Conference: The 24th Annual Symposium on Computational Geometry (SoCG 2008).

Comment: Full version of our paper is available *here*.

Abstract: We introduce a hierarchical partitioning scheme of the Euclidean plane, called *circular
partitions*. Such a partition consists of a hierarchy of convex polygons, each having small
aspect ratio, and satisfying specified volume constrains. We apply these partitions to obtain
a natural extension of the popular Treemap visualization method. Our proposed algorithm is not
constrained in using only rectangles, and can achieve provably better guarantees on the aspect
ratio of the constructed polygons.

Under relaxed conditions, we can also construct circular partitions in higher-dimensional
spaces. We can use these relaxed partitions to obtain improved approximation algorithms for
embedding ultrametrics into $d$-dimensional Euclidean space. In particular, we give
a $\operatorname{polylog}(\Delta)$-approximation algorithm for embedding $n$-point
ultrametrics into the $\mathbb R^d$ with minimum distortion ($\Delta$ denotes the spread of
the metric). The previously best-known approximation ratio for this problem was polynomial
in $n$ (Badoiu *et al.* SOCG 2006). This is the first algorithm for embedding
a non-trivial family of weighted graph metrics into a space of constant dimension that
achieves a polylogarithmic approximation ratio.

Nice example: [JPG]

Related webpage: History of treemaps