Area-efficient algorithms for straight-line tree drawings

Abstract. We consider straight-line drawings of trees on a hexagonal grid. The hexagonal grid is an extension of the common grid with inner nodes of degree six. We restrict the number of directions used for the edges from each node to its children from one to five, and to five patterns: straight, Y , ψ, X, and full. The ψ-drawings generalize hv-or strictly upward drawings to ternary trees.We show that complete ternary trees have a ψ-drawing on a square of size O(n 1.262 ) and general ternary trees can be drawn within O(n 1.631 ) area. Both bounds are optimal. Sub-quadratic bounds are also obtained for X-pattern drawings of complete tetra trees, and for full-pattern drawings of complete penta trees, which are 4-ary and 5-ary trees. These results parallel and complement the ones of Frati [8] for straight-line orthogonal drawings of ternary trees.Moreover, we provide an algorithm for compacted straight-line drawings of penta trees on the hexagonal grid, such that the direction of the edges from a node to its children is given by our patterns and these edges have the same length. However, drawing trees on a hexagonal grid within a prescribed area or with unit length edges is N P-hard.

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“…O(n log log n) [3,14] Straight-line orthogonal drawings of ternary trees were recently investigated by Frati [8]. Here the picture changes.…”

Section: Fig 2 the Five Patterns Of The Drawingsmentioning

confidence: 99%

“…The typical tree structure is not visible, since the algorithms wind edges and recursively fold subtrees. This poor behavior also holds for other tree drawing algorithms achieving good area bounds [3,9,10,14].…”

Section: Introductionmentioning

confidence: 99%

Tree Drawings on the Hexagonal Grid

et al. 2009

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“…Tree drawings have interested researchers for many decades: e.g., hierarchical drawings of binary trees date to the 1970's [31]. Many improvements have been proposed since this early work, using space efficiently and generalizing to non-binary trees [2,5,[17][18][19][28][29][30]. These drawings fail to meet the four constraints mentioned earlier, especially the constraint on angular resolution.…”

Section: Introductionmentioning

confidence: 99%

Drawing Trees with Perfect Angular Resolution and Polynomial Area

Duncan

Eppstein

Goodrich

et al. 2012

Discrete Comput Geom

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We study methods for drawing trees with perfect angular resolution, i.e., with angles at each node v equal to 2π/d(v). We show:1. Any unordered tree has a crossing-free straight-line drawing with perfect angular resolution and polynomial area.2. There are ordered trees that require exponential area for any crossing-free straight-line drawing having perfect angular resolution.3. Any ordered tree has a crossing-free Lombardi-style drawing (where each edge is represented by a circular arc) with perfect angular resolution and polynomial area.Thus, our results explore what is achievable with straight-line drawings and what more is achievable with Lombardi-style drawings, with respect to drawings of trees with perfect angular resolution.

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“…"(N) area for a complete binary tree and ! "(n log n) for a general binary tree, and straight-line orthogonal drawings [3,5,33] that achieve ! "(n loglog n) area.…”

Section: Graph Drawing Approachesmentioning

confidence: 99%

PedVis: A Structured, Space-Efficient Technique for Pedigree Visualization

Tuttle

Nonato²,

Silva

2010

IEEE Trans. Visual. Comput. Graphics

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A u g u s t 2 0 1 1 C o p y r i g h t © C l a u r i s s a T u t t l e 2 0 1 1 A l l R i g h t s R e s e r v e d T h e U n i v e r s i t y o f U t a h G r a d u a t e S c h o o l STATEMENT OF THESIS APPROVAL ABSTRACTPublic genealogical databases are becoming increasingly populated with historical data and records of the current population's ancestors. As this increasing amount of available information is used to link individuals to their ancestors, the resulting trees become deeper and denser, which justifies the need for using organized, space-efficient layouts to display the data. Existing layouts are often only able to show a small subset of the data at a time. As a result, it is easy to become lost when navigating through the data or to lose sight of the overall tree structure. On the contrary, leaving space for unknown ancestors allows one to better understand the tree's structure, but leaving this space becomes expensive and allows fewer generations to be displayed at a time. In this work, we propose that the H-tree based layout be used in genealogical software to display ancestral trees. We will show that this layout presents an increase in the number of displayable generations, provides a nicely arranged, symmetrical, intuitive and organized fractal structure, increases the user's ability to understand and navigate through the data, and accounts for the visualization requirements necessary for displaying such trees.Finally, user-study results indicate potential for user acceptance of the new layout.

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Area-efficient algorithms for straight-line tree drawings

Cited by 30 publications

References 20 publications

Tree Drawings on the Hexagonal Grid

Tree Drawings on the Hexagonal Grid

Drawing Trees with Perfect Angular Resolution and Polynomial Area

PedVis: A Structured, Space-Efficient Technique for Pedigree Visualization

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