Before revision:  Rather than scaling with the number of hierarchical clusterings of N elements (( 2N-3)!! ), our approach runs in time and space proportional to the significantly smaller powerset of N. Despite still being large, there are many important applications at the practically-computable range of N < 20. We demonstrate the advantages of exact inference on synthetic data of interest to Dasgupta's cost as well as on two real world applications, in particle physics at the Large Hadron Collider at CERN and in cancer genomics  .
After revision:  Rather than scaling with the number of hierarchical clusterings of N elements (( 2N-3)!! possible hierarchies. Also, for larger datasets where our exact algorithms become infeasible, we introduce an approximate algorithm based on a sparse trellis that compares well to other benchmarks. Exact methods are relevant to data analyses in particle physics at the Large Hadron Collider at CERN and in cancer genomics .