时求解\(f(k)\ n^{O (k^{1-1/(\delta - \varepsilon )})}\)对于任何可计算函数 f . 最著名的上限是\(n^{O(k^{1-1/\delta } \log n)}\)。

以上结果几乎与[op。引用。]，并概括了由 Marx 和 Sidiropoulos 引起的环境维度情况的类似下界（第 30 届计算几何年度研讨会（京都 2014 年），第 67-76 页。ACM，纽约，2014 年）。