On the Angular Resolution of Planar Graphs
SIAM Journal on Discrete Mathematics
Graph Drawing: Algorithms for the Visualization of Graphs
Graph Drawing: Algorithms for the Visualization of Graphs
Reconfigurable Disc Trees for Visualizing Large Hierarchical Information Space
INFOVIS '98 Proceedings of the 1998 IEEE Symposium on Information Visualization
Research report: Interacting with huge hierarchies: beyond cone trees
INFOVIS '95 Proceedings of the 1995 IEEE Symposium on Information Visualization
IEEE Transactions on Software Engineering
An approximation algorithm for a bottleneck traveling salesman problem
Journal of Discrete Algorithms
On Gilmore-Gomory's open question for the bottleneck TSP
Operations Research Letters
Pinning balloons with perfect angles and optimal area
GD'11 Proceedings of the 19th international conference on Graph Drawing
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In a balloon drawing of a tree, all the children under the same parent are placed on the circumference of the circle centered at their parent, and the radius of the circle centered at each node along any path from the root reflects the number of descendants associated with the node. Among various styles of tree drawings reported in the literature, the balloon drawing enjoys a desirable feature of displaying tree structures in a rather balanced fashion. For each internal node in a balloon drawing, the ray from the node to each of its children divides the wedge accommodating the subtree rooted at the child into two sub-wedges. Depending on whether the two sub-wedge angles are required to be identical or not, a balloon drawing can further be divided into two types: even sub-wedge and uneven sub-wedge types. In the most general case, for any internal node in the tree there are two dimensions of freedom that affect the quality of a balloon drawing: (1) altering the order in which the children of the node appear in the drawing, and (2) for the subtree rooted at each child of the node, flipping the two sub-wedges of the subtree. In this paper, we give a comprehensive complexity analysis for optimizing balloon drawings of rooted trees with respect to angular resolution, aspect ratio and standard deviation of angles under various drawing cases depending on whether the tree is of even or uneven sub-wedge type and whether (1) and (2) above are allowed. It turns out that some are NP-complete while others can be solved in polynomial time. We also derive approximation algorithms for those that are intractable in general.