On the Stretch Factor of Convex Delaunay Graphs

Let $S$ be a finite set of points in the plane and let $G$ be a graph with vertex set $S$, in which each edge $(p,q)$ has a weight equal to the Euclidean distance $|pq|$ between $p$ and $q$. For a real number $t\geq 1$, we say that $G$ is a $t$-spanner for $S$, if for any two points $p$ and $q$ of $S$, there exists a path in $G$ between $p$ and $q$ whose Euclidean length is at most $t|pq|$. The smallest such $t$ is called the stretch factor of $G$. The problem of constructing spanners has received much attention; see Narasimhan and Smid [12] for an extensive overview.

Spanners were introduced in computational geometry by Chew [3, 4], who proved the following two results. first, the $L_{1}$-Delaunay graph, i.e., the dual of the Voronoi diagram for the Manhattan metric, is a $\sqrt{10}$-spanner. Second, the Delaunay graph based on the convex distance function defined by an equilateral triangle, is a $2$-spanner. We remark that in both these results, the stretch factor is measured in the Euclidean metric. Chew also conjectured that the Delaunay graph based on the Euclidean metric, is a $t$-spanner, for some constant $t$. (If not all points of $S$ are on a line, and if no four points of $S$ are cocircular, then the Delaunay graph is the well-known Delaunay triangulation.) This conjecture was proved by Dobkin et al. [8], who showed that $t\leq\pi(1+\sqrt{5})/2$. The analysis was improved by Keil and Gutwin [9], who showed that $t\leq\frac{4\pi\sqrt{3}}{9}$.

In this paper, we unify these results by showing that the Delaunay graph based on any convex distance function has bounded stretch factor.

Throughout this paper, we fix a compact and convex set $C$ in the plane. We assume that the origin is in the interior of $C$. A homothet of $C$ is obtained by scaling $C$ with respect to the origin, followed by a translation. Thus, a homothet of $C$ can be written as

for some point $x$ in the plane and some real number $\lambda\geq 0$. We call $x$ the center of the homothet $x+\lambda C$.

For two points $x$ and $y$ in the plane, we define

If $x\neq y$, then this definition is equivalent to the following: Consider the translate $x+C$ and the ray emanating from $x$ that contains $y$. Let $y^{\prime}$ be the (unique) intersection between this ray and the boundary of $x+C$. Then

The function $d_{C}$ is called the convex distance function associated with $C$. Clearly, we have $d_{C}(x,x)=0$ and $d_{C}(x,y)>0$ for all points $x$ and $y$ with $x\neq y$. Chew and Drysdale [5] showed that the triangle inequality $d_{C}(x,z)\leq d_{C}(x,y)+d_{C}(y,z)$ holds. In general, the function $d_{C}$ is not symmetric, i.e., $d_{C}(x,y)$ is not necessarily equal to $d_{C}(y,x)$. If $C$ is symmetric with respect to the origin, however, then $d_{C}$ is symmetric.

Let $S$ be a finite set of points in the plane. For each point $p$ in $S$, we define

If $C$ is not strictly convex, then the set $V^{\prime}_{C}(p)$ may consist of a closed region of positive area with an infinite ray attached to it. For example, in figure 1, the set $V^{\prime}_{C}(a)$ consists of the set of all points that are on or to the left of the leftmost zig-zag line, together with the infinite horizontal ray that is at the same height as the point $a$. Also, the intersection of two regions $V^{\prime}_{C}(p)$ and $V^{\prime}_{C}(q)$, where $p$ and $q$ are distinct points of $S$, may have a positive area. As a result, the collection $V^{\prime}_{C}(p)$, where $p$ ranges over all points of $S$, does not necessarily give a subdivision of the plane in which the interior of each cell is associated with a unique point of $S$. In order to obtain such a subdivision, we follow the approach of Klein and Wood [10] (see also Ma [11]): first, infinite rays attached to regions of positive area are not considered to be part of the region. Second, a point $x$ in $\mathbb{R}^{2}$ that is in the interior of more than one region $V^{\prime}_{C}(p)$ is assigned to the region of the lexicographically smallest point $p$ in $S$ for which $x\in V^{\prime}_{C}(p)$.

To formally define Voronoi cells, let $\prec$ denote the lexicographical ordering on the set of all points in the plane. Let $p_{1}\prec p_{2}\prec\ldots\prec p_{n}$ be the points of $S$, sorted according to this order. Then the Voronoi cells $V_{C}(p_{i})$ of the points of $S$ are defined as

and, for $1<i\leq n$,

where $\mathord{\it cl}(X)$ and $\mathord{\it int}(X)$ denote the closure and the interior of the set $X\subseteq\mathbb{R}^{2}$, respectively.

Thus, in figure 1, the Voronoi cell $V_{C}(a)$ consists only of the set of all points that are on or to the left of the leftmost zig-zag line; the infinite horizontal ray that is at the same height as the point $a$ is not part of this cell.

The Voronoi diagram $\mathord{\it VD}_{C}(S)$ of $S$ with respect to $C$ is defined to be the collection of Voronoi cells $V_{C}(p)$, where $p$ ranges over all points of $S$. An example is given in figure 1.

As for the Euclidean case, the Voronoi diagram $\mathord{\it VD}_{C}(S)$ induces Voronoi cells, Voronoi edges, and Voronoi vertices. Each point in the plane is either in the interior of a unique Voronoi cell, in the relative interior of a unique Voronoi edge, or a unique Voronoi vertex. Each Voronoi edge $e$ belongs only to the two Voronoi cells that contain $e$ on their boundaries. Observe that Voronoi cells are closed.

The Delaunay graph is defined to be the dual of the Voronoi diagram:

For example, the Delaunay graph $\mathord{\it DG}_{C}(S)$ for the point set in figure 1 consists of the five edges $(a,b)$, $(a,d)$, $(b,c)$, $(b,d)$, and $(d,e)$.

We consider the Delaunay graph $\mathord{\it DG}_{C}(S)$ to be a geometric graph, which means that each edge $(p,q)$ is embedded as the closed line segment with endpoints $p$ and $q$.

Before we can state the main result of this paper, we introduce two parameters whose values depend on the shape of the set $C$. Let $x$ and $y$ be two distinct points on the boundary $\partial C$ of $C$. These points partition $\partial C$ into two chains. For each of these chains, there is an isosceles triangle with base $xy$ and whose third vertex is on the chain. Denote the base angles of these two triangles by $\alpha_{xy}$ and $\alpha^{\prime}_{xy}$; see figure 2 (left). We define

Consider again two distinct points $x$ and $y$ on $\partial C$, but now assume that $x$, $y$, and the origin are collinear. As before, $x$ and $y$ partition $\partial C$ into two chains. Let $\ell_{xy}$ and $\ell^{\prime}_{xy}$ denote the lengths of these chains; see figure 2 (right). We define

Clearly, the convex distance function $d_{C}$ and, therefore, the Voronoi diagram $\mathord{\it VD}_{C}(S)$, depends on the location of the origin in the interior of $C$. Surprisingly, the Delaunay graph $\mathord{\it DG}_{C}(S)$ does not depend on this location; see Ma [11, Section 2.1.6]. We define

In this paper, we will prove the following result:

We emphasize that we do not make any “general position” assumption; our proof of Theorem 1 is valid for any finite set of points in the plane.

Throughout the rest of this paper, we assume that the origin is chosen in the interior of $C$ such that $\kappa_{C}=\kappa_{C,0}$.

The rest of this paper is organized as follows. In Section 2, we prove some basic properties of the Delaunay graph which are needed in the proof of Theorem 1. In particular, we give a formal proof of the fact that this graph is plane. Even though this fact seems to be well known, we have not been able to find a proof in the literature. Section 3 contains a proof of Theorem 1. This proof is obtained by showing that the Delaunay graph satisfies the “diamond property” and a variant of the “good polygon property” of Das and Joseph [6]. The proof of the latter property is obtained by generalizing the analysis of Dobkin et al. [8] for the lengths of so-called one-sided paths.

Recall that in the Euclidean Delaunay graph, if two points $p$ and $q$ of $S$ are connected by an edge, then there exists a disk having $p$ and $q$ on its boundary that does not contain any point of $S$ in its interior. The next lemma generalizes this result to the Delaunay graph $\mathord{\it DG}_{C}(S)$.

To prove the second claim, let $x$ be an arbitrary point in $V_{C}(p)\cap V_{C}(q)$. Then $d_{C}(x,p)=d_{C}(x,q)$ and $d_{C}(x,r)\geq d_{C}(x,p)$ for all $r\in S$. Thus, if we define $\lambda:=d_{C}(x,p)$, then $\lambda>0$, both $p$ and $q$ are on the boundary of the homothet $x+\lambda C$, and no point of $S$ is in the interior of this homothet.   

As can be seen in figure 1, Voronoi cells are, in general, not convex. They are, however, star-shaped:

It is well known that the Euclidean Delaunay graph is a plane graph; see, for example, de Berg et al. [7, page 189]. The following lemma states that this is true for the Delaunay graph $\mathord{\it DG}_{C}(S)$ as well.

It remains to show that $\mathord{\it DG}_{C}(S)$ does not contain two edges $(p,q)$ and $(r,s)$ that cross properly. The proof is by contradiction. Thus, let $p$, $q$, $r$, and $s$ be four pairwise distinct points of $S$, no three of which are collinear, and assume that the line segments $(p,q)$ and $(r,s)$ are edges of $\mathord{\it DG}_{C}(S)$ that have exactly one point in common.

Since $(p,q)$ is an edge of $\mathord{\it DG}_{C}(S)$, there exists a point $x$ in the relative interior of $V_{C}(p)\cap V_{C}(q)$. Thus, by the second claim in Lemma 1, there exists a real number $\lambda>0$, such that the homothet $x+\lambda C$ contains $p$ and $q$ on its boundary and no point of $S$ is in the interior of this homothet. Observe that $x$ is in the interior of $x+\lambda C$. Let $D$ be a Euclidean disk centered at $x$ that is contained in the interior of $x+\lambda C$ and that is contained in $V_{C}(p)\cup V_{C}(q)$. We define $B$ to be the set of all 2-link polygonal chains $(p,z,q)$, with $z\in D$; see figure 3. Observe that $B$ has a positive area. Since $V_{C}(p)$ and $V_{c}(q)$ are star-shaped (by Lemma 2), we have $B\subseteq V_{C}(p)\cup V_{C}(q)$. Since $x+\lambda C$ is convex, we have $B\subseteq x+\lambda C$; in fact, the convex hull of $B$ is contained in $x+\lambda C$. Thus, neither $r$ nor $s$ is in the interior of the convex hull of $B$. Since $pq$ and $rs$ intersect in a point, the line segment $rs$ crosses the set $B$.

By a symmetric argument, since $(r,s)$ is an edge of $\mathord{\it DG}_{C}(S)$, there exist a point $y$ in the relative interior of $V_{C}(r)\cap V_{C}(s)$ and a real number $\mu>0$, such that $y+\mu C$ contains $r$ and $s$ on its boundary and no point of $S$ is in the interior of this homothet. Let $D^{\prime}$ be a Euclidean disk centered at $y$ that is contained in the interior of $y+\mu C$ and that is contained in $V_{C}(r)\cup V_{C}(s)$. We define $B^{\prime}$ to be the set of all 2-link polygonal chains $(r,z,s)$, with $z\in D^{\prime}$. The set $B^{\prime}$ has a positive area, the line segment $pq$ crosses this set, $B^{\prime}\subseteq V_{C}(r)\cup V_{C}(s)$, and neither $p$ nor $q$ is in the interior of the convex hull of $B^{\prime}$.

It follows that $B$ and $B^{\prime}$ overlap in a region of positive area. Since $B\subseteq V_{C}(p)\cup V_{C}(q)$ and $B^{\prime}\subseteq V_{C}(r)\cup V_{C}(s)$, however, the area of the intersection $B\cap B^{\prime}$ is equal to zero. This is a contradiction. It follows that the edges $(p,q)$ and $(r,s)$ do not cross.   

In this section, we will prove Theorem 1. first, we show that the Delaunay graph $\mathord{\it DG}_{C}(S)$ satisfies the diamond property and a variant of the good polygon property of Das and Joseph [6]. According to the results of Das and Joseph, this immediately implies that the stretch factor of $\mathord{\it DG}_{C}(S)$ is bounded. In fact, we will obtain an upper bound on the stretch factor which is better than the one that is implied by Das and Joseph’s result.