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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.

A coloring of a complete bipartite graph is shuffle-preserved if it is the case that assigning a color $c$ to edges $(u, v)$ and $(u', v')$ enforces the same color assignment for edges $(u, v')$ and $(u',v)$. (In words, the induced subgraph with respect to color $c$ is complete.) In this paper, we investigate a variant of the Ramsey problem for the class of complete bipartite multigraphs. (By a multigraph we mean a graph in which multiple edges, but no loops, are allowed.) Unlike the conventional m-coloring scheme in Ramsey theory which imposes a constraint (i.e., $m$) on the total number of colors allowed in a graph, we introduce a relaxed version called m-local coloring which only requires that, for every vertex $v$, the number of colors associated with $v$'s incident edges is bounded by $m$. Note that the number of colors found in a graph under $m$-local coloring may exceed m. We prove that given any $n \times n$ complete bipartite multigraph $G$, every shuffle-preserved $m$-local coloring displays a monochromatic copy of $K_{p,p}$ provided that $2(p-1)(m-1) < n$. Moreover, the above bound is tight when (i) $m=2$, or (ii) $n=2^k$ and $m=3\cdot 2^{k-2}$ for every integer $k\geq 2$. As for the lower bound of $p$, we show that the existence of a monochromatic $K_{p,p}$ is not guaranteed if $p> \lceil \frac{n}{m} \rceil$. Finally, we give a generalization for $k$-partite graphs and a method applicable to general graphs. Many conclusions found in $m$-local coloring can be inferred to similar results of $m$-coloring., Comment: 10 pages, 3 figures

Floor-planning is a fundamental step in VLSI chip design. Based upon the concept of orderly spanning trees, we present a simple O(n)-time algorithm to construct a floor-plan for any n-node plane triangulation. In comparison with previous floor-planning algorithms in the literature, our solution is not only simpler in the algorithm itself, but also produces floor-plans which require fewer module types. An equally important aspect of our new algorithm lies in its ability to fit the floor-plan area in a rectangle of size (n-1)x(2n+1)/3. Lower bounds on the worst-case area for floor-planning any plane triangulation are also provided in the paper., Comment: 13 pages, 5 figures, An early version of this work was presented at 9th International Symposium on Graph Drawing (GD 2001), Vienna, Austria, September 2001. Accepted to Journal of Algorithms, 2003