  Handling visual complexity is a challenging problem in visualization owing to
the subjectiveness of its definition and the difficulty in devising
generalizable quantitative metrics. In this paper we address this challenge by
measuring the visual complexity of two common forms of cluster-based
visualizations: scatter plots and parallel coordinatess. We conceptualize
visual complexity as a form of visual uncertainty, which is a measure of the
degree of difficulty for humans to interpret a visual representation correctly.
We propose an algorithm for estimating visual complexity for the aforementioned
visualizations using Allen's interval algebra. We first establish a set of
primitive 2-cluster cases in scatter plots and another set for parallel
coordinatess based on symmetric isomorphism. We confirm that both are the
minimal sets and verify the correctness of their members computationally. We
score the uncertainty of each primitive case based on its topological
properties, including the existence of overlapping regions, splitting regions
and meeting points or edges. We compare a few optional scoring schemes against
a set of subjective scores by humans, and identify the one that is the most
consistent with the subjective scores. Finally, we extend the 2-cluster measure
to k-cluster measure as a general purpose estimator of visual complexity for
these two forms of cluster-based visualization.
