Minimizing the continuous diameter when augmenting a geometric tree with a shortcut☆,☆☆
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 be an edge in G with weight that connects the vertices . For every value , we define a point p on edge uv that subdivides uv into two sub-edges up and pv of weights and , respectively. We write to indicate that p is a point along the edge uv, for some , and we write 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 . The continuous diameter of G is the largest network distance between any two points on G, and it is denoted by , i.e., . In contrast, for a network with vertex set V, the discrete diameter is the largest distance between any two vertices, i.e., . A pair is diametral when their distance is the continuous diameter, i.e., . A point is a diametral partner in G if there exists some point such that 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 . A line segment pq with endpoints 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 to G without introducing vertices at crossings between pq and other edges. We denote the resulting network by . 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 . 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., .
Our goal is to locate a shortcut pq for a geometric tree T that minimizes the continuous diameter of the augmented tree , as illustrated in Fig. 1. This means we seek two points with . We call a shortcut that minimizes the continuous diameter an optimal shortcut.
Let be the simple cycle in . The backbone of T, denoted by , is the intersection of all diametral paths of T. The absolute center of T is the unique point that minimizes the largest network distance from c, i.e., . Note that . Determining the absolute center—and, thus, the backbone—of a geometric tree with n vertices takes time [1], [16].
Our Contributions. We obtain the following structural and algorithmic results:
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A geometric tree T admits a useful shortcut if and only if its backbone is neither a straight-line segment nor a point (i.e., a degenerate line segment).
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Every geometric tree T has an optimal shortcut pq with both endpoints on the backbone, i.e., , and the absolute center c on the path from p to q in T.
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We develop an algorithm that produces an optimal shortcut for a geometric tree T with n vertices whose edges are straight-line segments in 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 with positive weights for the edges of G and the edges of its complement graph by inserting edges of as shortcuts to G. If the edges of G and have unit weight, then it is NP-hard to decide whether the diameter can be reduced below some target value 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 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 time using a parametric search technique. Recently, Wang [9] improved this algorithm to 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 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 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 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 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.
Section snippets
Usefulness
We say a shortcut pq is useful for T when , we say pq is indifferent for T when , and we say pq is useless for T when . In the discrete setting, every shortcut is useful or indifferent, as the discrete diameter only considers vertices of T. In the continuous setting, a shortcut may be useless for T, since the points on the shortcut pq matter as well, as in Fig. 2.
For two points , a shortcut pq is useful for the unordered pair when
Optimal shortcuts
Consider a geometric tree T whose backbone is a path from a to b. This path contains the absolute center c. We prove that there is an optimal shortcut pq for T such that p lies on the path from a to c and q lies on the path from c to b. This holds when consists only of c, since then T has no useful shortcuts and the degenerate shortcut cc is optimal. For the other cases, we establish our claim by proving the following statements.
- 1.
If pq is a useful shortcut for a geometric tree T, then every
Preparations for the algorithm
Our search for an optimal shortcut pq for T proceeds as follows. Initially, we place the endpoints of the shortcut, p and q, at the absolute center c of T. Conceptually, we move p and q continuously along the backbone balancing the diametral paths in . Throughout this continuous movement p remains along the path from a to c and q remains on the path from c to b, where a and b are the endpoints of . The diametral pairs in guide our search: each diametral pair in rules out some
Continuous algorithm
Inspired by the plane-sweep paradigm, we—conceptually—move the shortcut continuously while changing its speed and direction at certain events where the pair state or path state changes. To implement this approach, we discretize this movement such that the shortcut jumps from one event to the next.
Discretization
We view the trajectory of the shortcut throughout the continuous algorithm as a function of time. This function changes its shape at certain events, e.g., when one endpoint of the shortcut passes through a vertex or when the path state changes. We trace this trajectory from one event to the next in order to obtain a discrete simulation of the continuous algorithm. We describe this discretization for geometric trees whose edges are straight-line segments, where the trajectory is sufficiently
Conclusion
We discussed the problem of minimizing the continuous diameter when augmenting a geometric tree with a single shortcut. A natural extension of this problem would be to minimize the continuous diameter when augmenting a geometric network with multiple shortcuts. For instance, given a number , we would like to characterize the trees where at least k shortcuts are required to reduce the continuous diameter.
Even though our construction was specified for geometric trees whose edges are
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. We will submit the form, as requested.
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This work was partially supported by NSERC and FQRNT.
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A preliminary version of this work was presented at the 15th International Symposium on Algorithms and Data Structures (WADS 2017), July 31 to August 2, 2017, St. John's, NL, Canada.