Lightness is a fundamental parameter for Euclidean spanners; it is the ratio of the spanner weight to the weight of the minimum spanning tree of a finite set of points in R^d. In a recent breakthrough, Le and Solomon (2019) established the precise dependencies on \varepsilon>0 and d\in N of the minimum lightness of an (1+\varepsilon) -spanner , and observed that additional Steiner points can substantially improve the lightness. Le and Solomon (2020) constructed Steiner (1+\varepsilon)-spanners of lightness O(\varepsilon^{-1}\log\Delta) in the plane, where \Delta \in \Omega(\log n ) is the spread of the point set, defined as the ratio between the maximum and minimum distance between a pair of points. They also constructed spanners of lightness O(\varepsilon^{-(d+1)/2}) in dimensions d\geq 3. Recently, Bhore and T\'{o}th (2020) established a lower bound of \Omega(\varepsilon^{-d/2}) for the lightness of Steiner (1+\varepsilon)-spanners in R^d, for d\ge 2. The central open problem in this area is to close the gap between the lower and upper bounds in all dimensions d\geq 2. In this work, we show that for every finite set of points in the plane and every \varepsilon>0, there exists a Euclidean Steiner (1+\varepsilon)-spanner of lightness O(\varepsilon^{-1}); this matches the lower bound for d=2. We generalize the notion of shallow light trees, which may be of independent interest, and use directional spanners and a modified window partitioning scheme to achieve a tight weight analysis.
Lightness is a fundamental parameter for Euclidean spanners; it is the ratio of the spanner weight to the weight of the minimum spanning tree of a finite set of points in R^d. In a recent breakthrough, Le and Solomon (2019) established the precise dependencies on \varepsilon>0 and d\in N of the minimum lightness of (1+\varepsilon) -spanners , and observed that additional Steiner points can substantially improve the lightness. Le and Solomon (2020) constructed Steiner (1+\varepsilon)-spanners of lightness O(\varepsilon^{-1}\log\Delta) in the plane, where \Delta \geq \Omega(\sqrt{n ) is the spread of the point set, defined as the ratio between the maximum and minimum distance between a pair of points. They also constructed spanners of lightness O(\varepsilon^{-(d+1)/2}) in dimensions d\geq 3. Recently, Bhore and T\'{o}th (2020) established a lower bound of \Omega(\varepsilon^{-d/2}) for the lightness of Steiner (1+\varepsilon)-spanners in R^d, for d\ge 2. The central open problem in this area is to close the gap between the lower and upper bounds in all dimensions d\geq 2. In this work, we show that for every finite set of points in the plane and every \varepsilon>0, there exists a Euclidean Steiner (1+\varepsilon)-spanner of lightness O(\varepsilon^{-1}); this matches the lower bound for d=2. We generalize the notion of shallow light trees, which may be of independent interest, and use directional spanners and a modified window partitioning scheme to achieve a tight weight analysis.
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