  Comparison is a core task in visual analysis. Although there are numerous
guidelines to help users design effective visualizations to aid known
comparison tasks, there are few formalisms that define the semantics of
comparison operations in a way that can serve as the basis for a grammar of
comparison interactions. Recent work proposed a formalism called View
Composition Algebra (VCA) that enables ad hoc comparisons between any
combination of marks, trends, or charts in a visualization interface. However,
VCA limits comparisons to visual representations of data that have an identical
schema, or where the schemas form a strict subset relationship (e.g., comparing
price per state with price, but not with price per county). In contrast, the
majority of real-world data - temporal, geographical, organizational - are
hierarchical.
  To bridge this gap, this paper presents an extension to VCA (called VCAH)
that enables ad hoc comparisons between visualizations of hierarchical data.
VCAH leverages known hierarchical relationships to enable ad hoc comparison of
data at different hierarchical granularities. We illustrate applications to
hierarchical and Tableau visualizations.
