Treemaps with Bounded Aspect Ratio

TL;DR

This paper introduces orthoconvex treemaps, a new type of hierarchical visualization that guarantees a constant aspect ratio regardless of tree depth, improving upon traditional rectangular treemaps.

Contribution

The authors present orthoconvex treemaps that ensure bounded aspect ratio for any hierarchical data, regardless of tree depth or node weights, a significant advancement over previous methods.

Findings

Convex partitions achieve optimal aspect ratio of O(depth)

Orthoconvex treemaps have constant aspect ratio for any tree

Specialized results for single-level treemaps

Abstract

Treemaps are a popular technique to visualize hierarchical data. The input is a weighted tree $\tree$ where the weight of each node is the sum of the weights of its children. A treemap for $\tree$ is a hierarchical partition of a rectangle into simply connected regions, usually rectangles. Each region represents a node of $\tree$ and its area is proportional to the weight of the corresponding node. An important quality criterion for treemaps is the aspect ratio of its regions. One cannot bound the aspect ratio if the regions are restricted to be rectangles. In contrast, \emph{polygonal partitions}, that use convex polygons, have bounded aspect ratio. We are the first to obtain convex partitions with optimal aspect ratio $O(\depth(\tree))$. However, $\depth(\tree)$ still depends on the input tree. Hence we introduce a new type of treemaps, namely \emph{orthoconvex treemaps}, where regions representing leaves are rectangles, L-, and S-shapes, and regions representing internal nodes are orthoconvex polygons. We prove that any input tree, irrespective of the weights of the nodes and the depth of the tree, admits an orthoconvex treemap of constant aspect ratio. We also obtain several specialized results for single-level treemaps, that is, treemaps where the input tree has depth~1.