Orderly Spanning Trees with Applications

Abstract

We introduce and study the {\em orderly spanning trees} of plane graphs. This algorithmic tool generalizes {\em canonical orderings}, which exist only for triconnected plane graphs. Although not every plane graph admits an orderly spanning tree, we provide an algorithm to compute an {\em orderly pair} for any connected planar graph $G$, consisting of a plane graph $H$ of $G$, and an orderly spanning tree of $H$. We also present several applications of orderly spanning trees: (1) a new constructive proof for Schnyder's Realizer Theorem, (2) the first area-optimal 2-visibility drawing of $G$, and (3) the best known encodings of $G$ with O(1)-time query support. All algorithms in this paper run in linear time.Comment: 25 pages, 7 figures, A preliminary version appeared in Proceedings of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2001), Washington D.C., USA, January 7-9, 2001, pp. 506-51

Complexity Measures for Mosaic Drawings

“…As was shown in [4], we can always obtain such drawings with O(n 2 ) area. Here we show that this bound is tight: there exist outerplane graphs that require Ω(n 2 ) area.…”

“…More specifically, there exist outerplane graphs with k ears, such that any mosaic drawing of these graphs has either Ω(n 2 /k 2 ) area or total channel complexity Ω(k). This bound is tight, since the algorithm by Chiang et al [4] can be used to construct mosaic drawings without channels in O(n 2 ) area. In Section 4 we consider straight channels.…”

“…However, in scenarios where the shapes representing vertices are not fixed, the complexity of these shapes clearly contributes to the visual quality of the drawing. For example, there are several sequences of papers which strive to represent (weighted) planar triangulated graphs with rectilinear polygons of the lowest possible complexity, which is 8 in both the weighted [1] and in the unweighted case [4].…”

Balanced parentheses strike back

“…-Balanced parentheses Chuang et al 1998;He et al 1999;Chiang et al 2005;Munro and Rao 2004;Bonichon et al 2006], a folklore encoding consisting of a balanced string of parentheses representing the counterclockwise depth-first traversal of T , where an open (respectively, closed) parenthesis denotes a descending (respectively, ascending) edge traversal. For technical reason, one usually adds a pair of enclosing parentheses to the above 2n − 2 parentheses, resulting in a representation consisting of 2n parentheses.…”

“…Munro, Raman, and Rao showed an o(n)-bit auxiliary string to support O(1)-time query for leaf-rank, leaf-select, and leaf-size. Chiang, Lin, and Lu [Chiang et al 2005] showed an o(n)-bit auxiliary string to support O(1)-time degree query. Munro and Rao [Munro and Rao 2004] further gave an o(n)-bit auxiliary string to support O(1)-time level-ancestor query.…”