Tree Drawings on the Hexagonal Grid

Abstract

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.

Drawing Unordered Trees on k-Grids

“…Similar to the results of [1], we establish an upper and lower bound for the area for complete trees. Proof.…”

“…We extend the 4-grid with its four directions to the hexagonal or 6-grid [1,5,16] with six directions by adding an edge {u, v} for each u ∈ V on coordinates (x, y) and v ∈ V on (x + 1, y − 1). The octagonal or 8-grid is a 6-grid with additional edges {u, v} between each u ∈ V on (x, y) and v ∈ V on (x+1, y+1).…”

“…N P-hardness results for minimum area were presented in [4,18]. We presented an equivalent result for the 6-grid for trees with degree up to six [1]. In the plane only two grid axes are linear independent.…”

“…This grid was widely investigated in literature [2,7,8,11,13,14,21,23]. The 6-and the 8-grid with additional axes are used to draw trees with higher degree [1,5,16]. A motivation for such grids are discrete representations of radial drawings [9].…”

Pinning balloons with perfect angles and optimal area

“…For the special case of orthogonal straight-line drawings of ternary trees (they automatically guarantee perfect angular resolution) Frati [7] provided an algorithm whose drawings require O(n 1.6131 ) area; the drawing of the complete ternary tree requires O(n 1.262 ) area. Bachmaier et al obtained a drawing of the complete 6-regular tree with perfect angular resolution with area O(n 1.37 ) [1]. In contrast to our setting the so-called balloon drawings [9,10] place all balloons at the same distance.…”