Lee-Ad Gottlieb
ABSTRACT
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.
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Index Terms
- Light spanners for Snowflake Metrics
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