具有有界纵横比的树图
Treemaps with Bounded Aspect Ratio
摘要：
树图是可视化分层数据的一种流行技术。输入是加权树$\tree$，其中每个节点的权重是其子节点的权重之和。$\tree$的树图是将矩形分层划分为单连通区域（通常为矩形）。每个区域表示$\tree$的一个节点，并且其面积与对应节点的权重成比例。树图的一个重要质量标准是其区域的纵横比。如果区域被限制为矩形，则不能限制长宽比。相反，使用凸多边形的\emph{polygonal partitions}具有有界的纵横比。我们是第一个获得凸分区的最佳纵横比$O(\depth(\tree))$。但是，$\depth(\tree)$仍然依赖于输入树。因此，我们引入了一种新类型的树图，即\emph{orthoconvex treemaps}，其中表示叶子的区域是矩形，L-和S-形状，表示内部节点的区域是正凸多边形。我们证明，任何输入树，不管节点的权重和树的深度，承认一个正交凸树图的恒定纵横比。我们还获得了一些专门的结果，单级树图，即输入树的深度为1的树图。
英文摘要：
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.
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