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- W4300659878 abstract "We study the complexity of geometric problems on spaces of low fractal dimension. It was recently shown by [Sidiropoulos & Sridhar, SoCG 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 set of $n$ points in $d$-dimensional Euclidean space, of fractal dimension $deltain (1,d)$, for any $epsilon >0$ and $cgeq 1$, any $c$-spanner must have treewidth at least $Omega left( frac{n^{1-1/(delta - epsilon)}}{c^{d-1}} right)$, matching 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 $epsilon >0$ we show that: 1) $d$-dimensional Euclidean TSP on $n$ points with fractal dimension at most $delta$ cannot be solved in time $2^{Oleft(n^{1-1/(delta - epsilon)} right)}$. The best-known upper bound is $2^{O(n^{1-1/delta} log n)}$. 2) 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 $f(k)n^{O left(k^{1-1/(delta - epsilon)}right)}$ 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 [Sidiropoulos & Sridhar, SoCG 2017], and generalize analogous lower bounds for the case of ambient dimension due to [Marx & Sidiropoulos, SoCG 2014]." @default.
- W4300659878 created "2022-10-03" @default.
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- W4300659878 date "2017-12-12" @default.
- W4300659878 modified "2023-10-16" @default.
- W4300659878 title "Fractal dimension and lower bounds for geometric problems" @default.
- W4300659878 doi "https://doi.org/10.48550/arxiv.1712.04595" @default.
- W4300659878 hasPublicationYear "2017" @default.
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