  Hierarchical clustering over graphs is a fundamental task in data mining and
machine learning with applications in domains such as phylogenetics, social
network analysis, and information retrieval. Specifically, we consider the
recently popularized objective function for hierarchical clustering due to
Dasgupta. Previous algorithms for (approximately) minimizing this objective
function require linear time/space complexity. In many applications the
underlying graph can be massive in size making it computationally challenging
to process the graph even using a linear time/space algorithm. As a result,
there is a strong interest in designing algorithms that can perform global
computation using only sublinear resources. The focus of this work is to study
hierarchical clustering for massive graphs under three well-studied models of
sublinear computation which focus on space, time, and communication,
respectively, as the primary resources to optimize: (1) (dynamic) streaming
model where edges are presented as a stream, (2) query model where the graph is
queried using neighbor and degree queries, (3) MPC model where the graph edges
are partitioned over several machines connected via a communication channel.
  We design sublinear algorithms for hierarchical clustering in all three
models above. At the heart of our algorithmic results is a view of the
objective in terms of cuts in the graph, which allows us to use a relaxed
notion of cut sparsifiers to do hierarchical clustering while introducing only
a small distortion in the objective function. Our main algorithmic
contributions are then to show how cut sparsifiers of the desired form can be
efficiently constructed in the query model and the MPC model. We complement our
algorithmic results by establishing nearly matching lower bounds that rule out
the possibility of designing better algorithms in each of these models.
