Lightness and sparsity are two natural parameters for Euclidean (1+ \epsilon )-spanners. Classical results show that, when the dimension d\in N and \epsilon >0 are constant, every set S of n points in d-space admits an (1+ \epsilon )-spanners with O(n) edges and weight proportional to that of the Euclidean MST of S. Tight bounds on the dependence on \epsilon >0 for constant d\in N have been established only recently. Le and Solomon (FOCS 2019) showed that Steiner points can substantially improve the lightness and sparsity of a (1+ \epsilon )-spanner. They gave upper bounds of O( \epsilon ^{-(d+1)/2}) for the minimum lightness in dimensions d\geq 3, and O( \epsilon ^{-(d-1))/2}) for the minimum sparsity in d-space for all d\geq 1. They obtained lower bounds only in the plane (d=2). Le and Solomon (ESA 2020) also constructed Steiner (1+ \epsilon )-spanners of lightness O( \epsilon ^{-1}\log\Delta) in the plane, where \Delta\in \Omega(\log n) is the \emph{spread} of S, defined as the ratio between the maximum and minimum distance between a pair of points. In this work, we improve several bounds on the lightness and sparsity of Euclidean Steiner (1+ \epsilon )-spanners. Using a new geometric analysis, we establish lower bounds of \Omega( \epsilon ^{-d/2}) for the lightness and \Omega( \epsilon ^{-(d-1)/2}) for the sparsity of such spanners in Euclidean d-space for all d\geq 2. We use the geometric insight from our lower bound analysis to construct Steiner (1+ \epsilon )-spanners of lightness O( \epsilon ^{-1}\log n) for n points in Euclidean plane.
Lightness and sparsity are two natural parameters for Euclidean (1+ \varepsilon )-spanners. Classical results show that, when the dimension d\in N and \varepsilon >0 are constant, every set S of n points in d-space admits an (1+ \varepsilon )-spanners with O(n) edges and weight proportional to that of the Euclidean MST of S. Tight bounds on the dependence on \varepsilon >0 for constant d\in N have been established only recently. Le and Solomon (FOCS 2019) showed that Steiner points can substantially improve the lightness and sparsity of a (1+ \varepsilon )-spanner. They gave upper bounds of O( \varepsilon ^{-(d+1)/2}) for the minimum lightness in dimensions d\geq 3, and O( \varepsilon ^{-(d-1))/2}) for the minimum sparsity in d-space for all d\geq 1. They obtained lower bounds only in the plane (d=2). Le and Solomon (ESA 2020) also constructed Steiner (1+ \varepsilon )-spanners of lightness O( \varepsilon ^{-1}\log\Delta) in the plane, where \Delta\in \Omega(\log n) is the \emph{spread} of S, defined as the ratio between the maximum and minimum distance between a pair of points. In this work, we improve several bounds on the lightness and sparsity of Euclidean Steiner (1+ \varepsilon )-spanners. Using a new geometric analysis, we establish lower bounds of \Omega( \varepsilon ^{-d/2}) for the lightness and \Omega( \varepsilon ^{-(d-1)/2}) for the sparsity of such spanners in Euclidean d-space for all d\geq 2. We use the geometric insight from our lower bound analysis to construct Steiner (1+ \varepsilon )-spanners of lightness O( \varepsilon ^{-1}\log n) for n points in Euclidean plane.
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