TY - GEN

T1 - Light spanners for snowflake metrics

AU - Gottlieb, Lee Ad

AU - Solomon, Shay

PY - 2014

Y1 - 2014

N2 - A classic result in the study of spanners is the existence of light low-stretch spanners for Euclidean spaces. These spanners have arbitrary low stretch, and weight only a constant factor greater than that of the minimum spanning tree of the points (with dependence on the stretch and Euclidean dimension). A central open problem in this field asks whether other spaces admit low weight spanners as well - for example metric space with low intrinsic dimension - yet only a handful of results of this type are known. In this paper, we consider snowflake metric spaces of low intrinsic dimension. The α-snowflake of a metric (X, δ) is the metric (X, δα), for 0 < α < 1. By utilizing an approach completely different than those used for Euclidean spaces, we demonstrate that snowflake metrics admit light spanners. Further, we show that the spanner is of diameter O(log n), a result not possible for Euclidean spaces. As an immediate corollary to our spanner, we obtain dramatic improvements in algorithms for the traveling salesman problem in this setting, achieving a polynomial-time approximation scheme with near-linear runtime. Along the way, we also show that all ℓp spaces admit light spanners, a result of interest in its own right.

AB - A classic result in the study of spanners is the existence of light low-stretch spanners for Euclidean spaces. These spanners have arbitrary low stretch, and weight only a constant factor greater than that of the minimum spanning tree of the points (with dependence on the stretch and Euclidean dimension). A central open problem in this field asks whether other spaces admit low weight spanners as well - for example metric space with low intrinsic dimension - yet only a handful of results of this type are known. In this paper, we consider snowflake metric spaces of low intrinsic dimension. The α-snowflake of a metric (X, δ) is the metric (X, δα), for 0 < α < 1. By utilizing an approach completely different than those used for Euclidean spaces, we demonstrate that snowflake metrics admit light spanners. Further, we show that the spanner is of diameter O(log n), a result not possible for Euclidean spaces. As an immediate corollary to our spanner, we obtain dramatic improvements in algorithms for the traveling salesman problem in this setting, achieving a polynomial-time approximation scheme with near-linear runtime. Along the way, we also show that all ℓp spaces admit light spanners, a result of interest in its own right.

KW - Metric spanners

KW - Snowflake metrics

UR - http://www.scopus.com/inward/record.url?scp=84904435686&partnerID=8YFLogxK

U2 - 10.1145/2582112.2582140

DO - 10.1145/2582112.2582140

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AN - SCOPUS:84904435686

SN - 9781450325943

T3 - Proceedings of the Annual Symposium on Computational Geometry

SP - 387

EP - 395

BT - Proceedings of the 30th Annual Symposium on Computational Geometry, SoCG 2014

T2 - 30th Annual Symposium on Computational Geometry, SoCG 2014

Y2 - 8 June 2014 through 11 June 2014

ER -