  Nucleus decompositions have been shown to be a useful tool for finding dense
subgraphs. The coreness value of a clique represents its density based on the
number of other cliques it is adjacent to. One useful output of nucleus
decomposition is to generate a hierarchy among dense subgraphs at different
resolutions. However, existing parallel algorithms for nucleus decomposition do
not generate this hierarchy, and only compute the coreness values. This paper
presents a scalable parallel algorithm for hierarchy construction, with
practical optimizations, such as interleaving the coreness computation with
hierarchy construction and using a concurrent union-find data structure in an
innovative way to generate the hierarchy. We also introduce a parallel
approximation algorithm for nucleus decomposition, which achieves much lower
span in theory and better performance in practice. We prove strong theoretical
bounds on the work and span (parallel time) of our algorithms.
  On a 30-core machine with two-way hyper-threading on real-world graphs, our
parallel hierarchy construction algorithm achieves up to a 58.84x speedup over
the state-of-the-art sequential hierarchy construction algorithm by Sariyuce et
al. and up to a 30.96x self-relative parallel speedup. On the same machine, our
approximation algorithm achieves a 3.3x speedup over our exact algorithm, while
generating coreness estimates with a multiplicative error of 1.33x on average.
