On Balloon Drawings of Rooted Trees

Among various styles of tree drawing reported in the literature, balloon drawing enjoys a desirable feature of displaying tree structures in a rather balanced fashion. Each subtree in the balloon drawing of a tree is enclosed in a circle. The radius of each circle is proportional to the number of descendents associated with the root node of the subtree. In this paper, we investigate various issues related to balloon drawing of rooted trees from both algorithmic and practical viewpoints. First, we design an efficient algorithm to optimize angular resolution and aspect ratio for the balloon drawing of rooted unordered trees. For the case of ordered trees for which the center of the enclosing circle of a subtree need not coincide with the root of the subtree, flipping the drawing of a subtree (along the axis from the parent to the root of the subtree) might change both the aspect ratio and the angular resolution of the drawing. We show that optimizing the angular resolution as well as the aspect ratio with respect to this type of rooted ordered trees is reducible to the perfect matching problem for bipartite graphs, which is solvable in polynomial time. Aside from studying balloon drawing from an algorithmic viewpoint, we also propose a local magnetic spring model (which can be thought of as a variant of the popular force-directed strategies) for producing dynamic balloon drawings for rooted trees. In our framework, each edge is modelled by a magnetized spring, while each vertex is placed on a local polar magnetic field which does not interact with other magnetic fields. Our approach facilitates various operations, including interaction and navigation, on trees. With a slight modification to our force-directed based balloon drawing algorithm, we are able to apply our work to the drawing of galaxy systems, H-trees, and sparse graphs, which are of practical interest