We give a generalized definition of stretch that simplifies the efficient construction of low-stretch embeddings suitable for graph algorithms. The generalization, based on discounting highly stretched edges by taking their p-th power for some 0<p<1, is directly related to performances of existing algorithms. This discounting of high-stretch edges allows us to treat many classes of edges with coarser granularity. It leads to a two-pass approach that combines bottom-up clustering and top-down decompositions to construct these embeddings in O(mloglogn) time. Our algorithm parallelizes readily and can also produce generalizations of low-stretch subgraphs.
@article{arxiv.1401.2454,
title = {Stretching Stretch},
author = {Michael B. Cohen and Gary L. Miller and Jakub W. Pachocki and Richard Peng and Shen Chen Xu},
journal= {arXiv preprint arXiv:1401.2454},
year = {2014}
}