  In many models for large-scale computation, decomposition of the problem is
key to efficient algorithms. For distance-related graph problems, it is often
crucial that such a decomposition results in clusters of small diameter, while
the probability that an edge is cut by the decomposition scales linearly with
the length of the edge. There is a large body of literature on low diameter
graph decomposition with small edge cutting probabilities, with all existing
techniques heavily building on single source shortest paths (SSSP)
computations. Unfortunately, in many theoretical models for large-scale
computations, the SSSP task constitutes a complexity bottleneck. Therefore, it
is desirable to replace exact SSSP computations with approximate ones. However
this imposes a fundamental challenge since the existing constructions of such
decompositions inherently rely on the subtractive form of the triangle
inequality. The current paper overcomes this obstacle by developing a technique
termed blurry ball growing. By combining this technique with a clever
algorithmic idea of Miller et al. (SPAA 13), we obtain a construction of low
diameter decompositions with small edge cutting probabilities which replaces
exact SSSP computations by (a small number of) approximate ones. The utility of
our approach is showcased by deriving efficient algorithms that work in the
Congest, PRAM, and semi-streaming models of computation. As an application, we
obtain metric tree embedding algorithms in the vein of Bartal (FOCS 96) whose
computational complexities in these models are optimal up to polylogarithmic
factors. Our embeddings have the additional useful property that the tree can
be mapped back to the original graph such that each edge is "used" only O(log
n) times, which is of interest for capacitated problems and simulating Congest
algorithms on the tree into which the graph is embedded.
