Aspect Ratio Universal Rectangular Layouts

TL;DR

The paper addresses the problem of characterizing and recognizing aspect ratio universal layouts (ARU) for axis-aligned rectangular subdivisions. It shows that weak ARU is equivalent to sliceable layouts, while strong ARU is equivalent to layouts that are both one-sided and sliceable, with these properties tied to the uniqueness of transversal structures on the extended dual graph $G^*(\mathcal{L})$. The authors provide a combinatorial framework linking geometry and transversal structures, and they present a quadratic-time algorithm to decide whether a given graph is the dual of a strongly ARU layout and to construct such a layout when possible. The results have practical implications for data visualization and cartography, enabling flexible realization of arbitrary aspect ratios while preserving the underlying adjacency semantics. The work also highlights structural properties (e.g., presence of cuts and pivots) that guide efficient recognition and suggests open questions about improving runtimes for broader classes of layouts.

Abstract

A \emph{generic rectangular layout} (for short, \emph{layout}) is a subdivision of an axis-aligned rectangle into axis-aligned rectangles, no four of which have a point in common. Such layouts are used in data visualization and in cartography. The contacts between the rectangles represent semantic or geographic relations. A layout is weakly (strongly) \emph{aspect ratio universal} if any assignment of aspect ratios to rectangles can be realized by a weakly (strongly) equivalent layout. We give combinatorial characterizations for weakly and strongly aspect ratio universal layouts. Furthermore, we describe a quadratic-time algorithm that decides whether a given graph is the dual graph of a strongly aspect ratio universal layout, and finds such a layout if one exists.

Figures (12)

• Figure 1: (a--b) A layout and its dual graph. (c--d) A sliceable layout and its dual graph. The two layouts are neither strongly nor weakly equivalent, but their dual graphs are isomorphic.
• Figure 2: (a--c) Three generic layouts that are weakly equivalent. The layouts in (a) and (b) are strongly equivalent, but not strongly equivalent to the layout in (c), as the adjacencies between the pairs of rectangles $\{r_3,r_6\}$ and $\{r_4,r_5\}$ are different. (d) A nongeneric layout that is in the closure of the strong equivalence classes of the layouts in (a--b) and (c).
• Figure 3: (a) A layout $\mathcal{L}$, the red maximal segments are not one-sided. (b) Three sublayouts of $\mathcal{L}$, one of which is trivial. (c) An irreducible layout. (d) A rectangular arrangement.
• Figure 4: (a) A layout $\mathcal{L}$ bounded by edges $e_1,\ldots , e_4$. (b) Extended dual graph $G^*(\mathcal{L})$ with an outer 4-cycle $(e_1,\ldots ,e_4)$. (c) A transversal structure.
• Figure 5: A flip of an empty (left) and a nonempty (right) alternating cycle.

Theorems & Definitions (46)

• Lemma 1
• proof
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 5
• proof
• Lemma 6
• proof
• Corollary 7