Minimizing the continuous diameter when augmenting a geometric tree with a shortcut

Abstract

We augment a tree T with a shortcut pq to minimize the largest distance between any two points along the resulting augmented tree $T+pq$. We study this problem in a continuous and geometric setting where T is a geometric tree in the Euclidean plane, a shortcut is a line segment connecting any two points along the edges of T, and we consider all points on $T+pq$ (i.e., vertices and points along edges) when determining the largest distance along $T+pq$. The continuous diameter is the largest distance between any two points along edges. We establish that a single shortcut is sufficient to reduce the continuous diameter of a geometric tree T if and only if the intersection of all diametral paths of T is neither a line segment nor a point. We determine an optimal shortcut for a geometric tree with n straight-line edges in $O(n\log n)$ time.

Introduction

A network is a connected, undirected graph with positive edge weights. A curve is rectifiable if it has a finite length. A geometric network is a network that is embedded in the Euclidean plane whose edges are rectifiable curves weighted with their length. We describe our algorithmic results for straight-line edges, even though they extend to more general edges. We define points along the edges of a network as follows. Let $uv\in E$ be an edge in G with weight $w_{uv}$ that connects the vertices $u,v\in V$. For every value $\lambda \in [0,1]$, we define a point p on edge uv that subdivides uv into two sub-edges up and pv of weights $w_{up}=\lambda w_{uv}$ and $w_{pv}=(1-\lambda )w_{uv}$, respectively. We write $p\in uv$ to indicate that p is a point along the edge uv, for some $\lambda \in [0,1]$, and we write $p\in G$ to denote that p is a point along some edge of the network G. There is no ambiguity if two edges cross: there are two points along the network that correspond to the crossing in the plane, since points along edges are specified by their relative position to the endpoints of their containing edge, expressed by λ, and not by coordinates in the plane.

The network distance between any two points p and q on a geometric network G is the length of a shortest weighted path from p to q in G and it is denoted by $d_G(p,q)$. The continuous diameter of G is the largest network distance between any two points on G, and it is denoted by $\mathrm{diam}(G)$, i.e., $\mathrm{diam}(G)={\max }_{p,q\in G}{d}_G(p,q)$. In contrast, for a network with vertex set V, the discrete diameter is the largest distance between any two vertices, i.e., ${\max }_{u,v\in V}{d}_G(u,v)$. A pair $p,q\in G$ is diametral when their distance is the continuous diameter, i.e., $\mathrm{diam}(G)=d_G(p,q)$. A point $p\in G$ is a diametral partner in G if there exists some point $q\in G$ such that $p,q$ is a diametral pair of G. A diametral path in G is a shortest weighted path in G that connects a diametral pair of G.

We denote the Euclidean distance between two points p and q by $|pq|$. A line segment pq with endpoints $p,q\in G$ is a shortcut for G. We augment a geometric network G with a shortcut pq: If they do not exist already, we introduce vertices at p and at q, thereby subdividing the edges containing p and q. We add the line segment pq as an edge of length $|pq|$ to G without introducing vertices at crossings between pq and other edges. We denote the resulting network by $G+pq$. In this work, we move a shortcut along a network and some of the intermediate shortcuts may coincide with edges or parts of edges. To simplify this discussion, we allow shortcuts pq for a network G with $d_G(p,q)=|pq|$. Instead a local constraint on the definition of a shortcut, we consider a shortcut to be useful for G when its addition to G reduces the continuous diameter, i.e., $\mathrm{diam}(G+pq)<\mathrm{diam}(G)$.

Our goal is to locate a shortcut pq for a geometric tree T that minimizes the continuous diameter of the augmented tree $T+pq$, as illustrated in Fig. 1. This means we seek two points $p,q\in T$ with $\mathrm{diam}(T+pq)={\min }_{r,s\in T}\mathrm{diam}(T+rs)$. We call a shortcut that minimizes the continuous diameter an optimal shortcut.

Let $C(p,q)$ be the simple cycle in $T+pq$. The backbone of T, denoted by $\mathcal{B}$, is the intersection of all diametral paths of T. The absolute center of T is the unique point $c\in T$ that minimizes the largest network distance from c, i.e., ${\max }_{q\in T}{d}_T(c,q)={\min }_{p\in T}{\max }_{q\in T}{d}_T(p,q)$. Note that $c\in \mathcal{B}$. Determining the absolute center—and, thus, the backbone—of a geometric tree with n vertices takes $O(n)$ time [1], [16].

Our Contributions. We obtain the following structural and algorithmic results:

• A geometric tree T admits a useful shortcut if and only if its backbone $\mathcal{B}$ is neither a straight-line segment nor a point (i.e., a degenerate line segment).

• Every geometric tree T has an optimal shortcut pq with both endpoints on the backbone, i.e., $p,q\in \mathcal{B}$, and the absolute center c on the path from p to q in T.

• We develop an algorithm that produces an optimal shortcut for a geometric tree T with n vertices whose edges are straight-line segments in $O(n\log n)$ time.

We summarize related work on minimum-diameter network augmentation.

In the abstract and discrete setting, the goal is to minimize the discrete diameter of an abstract graph $G=(V,E)$ with positive weights for the edges of G and the edges of its complement graph $\overline{G}=(V,\left(\begin{array}{c}V\\ 2\end{array}\right)-E)$ by inserting edges of $\overline{G}$ as shortcuts to G. If the edges of G and $\overline{G}$ have unit weight, then it is NP-hard to decide whether the diameter can be reduced below some target value $D\ge 2$ by adding at most k shortcuts [2], [3], [4]. This problem remains NP-hard when the number of shortcuts is variable, even for trees [2]. Its weighted version falls into the parameterized complexity class W[2]-hard [5], [6]. On the other hand, Oh and Ahn [7] determine an optimal vertex-to-vertex shortcut that minimizes the discrete diameter of an abstract n-vertex tree with positive edge weights in $O(n^2{\log }^{3}n)$ time. Minimum-diameter augmentation has also been studied as a bicriteria optimization in which both the diameter and the number (or cost) of the additional edges are minimized. For instance, Frati et al. [5] summarize the literature regarding the research on bicriteria optimization in minimum-diameter network augmentation.

Große et al. [8] introduce the geometric and discrete setting in which the problem is to minimize the discrete diameter of a geometric network by connecting vertices with line segments. Große et al. [8] determine an optimal shortcut for a polygonal path with n vertices in $O(n{\log }^{3}n)$ time using a parametric search technique. Recently, Wang [9] improved this algorithm to $O(n\log n)$ time. Apart from the discrete diameter, the stretch factor, i.e., the largest ratio of the network distance between any two vertices and their Euclidean distance, has also been considered as a target function [10], [11].

In the geometric and continuous setting [12], the task is to minimize the continuous diameter of a geometric network by inserting line segments that may connect any two points along the edges. For a polygonal path of length n, one can determine an optimal shortcut in $O(n)$ time. For a cycle, one shortcut can never decrease the continuous diameter while two always suffice. For convex cycles with n vertices, one can determine an optimal pair of shortcuts in $O(n)$ time. In the model studied in this work, a crossing of a shortcut with an edge or another shortcut is not a vertex: a path may only enter edges at their endpoints. In the planar model [13], [14], every crossing is a vertex of the resulting network, which leads to a different graph structure and, thus, continuous diameter. In the planar model, Yang [14] characterizes optimal shortcuts for a polygonal path. Cáceres et al. [13] determine in polynomial time whether the continuous diameter of a plane geometric network can be reduced with a single shortcut.

This work is structured as follows. In Section 2, we establish our structural results: We observe that—unlike in the discrete version of this problem—adding a shortcut to a tree might increase the continuous diameter. We characterize the trees that have a useful shortcut, i.e., a shortcut that reduces the continuous diameter, as precisely those trees where the intersection of all diametral paths is neither a straight-line segment nor a point (i.e., a degenerate line segment). The intersection of all diametral paths, called the backbone $\mathcal{B}$ of T plays a key role when locating an optimal shortcut. In the discrete setting, Große et al. [15] show that there exists an optimal shortcut for a tree with both endpoints along the backbone. We prove that this result carries over to the continuous setting and strengthen it: we show that every geometric tree has an optimal shortcut pq with both endpoints along the backbone such that the absolute center c of T lies on the path from p to q in T. This yields a restriction of the search space that allows us to find an optimal shortcut for a geometric tree with n straight-line edges in $O(n\log n)$ time. We develop this algorithm in three steps: In Section 4, we examine how the diametral paths in the augmented tree rule out certain directions for the search. In Section 5, we develop a set of rules that inform us how to continuously slide a shortcut along the backbone until we eventually reach an optimal shortcut. In Section 6, we simulate this conceptual continuous algorithm with a discretization that achieves the desired running time.

The structural results hold for a geometric tree whose edges are rectifiable curves, i.e., curves that have a well defined length. We describe the algorithmic result for trees with straight-line edges; the techniques carry over to more general types of edges, e.g., algebraic curves of constant degree. Our model of computation is the real RAM.