Stretching Stretch

We give a generalized definition of stretch that simplifies the efficient construction of lowstretch embeddings suitable for graph algorithms. The generalization, based on discounting highly stretched edges by taking their pth 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(m log log n) time. Our algorithm parallelizes readily and can also produce generalizations of low-stretch subgraphs.