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- W4312991750 abstract "Given a set S of n points in the plane and a parameter $$varepsilon >0$$ , a Euclidean $$(1,+,varepsilon )$$ -spanner is a geometric graph $$G=(S,E)$$ that contains a path of weight at most $$(1+varepsilon )Vert pqVert _2$$ for all $$p,qin S$$ . We show that the minimum weight of a Euclidean $$(1+varepsilon )$$ -spanner for n points in the unit square $$[0,1]^2$$ is $$O(varepsilon ^{-3/2},sqrt{n})$$ , and this bound is the best possible. The upper bound is based on a new spanner algorithm that sparsifies Yao-graphs. It improves upon the baseline $$O(varepsilon ^{-2}sqrt{n})$$ , obtained by combining a tight bound for the weight of an MST and a tight bound for the lightness of Euclidean $$(1+varepsilon )$$ -spanners, which is the ratio of the spanner weight to the weight of the MST. The result generalizes to d-space for all $$din mathbb {N}$$ : The minimum weight of a Euclidean $$(1,+,varepsilon )$$ -spanner for n points in the unit cube $$[0,1]^d$$ is $$O_d(varepsilon ^{(1-d^2)/d}n^{(d-1)/d})$$ , and this bound is the best possible. For the $$ntimes n$$ section of the integer lattice, we show that the minimum weight of a Euclidean $$(1+varepsilon )$$ -spanner is between $$varOmega (varepsilon ^{-3/4}n^2)$$ and $$O(varepsilon ^{-1}log (varepsilon ^{-1}), n^2)$$ . These bounds become $$varOmega (varepsilon ^{-3/4}sqrt{n})$$ and $$O(varepsilon ^{-1}log (varepsilon ^{-1})sqrt{n})$$ when scaled to a grid of n points in $$[0,1]^2$$ ." @default.
- W4312991750 created "2023-01-05" @default.
- W4312991750 creator A5030799674 @default.
- W4312991750 date "2022-01-01" @default.
- W4312991750 modified "2023-09-26" @default.
- W4312991750 title "Minimum Weight Euclidean $$(1+varepsilon )$$-Spanners" @default.
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- W4312991750 doi "https://doi.org/10.1007/978-3-031-15914-5_32" @default.
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