TY - GEN
T1 - Towards Instance-Optimal Euclidean Spanners
AU - Le, Hung
AU - Solomon, Shay
AU - Than, Cuong
AU - Toth, Csaba D.
AU - Zhang, Tianyi
N1 - Publisher Copyright:
© 2024 IEEE.
PY - 2024
Y1 - 2024
N2 - Euclidean spanners are important geometric objects that have been extensively studied since the 1980s. The two most basic 'compactness' measures of a Euclidean spanner E 12We shall identify a graph H = (X,E) with its edge set E. All edge weights are given by the Euclidean distances. are the size (number of edges) |E| and the weight (sum of edge weights) |E|. The state-of-the-art constructions of Euclidean (1+ϵ)-spanners in Rd have od(n·ϵ-d+1) edges (or sparsity od(ϵ-d+1)) and weight od(ϵ-d log ϵ-1) ·|Emst}| (or lightness od(ϵ-dlogϵ-1)); here od suppresses a factor of dO(d) and |Emst| denotes the weight of a minimum spanning tree of the input point set. Importantly, these two upper bounds are (near-)optimal (up to the dO(d) factor and disregarding the factor of log(ϵ-1) in the lightness bound) for some extremal instances [Le and Solomon, 2019], and therefore they are (near-)optimal in an existential sense. Moreover, both these upper bounds are attained by the same construction-the classic greedy spanner, whose sparsity and lightness are not only existentially optimal, but they also significantly outperform those of any other Euclidean spanner construction studied in an experimental study by [Farshi-Gudmundsson, 2009] for various practical point sets in the plane. This raises the natural question of whether the greedy spanner is (near-) optimal for any point set instance? Motivated by this question, we initiate the study of instance optimal Euclidean spanners. Our results are two-fold. •Rather surprisingly (given the aforementioned experimental study), we demonstrate that the greedy spanner is far from being instance optimal, even when allowing its stretch to grow. More concretely, we design two hard instances of point sets in the plane, where the greedy (1+xϵ)-spanner (for basically any parameter x ≥ 1) has Ωx(ϵ-1/2)·|Espa| edges and weight Ωx(ϵ-1)·|Elight|, where Espa and Elight denote the per-instance sparsest and lightest (1 +ϵ)-spanners, respectively, and the Ωx notation suppresses a polynomial dependence on 1/x. •As our main contribution, we design a new construction of Euclidean spanners, which is inherently different from known constructions, achieving the following bounds: a stretch of 1+ϵ· 2O(log∗(d/ϵ) with O(1)·|Espa| edges and weight O(1). |Elight|. In other words, we show that a slight increase to the stretch suffices for obtaining instance optimality up to an absolute constant for both sparsity and lightness. Remarkably, there is only a log-star dependence on the dimension in the stretch, and there is no dependence on it whatsoever in the number of edges and weight. In general, for any integer k≥ 1, we can construct a Euclidean spanner in Rd of stretch 1+ϵ· 2O(k) with O(log(k)}(ϵ-1)+log(k-1)}(d))·|Espa| edges and weight O(log(k)(ϵ-1)+log(k-1)(d))·|Elight|, where log(k) denotes the k-iterated logarithm.
AB - Euclidean spanners are important geometric objects that have been extensively studied since the 1980s. The two most basic 'compactness' measures of a Euclidean spanner E 12We shall identify a graph H = (X,E) with its edge set E. All edge weights are given by the Euclidean distances. are the size (number of edges) |E| and the weight (sum of edge weights) |E|. The state-of-the-art constructions of Euclidean (1+ϵ)-spanners in Rd have od(n·ϵ-d+1) edges (or sparsity od(ϵ-d+1)) and weight od(ϵ-d log ϵ-1) ·|Emst}| (or lightness od(ϵ-dlogϵ-1)); here od suppresses a factor of dO(d) and |Emst| denotes the weight of a minimum spanning tree of the input point set. Importantly, these two upper bounds are (near-)optimal (up to the dO(d) factor and disregarding the factor of log(ϵ-1) in the lightness bound) for some extremal instances [Le and Solomon, 2019], and therefore they are (near-)optimal in an existential sense. Moreover, both these upper bounds are attained by the same construction-the classic greedy spanner, whose sparsity and lightness are not only existentially optimal, but they also significantly outperform those of any other Euclidean spanner construction studied in an experimental study by [Farshi-Gudmundsson, 2009] for various practical point sets in the plane. This raises the natural question of whether the greedy spanner is (near-) optimal for any point set instance? Motivated by this question, we initiate the study of instance optimal Euclidean spanners. Our results are two-fold. •Rather surprisingly (given the aforementioned experimental study), we demonstrate that the greedy spanner is far from being instance optimal, even when allowing its stretch to grow. More concretely, we design two hard instances of point sets in the plane, where the greedy (1+xϵ)-spanner (for basically any parameter x ≥ 1) has Ωx(ϵ-1/2)·|Espa| edges and weight Ωx(ϵ-1)·|Elight|, where Espa and Elight denote the per-instance sparsest and lightest (1 +ϵ)-spanners, respectively, and the Ωx notation suppresses a polynomial dependence on 1/x. •As our main contribution, we design a new construction of Euclidean spanners, which is inherently different from known constructions, achieving the following bounds: a stretch of 1+ϵ· 2O(log∗(d/ϵ) with O(1)·|Espa| edges and weight O(1). |Elight|. In other words, we show that a slight increase to the stretch suffices for obtaining instance optimality up to an absolute constant for both sparsity and lightness. Remarkably, there is only a log-star dependence on the dimension in the stretch, and there is no dependence on it whatsoever in the number of edges and weight. In general, for any integer k≥ 1, we can construct a Euclidean spanner in Rd of stretch 1+ϵ· 2O(k) with O(log(k)}(ϵ-1)+log(k-1)}(d))·|Espa| edges and weight O(log(k)(ϵ-1)+log(k-1)(d))·|Elight|, where log(k) denotes the k-iterated logarithm.
KW - instance optimal
KW - spanner
UR - http://www.scopus.com/inward/record.url?scp=85213048816&partnerID=8YFLogxK
U2 - 10.1109/FOCS61266.2024.00099
DO - 10.1109/FOCS61266.2024.00099
M3 - ???researchoutput.researchoutputtypes.contributiontobookanthology.conference???
AN - SCOPUS:85213048816
T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
SP - 1579
EP - 1609
BT - Proceedings - 2024 IEEE 65th Annual Symposium on Foundations of Computer Science, FOCS 2024
PB - IEEE Computer Society
T2 - 65th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2024
Y2 - 27 October 2024 through 30 October 2024
ER -