Near-linear time approximation schemes for Steiner tree and forest in low-dimensional spaces

04/07/2019

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by   Lee-Ad Gottlieb, et al.

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We give an algorithm that computes a (1+ϵ)-approximate Steiner forest in near-linear time n · 2^(1/ϵ)^O(ddim^2) (loglog n)^2. This is a dramatic improvement upon the best previous result due to Chan et al., who gave a runtime of n^2^O(ddim)· 2^(ddim/ϵ)^O(ddim)√(log n). For Steiner tree our methods achieve an even better runtime n (log n)^(1/ϵ)^O(ddim^2) in doubling spaces. For Euclidean space the runtime can be reduced to 2^(1/ϵ)^O(d^2) n log n, improving upon the result of Arora in fixed dimension d.