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Low Ply Drawings of Trees and 2-Trees

Ply number is a recently developed graph drawing metric inspired by studying road networks. Informally, for each vertex v, which is associated with a point in the plane, a disk is drawn centered on v with a radius that is alpha times the length of the longest edge incident to v, for some constant alpha in (0, 0.5]. The ply number is the maximum number of disks that overlap at a single point. We show that any tree with maximum degree Delta has a 1-ply drawing when alpha = O(1 / Delta). We also show that when alpha = 1/2, trees can be drawn with logarithmic ply number, with an area that is polynomial for bounded-degree trees. Lastly, we show that this logarithmic upper bound does not apply to 2-trees, by giving a lower bound of Omega(sqrt(n / log n)) ply for any value of alpha.

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08/09/2018

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1 Introduction

A useful paradigm for drawing graphs involves visualizing them as maps or road networks, allowing a visualizer to “zoom” in and out of the graph based on known techniques that apply to maps. For example, Gansner et al. [10] describe a GMap system for visualizing clusters in graphs as countries with nearby clusters drawn as neighboring countries. In addition, Nachmanson et al. [18, 19] describe a GraphMaps system for visualizing graphs as embedded road networks, so as to leverage the drawing and zooming capabilities of a roadmap viewer to explore the graph. Thus, a natural question arises as to which graphs are amenable to being drawn as road networks.

To answer this question, we formulate a precise definition of what we mean by a graph that could be drawn as a road network. One might at first suggest that graph planarity would be a good choice for such a formalism. But the class of planar graphs includes several graph instances that are difficult to visualize as road networks, such as the so-called “nested triangles” graph (e.g., see [6, 9, 12]). In addition, as shown by Eppstein and Goodrich [7], the class of planar graphs is not general enough to include all real-world road networks, as road networks are often not planar. For example, the California highway system alone has over 6,000 crossings. Instead of using planarity, then, Eppstein and Goodrich [7] introduce the concept of the ply number of an embedded graph, and they demonstrate experimentally that real-world road networks tend to have small ply. Intuitively, the ply concept tries to capture how road networks have features that are well-separated at multiple scales. The formal definition of the ply number of a graph is derived from the definition of ply for a set of disks (which captures the depth of coverage for such a set of disks) [17]; hence, the ply number of an embedded graph is defined in terms of the ply of a set of disks defined with respect to this embedding.

Let us therefore formally define the ply number of an embedded geometric graph. Let be a straight-line drawing of a graph . For every vertex , let be the open disk centered at and whose radius is times the length of the longest edge incident to . The set of ply disks containing a point is then . The -ply number of this drawing is defined as

Usually, is chosen in the range . In this range, a graph with two vertices and a single edge connecting them has ply number 1, because the ply disks for the two vertices will not overlap.

There are two natural optimization problems for the ply number when constructing a drawing of a given graph. One is to is to fix a constant ply number, and try to find a drawing that maximizes the value of . The other is to fix a value for , typically , and try to find a drawing that minimizes the ply number. In this paper, we provide new results for both of these cases.

1.1 Previous related work

As an empirical justification of the use of ply numbers, De Luca et al.’s experimental study [4] found that some force-directed algorithms, including Kamada-Kawai [16], stress majorization [11], and the fast multipole method [13] all tend to produce drawings with low ply number. Their experiments also suggest that even trees with at most three children per node can have unbounded ply number when .

The problem of drawing graphs with ply number equal to 1 is related to that of constructing circle-contact representations. A circle-contact representation for a graph is a collection of interior-disjoint circles, in which each circle represents a single vertex, and two vertices are adjacent if and only if their circles are tangent to one another [14, 15]. Di Giacomo et al. [5] show that graphs with ply number 1 are equivalent to graphs with weak unit disk contact representations, which are known to be NP-hard to recognize [3]. They also show that binary trees have drawings with ply number 2 when , or with ply number 1 when . Their drawing is reproduced in Figure 1.

Figure 1: Di Giacomo et al.’s drawing of a binary tree with -ply number 1, for . The edge lengths decrease by a factor of 2 at each level.

Angelini et al. [2] relax our definition of ply number to define the vertex-ply of a drawing, which is the maximum number of intersecting disks at any vertex of the drawing. Graphs with vertex-ply number 1 can then be interpreted as a new variant of proximity drawings.

In an earlier paper, Angelini et al. [1] show that 10-ary trees have unbounded ply number. Furthermore, they prove that 5-ary trees can be drawn with logarithmic ply number and polynomial area. The ply number of drawings of trees with between three and nine children per node remains an interesting and surprisingly daunting open problem.

1.2 Our results

In this paper, we study a number of related problems concerning low-ply drawings of bounded-degree trees. We first answer an open question proposed by Di Giacomo et al. [5], which asks whether all trees with maximum degree have 1-ply drawings for a sufficiently small . We show in Section 2 that a simple fractal drawing pattern can achieve this when .

In Section 3, we show that all trees (not just 5-ary trees) can be drawn with logarithmic ply number, for . Furthermore, the area is polynomial for trees with bounded degree. These results depend on some careful arguments about geometric configurations and fractal-like geometric constructions, as well as yet another use of the heavy-path decomposition technique of Sleator and Tarjan [20].

It is then natural to consider whether any planar graph classes larger than trees can be drawn with logarithmic ply number. In Section 4, we show that this is not the case for -trees, by constructing a family of -trees that require a ply number of for any fixed . Previous lower bounds have only applied for planar drawings, while non-planar drawings are known to sometimes have better ply number.

2 1-ply Drawings

In this section, we fix our drawings to have ply number 1, and provide conditions on such that we can construct drawings of trees of any bounded degree. At a high level, our drawings are constructed as follows. For a tree with maximum degree , we divide the area around each parent vertex radially into equal wedges, so that all of the angles are . Then we assign each subtree to a different wedge, and draw it within that wedge. The distance from each node to its children is chosen to be a constant fraction of its distance from its own parent. When , this produces the drawing shown in Figure 1. Note that for a non-root vertex, one of the wedges will contain the edge from the parent vertex, and will not contain a subtree.

This produces a drawing that is highly symmetric, in a fashion that would produce a fractal if continued in the limit.111See Falconer [8] for further reading about fractal geometry. Thus, any bounded-degree tree is a subtree of this infinite tree; hence, this drawing algorithm can produce a drawing of any bounded-degree tree. Filling in the details of this construction requires setting the values of two parameters: , the ratio between outgoing and incoming edge lengths; and , the ratio between the radius of a ply disk for a vertex and the length of its longest incident edge. We provide constraints for the following three cases, which taken together ensure that there are no overlaps, so that the ply number of our drawings is 1. We then maximize such that all of these constraints are satisfied.

  1. Ply disks for adjacent vertices must not overlap.

  2. Ply disks for vertices in separate subtrees must not overlap.

  3. A ply disk for a vertex must never overlap a ply disk for one of its descendants.

It is easily verified that these three conditions are necessary and sufficient for a tree to have a 1-ply drawing.

2.1 Condition 1: Separate adjacent vertices

Except for the root vertex, which has no incoming edge, we proportion the lengths of the edges for each vertex as shown in Figure 2.

Figure 2: Our edges decrease by a factor of at each level, and the ply disks have radius times the length of the incoming edge.

That is, taking the length of the reference edge as 1 (illustrated in Figure 2 going from parent to child in a left-to-right orientation), then, based on our definition of the -ply number, the radius of the larger circle is , the radius of the smaller circle is , and their distance is 1. Thus, we have our first condition relating and .

2.2 Condition 2: Separate subtrees with the same root

We require that the ply disks for any subtree all be contained within a wedge of angle around its parent vertex, where is the degree. Since our wedges for each subtree are disjoint, this ensures that the ply disks for two adjacent subtrees cannot overlap.

As illustrated in Figure 3, the distance from a child vertex to the boundary of its containing wedge is . Note also that the lengths of edges along a path in this subtree form a geometric sequence with ratio . So the maximum distance from a child vertex to any vertex in its subtree is .

Figure 3: Our constraint on a wedge containing a subtree of a central vertex.

Therefore, to confine each subtree within its wedge, we must set

Solving for , we get

(1)

2.3 Condition 3: Separate each vertex from its descendants

Our last condition is that the ply disk for a vertex cannot overlap any of its descendants. The closest descendants will be those in the wedges on either side of the edge between their parent and grandparent, which are at an angle of from their parent, as in Figure 4.

Figure 4: Our layout leaves a gap of angle for the edge from the parent vertex. The descendants on either side must not overlap their ancestors.

Once again we normalize , as having length 1. We then perform a rigid transformation that takes the grandparent, , to the origin so that the edge is along the -axis, ’s closest grandchild, which we call , is located at the point . We require that the distance from to its descendants be no greater than the distance from to the boundary of the ply disk for . Recall that our wedge angle . We apply the following constraint:

After simplifying and solving for , our condition is

Let us now compare our three conditions. We see that equation 2 gives us an upper bound for , while equations 1 and 3 give us upper bounds for that both increase as gets larger. So to maximize , we let be equal to its upper bound. This gives us the following theorem.

Theorem 1.

Let be a tree with maximum degree , and let

has a 1-ply drawing if

Corollary 1.

A tree with maximum degree has a 1 ply drawing when .

Proof.

First, recall that we defined:

Now we will consider the limiting value of .

Therefore, . So as , .

Secondly, recall that in our theorem we showed:

Suppose that we use the first condition, . Then . So .

Then suppose that we use the second condition:

Again, , so

Note, however, that some of our conditions are not tight. For condition 2, we assumed that the branches of our subtrees would approach the sides of their wedge directly. But when the degree of our tree is 4, the angle between two subtrees is . Therefore, every edge in our tree is either horizontal or vertical, so we can measure the distance to the boundary of the wedge using Manhattan distance instead of Euclidean distance. (See Figure 5.)

So for a tree with degree 4, we replace condition 2 with the following equation:

This implies , and our other conditions imply . In this case, our bound is tight. (See Figure 6.)

Figure 5: An improved bound for Condition 2. The Manhattan distance is sufficient to confine subtrees within a wedge when all edges are either horizontal or vertical.
Figure 6: A 1-ply drawing of a tree with maximum degree four, for which , .

3 Polynomial area, logarithmic ply number

In this section, we prove the following theorem.

Theorem 2.

For , a tree with maximum degree can be drawn with ply number in area .

Note that for a bounded-degree tree, is a constant, so our area is polynomial in . We first give a simple fractal layering algorithm that proves our theorem for balanced trees. Then we extend it to all trees by using a heavy path decomposition. A similar approach was used by Angelini et. al. [1] for drawing trees up to maximum degree six, but we add our layering technique to make their algorithm work for all trees.

3.1 Radially layered drawings

We begin with a simple algorithm for drawing trees by layering their children. For each vertex, we choose a sequence of distances for the layers, such that vertices in adjacent layers have disjoint ply disks.

Lemma 1.

Suppose that is the root of a star graph. Let be children at distances , respectively. If , then the ply disks for and are disjoint.

Proof.

The distance to is , so since , its ply disk will have radius , and will be contained within an open disk of radius centered at . The distance to is , so its ply disk will have radius . Its closest approach to will be at distance . Thus, the ply disks for and are disjoint. (See Figure 7.) ∎

Figure 7: If each layer in a tree drawing is at least three times as far as the previous layer, the ply disks for the layers will not overlap. In this figure, and , so our condition holds.

Next, note that we can put up to six vertices in each layer without overlaps. So for a tree with degree , we need layers. We pick any desired size for the initial layer around our root, then draw the subtrees for each child vertex recursively within their own ply disks. Therefore, the size of the smallest layer must shrink by a factor of each time we add a level to our tree.

Since our tree is balanced, its total height is . Thus the ratio of the longest to the smallest edge is . The area will then also be , for a larger constant.

This completes our proof for balanced trees. Figure 8 provides an example drawing of such a tree with degree 18 using three layers.

Figure 8: A tree with degree 18, where the children of each vertex are drawn in three layers.

3.2 Heavy path decomposition

When our trees are not balanced, we will use the heavy path decomposition [20] to still produce drawings with logarithmic ply number. This decomposition partitions the vertices in our tree into paths that each end at a leaf. To choose the first path, we begin at the root. Then from its child subtrees, we choose the largest one and add its root to our path. We continue downward until we reach a leaf.

We next remove the vertices on this path from our tree, creating a new set of subtrees, and repeat the same process for each subtree. That is, the root vertex for each of these subtrees will become the starting point for a new path constructed by the same process. We recurse until every vertex in our tree is assigned to some path. The subtrees that are rooted at a child of a vertex and whose root is not on the same path as are are said to be anchored at . The path containing the root of each of those subtrees is also said to be anchored at .

The set of paths constructed by this process now itself forms a new tree (see Figure 9), in which the path is a parent of if one of the vertices in is an anchor for . We will show that the ply number of our drawings is proportional to the height of this decomposition tree, which is known to be .

Now we describe how to draw each path in the decomposition tree. First, we define a 2-drawing of a path as a straight-line drawing of along a single segment that satisfies the following properties.

  • All of the vertices appear in the line segment in the same order as they appear in .

  • For each we have .

Lemma 2.

A 2-drawing of a path has ply number at most 2.

Proof.

See Lemma 5 in Angelini et al. [1]. ∎

Figure 9: A tree and its heavy path decomposition.
Figure 10: Labels for different sizes in a heavy path decomposition tree.

Now suppose that we have a path in our heavy path decomposition, and let be anchored at vertex , so that is the parent of . Let be the total size of the subtrees anchored at , and let be the total size of the subtrees anchored at (Figure 10). Lastly, we denote the length of the edge as .

Intuitively, we want to draw each path so that more space is available for vertices that have larger subtrees. At the same time, we want to ensure that the lengths of the two edges for a vertex are within a factor of two, so that our path is a 2-drawing. This can be achieved using the following algorithm DrawPath.

To draw the path , we first set and , for each . Next we visit the edges of our path in decreasing order of length. When an edge is visited, we make sure that both of its neighboring edges are at least half as long. That is, we set:

Lemma 3.

The algorithm DrawPath constructs a 2-drawing of such that , , and for each , and .

Proof.

See Lemma 6 in Angelini et al. [1]. ∎

Figure 11: Three vertices along a path in our decomposition, along with their drawing disks (not the ply disks). For the center vertex , we show three paths in different layers around it, which would be drawn recursively.

We now perform a bottom-up construction of our tree, drawing each path using the DrawPath algorithm. Once all of the paths anchored at vertices in have been drawn, we construct a drawing of with each path in a separate layer (Figure 11). This translation may increase the ply radius of the first vertex in each of these paths, so the ply number of the drawing for each path may increase from 2 to 3.

In the appendix, we prove the following properties of this drawing.

lemmadrawinglemma For each vertex we can associate a drawing disk (which is distinct from the ply disk for ) that satisfies the following properties.

  1. If are two distinct vertices on the same path, then their disks are disjoint.

  2. The ply disks for the subtrees anchored at are all contained within , and are within disjoint layers.

  3. Each path is scaled by a factor of larger than the paths that are anchored at its vertices.

Together, these properties imply that the ply disks for a path can only overlap with ply disks for their ancestor paths in the heavy path decomposition tree. Therefore, since each path is drawn with ply number at most 3, the total ply number is at most , where is the height of the heavy path decomposition tree. Since , the ply number is .

Lastly, if is a constant, then the total scaling for our largest disk is , which simplifies to . This completes our proof of Theorem 2.

4 Lower bound for 2-trees

Since all trees can be drawn with ply number, it is natural to consider larger planar graph classes. We show that a 2-tree can require at least ply, for any fixed .

First, let us informally describe a 2-tree. A tree can be constructed by beginning with a root vertex, and adding vertices one a time, attaching each new vertex to a single parent. A 2-tree can be constructed by beginning with two root vertices connected by an edge. Then each time we add a new vertex, we attach it to two different parents.

We know that a star can be drawn with ply number 2 when the distance to successive vertices increases exponentially  [1]. A tree can be drawn with ply number when the distances from parents to their children decrease exponentially as we move down the tree. Intuitively, combining these two graphs produces a graph that requires large ply, since it is impossible to satisfy both conditions simultaneously.

Accordingly, we begin with disjoint complete binary trees of height , which we label , , where and will be determined later. Then we add one vertex connected to every vertex in each tree. Note this graph can be constructed as a subgraph of a 2-tree.

Now we have two possible types of drawings for our graph. In one case, every tree has some vertex whose ply disk contains . Therefore, the ply number of our graph is at least , since there are at least ply disks that all contain . In the second case, there is some tree for which none of the ply disks for its vertices contain . To analyze this case, we make use of the following lemma.

Lemma 4.

If two adjacent vertices and satisfy , then the ply disk for contains .

Proof.

Suppose that , so that . By the triangle inequality, . Then the ply radius for is at least . Therefore, the ply disk for contains . (See Figure 12.) ∎

Figure 12: If two adjacent vertices and are both attached to , and is much further from , then the ply disk for will contain .

Assume without loss of generality that the distance from to the root of is 1, and let . We can then show by induction that if no ply disk in contains , then the nodes at the th level of our tree are at distance at most from , and at least .

Now partition our drawing into annulae , where the inner radius of is , and the outer radius is , for . Next choose to be the index of the annulus containing the maximum number of vertices. We have more than vertices, and only annulae to distribute our vertices, so must contain at least vertices. Since each of these vertices is at a distance of at least , each has a ply radius of at least .

A vertex at a distance of from will have a ply radius of at least , so the outer edge of its ply disk will be at a distance of . Therefore, let be the disk centered at with a radius of , so that all of the ply disks for vertices in are contained in . Now we compute the ratio of the areas of the ply disks in to its own area, which is a lower bound for the ply number. Note that contains at least ply disks that each have a radius of at least . Therefore, this ratio is at least:

Now let , and let . Note that the total number of vertices in each tree is . The total number of vertices overall is then .

If every tree has a vertex whose ply disk contains , then the ply number is at least . Otherwise, if some tree does not have such a vertex, then that tree’s ply number is . This gives us the following theorem.

Theorem 3.

There is a 2-tree with vertices for which any drawing has ply number , for any fixed .

5 Conclusion

We have shown that all trees have 1-ply drawings when , or logarithmic ply number when , and that 2-trees may require ply for any .

There are many open questions left to resolve, but we are especially interested in closing the gap between constant and logarithmic ply for trees with between three and nine children per node. We would also like to consider intermediate planar graph classes between trees and 2-trees, such as outerplanar graphs, and determine whether they can be drawn with ply.

Acknowledgements

This research was supported by DARPA agreement no. AFRL FA8750-15-2-0092 and NSF grants 1526631 and 1815073. The views expressed are those of the authors and do not reflect the official policy or position of the Department of Defense or the U.S. Government.

References

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Appendix

Here we include a proof of Lemma 11.

*

Proof.

We prove each part of our lemma as follows.

  1. Suppose that our heavy path decomposition tree has a total height of , and the path is at height . Then we use the DrawPath algorithm to construct a drawing of . We set the drawing disk for a vertex in to have radius , that is, the size of the subtrees anchored at . Since the length of the edge is at least (by Lemma 3), the drawing disks for any two adjacent vertices in our path will not overlap.

  2. Next we scale the drawing of by . Note that each path anchored at a vertex in is scaled by , so the difference in the scaling factor is . We show that at least paths can be anchored in different layers around each vertex in .

    From Lemma 3, we know that each path anchored at has an unscaled length of at most , where is the total size of the subtrees anchored at . We also know by Lemma 1 that the ply disks for vertices in two different paths will not overlap if their distance from differs by at least a factor of three.

    So we will draw the th path anchored at is drawn between and , where satisfies the following recurrence:

    Solving the recurrence, we find that . Since we have at most layers, the largest layer will have an outer radius less than . Since the unscaled drawing disk for had a radius of , a relative scaling factor of is sufficient to fit the paths that are anchored at it.

  3. Since our heavy path decomposition has height , the largest path will be scaled by a factor of from its original length of . So the diameter of our drawing is , which simplifies to . The total area is then also , for a larger constant.