Given a point set with points, the Euclidean Minimum Spanning Tree (EMST) of , i.e. the spanning geometric graph of where the sum of the Euclidean lengths of its edges is the minimum, can be obtained in time . A Euclidean Minimum Spanning Tree of a point set is of maximum degree at most six  and a Euclidean Minimum Spanning Tree of maximum degree at most five can always be found . The problem of drawing a tree of maximum degree at most five as a EMST can be efficiently solved . On the other hand, for a tree of maximum degree six the same problem is NP-hard . Regarding the area requirement of the drawing of a tree of maximum degree five as a EMST, the algorithm of Monma and Suri  produces a drawing in a grid of exponential area. Additionally, there exist trees of maximum degree at most five for which no drawing as EMST lies on a grid of polynomial area .
Given a rooted point set of size with root , the spanning geometric graph of in which each point of is connected with by a monotone path and the sum of the Euclidean lengths of its edges is the minimum is a tree , denoted as the rooted Monotone Minimum Spanning Tree (rooted MMST) of in , and can be obtained in time . In contrast to the case of Euclidean Minimum Spanning Tree, it is not known whether the maximum degree of a rooted MMST is bounded by a constant number, whether a drawing of a rooted tree as a rooted MMST can be efficiently produced and if such a drawing necessarily lies on a grid of exponential area.
The restricted fathers tree problem which was studied by Guttman-Beck and Hassin  is related to the problem of obtaining the rooted MMST of a rooted point set. The goal of the restricted fathers tree problem is to obtain the minimum spanning tree of a weighted rooted graph where each vertex of contains a key, where in the root is connected to all other vertices by paths in which the keys of the traversed vertices form a decreasing sequence. The restricted fathers tree problem is greedily solvable [8, Corollary 2.6].
Much research has been done in drawing a rooted tree under several aesthetical drawing conventions [5, Section 3.1]. The problem of finding a drawing w.r.t. minimizing the area is widely studied . Recently, Chan  improved the area requirements of several types of tree drawings. As far as monotonicity is concerned, drawing a rooted tree such that each child vertex is mapped to a point with coordinate less than or equal to the coordinate of the point to which its parent is mapped with the goal of optimizing the area of the drawing is widely investigated, e.g. see [15, Chapter 3], , . Furthermore, drawing a rooted tree in the plane such that each pair of points is connected by a path that is monotone in some direction, in a grid of small area, is thoroughly investigated [2, 9, 13].
In this article, we show that the maximum degree of a rooted MMST is not bounded by any constant number. We give a linear time recursive algorithm that draws a rooted tree as a rooted MMST and we show that there exist rooted trees that can be drawn as a rooted MMST only in a grid of exponential area (and not in a grid of polynomial area).
points of the plane. The vector fromto is denoted as .
A geometric graph is a pair of (i) a point set which is its vertex set and (ii) a set of line segments connecting points of which is its edge set. A geometric path , , …, is a geometric graph with , , …, as its vertex set and , , …, as its edge set. The geometric path , , …, is monotone if either (i) for each , , …, the coordinate of is greater than or equal to the coordinate of , or (ii) for each , , …, the coordinate of is less than or equal to the coordinate of .
Lee and Preparata  showed that in a geometric graph where all vertices have different coordinates, each vertex is connected to the vertex with the smallest coordinate by some monotone path if and only if each vertex is connected to some vertex where the coordinate of is smaller than the coordinate of .
Fact 1 (Corollary 3 of 111It was recently brought to our attention that Fact 1 can also be obtained as a Corollary of [8, Corollary 2.6].).
Given a geometric tree with root where all vertices have different coordinates and each vertex different from the root has coordinate greater than the coordinate of the root, is the rooted MMST of its vertex set if and only if each vertex different from the root is linked to a single vertex below it, its nearest vertex.
The depth of a rooted tree is the maximum number of edges traversed in a path from the root of to another vertex of .
A straight-line drawing of a graph is a function that associates each to a point of the plane and each to a line segment of the plane connecting the images of and . In this article, we only deal with straight-line drawings, hence we omit the term straight-line from now on.
3 Drawing a Rooted Tree as a Rooted Monotone Minimum Spanning Tree
In this section we show that for each natural number we can find a rooted point set for which the rooted MMST of is of maximum degree . We also give a linear time recursive algorithm that draws a tree with root as a rooted MMST. Finally, we prove that there exist rooted trees that can be drawn as rooted MMSTs only in a grid of exponential area.
The maximum degree of a rooted MMST is not bounded by a constant number.
For each , we consider a rooted point set with points, in which all points different from the root are higher than and in increasing order w.r.t. the coordinate and decreasing order w.r.t. the coordinate. More specifically, the point set consists of the points , , , …, , …, where , , , …, . Then, for each , , , …, , the closest point of that has coordinate less than the coordinate of is the root, since for each , and for the root is the single point of with less coordinate than it. Hence, by Fact 1 the rooted MMST of is the tree where each , , is connected just to . ∎
We now give our algorithm that draws a rooted tree as a rooted MMST. Our algorithm is recursive. It first draws the subtrees , , …, with roots , , …, , that are connected to the root , as rooted MMSTs and then computes appropriate vectors , , …, such that the final drawing is a rooted MMST. In each vertex of we assign the vector , where denotes the parent of in . We also store in each vertex of the width (width) and the height (height) of the bounding box, i.e. the box of the minimum area in which the drawing is included, of the subtree rooted at .
If has depth , and the children of are the vertices , , …, , then we use the same construction that we used in Proposition 1. In more detail, we assign , , …, , …, where , . Furthermore, height and width. For an example see Figure 1. In this way, as shown in Proposition 1, the root is the closest point below for each , , , …, , and hence by Fact 1 our drawing is a rooted MMST.
If the tree has depth at least , let , , …, be the children of and , , …, be the corresponding subtrees. We recursively draw the subtrees , , …, and then we assign the values of the vectors , , …, in analogous way to the case that the tree has depth , but this time the and coordinates of the vectors are larger such that the final drawing is a rooted MMST. More specifically, let , , ,…, then , height, , , …, , since each is drawn above . Regarding the coordinates of the vectors, in order that the closest point below for the vertices in the final drawing is their parent and hence by Fact 1 the final drawing is a rooted MMST, the coordinates of the vectors satisfy the following recursive formula: and , width . The coordinates of the vectors satisfy this recursive formula because (i) has to be greater than or equal to , , ,…, , such that for each vertex in it holds that and hence the closest point to below is , and (ii) has to be greater than or equal to width + , , ,…, , such that for each vertex in and each in it holds that and hence the closest point below to each vertex in remains its parent which lies inside the drawing of . For an example see Figure 2. From the above discussion, we get the following Theorem.
We can draw a rooted tree as a rooted Monotone Minimum Spanning Tree in linear time.
We now show that drawing some rooted trees as rooted MMSTs can not be performed in a grid of polynomial area.
There exist rooted trees which can be drawn as rooted Monotone Minimum Spanning Trees only in a grid of exponential area.
We show that drawing the rooted tree , i.e. the tree consisting of a root center vertex and leaf vertices which are adjacent only to , as a rooted MMST requires a grid of exponential width. Consider, w.l.o.g. that is drawn at the origin of the Cartesian Coordinate System. Then, by the pigeonhole principle at least of the vertices are all drawn in the same quadrant of the Cartesian Coordinate System. Assume w.l.o.g. that this quadrant is the first quadrant, i.e. the quadrant containing the points with positive and coordinates. By Fact 1 it follows that no two vertices can be mapped to points of the first quadrant with the same coordinate. Let the vertices, mapped in the first quadrant, be mapped to the points , , …, , with , , , …, . By Fact 1 and since all the points , , …, belong to the first quadrant, it follows that and hence . We now show that the drawing restricted to the first quadrant has exponential area. We do this as follows: We first show that has more area than another drawing , where is a rooted MMST and that has exponential width. The intuition is to “translate” the points , , , …, , only in the coordinate as low as possible in order to obtain . More specifically, consists of the root mapped at the origin, and where and , and the line segments , . See for example Figure 3(a). Clearly, has less area than . Furthermore, since the circle with center and radius contains in its interior the part of the circle with center and radius that is inside the first quadrant and both and are outside , then (see for example Figure 3(b)). Hence, by Fact 1 it follows that is a rooted MMST. In the , it is easy to show that the width is exponential. Since it follows that , and . ∎
In this article, we showed that the maximum degree of a rooted MMST is not bounded by any constant. We gave a linear time recursive algorithm that draws a rooted tree as a rooted MMST in a grid of exponential area. We finally proved that there exist rooted trees that can be drawn as rooted MMSTs only in a grid of exponential area.
Acknowledgments. The author would like to thank the Professor Alexander Arvanitakis and the Professor Aris Pagourtzis for the thoughtful discussions. This research was financially supported by the Special Account for Research Grants of the National Technical University of Athens.
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