We seek to augment a geometric network in the Euclidean plane with shortcuts to minimize its continuous diameter, i.e., the largest network distance between any two points on the augmented network. Unlike in the discrete setting where a shortcut connects two vertices and the diameter is measured between vertices, we take all points along the edges of the network into account when placing a shortcut and when measuring distances in the augmented network. We study this network augmentation problem for paths and cycles. For paths, we determine an optimal shortcut in linear time. For cycles, we show that a single shortcut never decreases the continuous diameter and that two shortcuts always suffice to reduce the continuous diameter. Furthermore, we characterize optimal pairs of shortcuts for convex and non-convex cycles. Finally, we develop a linear time algorithm that produces an optimal pair of shortcuts for convex cycles. Apart from the algorithms, our results extend to rectifiable curves. Our work reveals some of the underlying challenges that must be overcome when addressing the discrete version of this network augmentation problem, where we minimize the discrete diameter of a network with shortcuts that connect only vertices.  ACM Subject Classification The minimum-diameter network augmentation problem is concerned with minimizing the largest distance between two vertices of an edge-weighted graph by introducing new edges as shortcuts. We study the this problem in a continuous and geometric setting where the network is a geometric graph embedded into the Euclidean plane, the weight of a shortcut is the Euclidean distance of its endpoints, and shortcuts can be introduced between any two points along the network that may be vertices or points along edges. As a sample application, consider a network of highways where we measure the distance between two locations in terms of the travel time. An urban engineer might want to improve the worst-case travel time along a highway or along a ring road by introducing shortcuts. Our work advises where these shortcuts should be build. For example, we show where to * This work was partially supported by NSERC.  A network is an undirected graph that is embedded into the Euclidean plane and whose edges are weighted with their Euclidean length. We say a point p lies on a network G and write p ∈ G when there is an edge e of G such that p is a point along the embedding of e. A point p on an edge e of length l subdivides e into two sub-edges lengths (1 − λ) · l and λ · l for some value λ ∈ [0, 1]. By specifying the points on G in terms of their relative position (expressed by λ) along their containing edge, we avoid any ambiguity in case of crossings. The network distance between two points p and q on a network G is the length of a weighted shortest path from p to q in G. We denote the network distance between p and q by d G (p, q) and we omit the subscript when the network is understood. The largest network distance between any two points on G is the The term continuous distinguishes this notion from the discrete diameter that measures the largest network distance between any two vertices. We denote the Euclidean distance between p and q by |pq|. A line segment pq, with We augment a network G with a shortcut pq as follows. We introduce new vertices at p and at q in G, subdividing their containing edges, and we add a edge from p to q of length |pq|. The resulting network is denoted by G + pq. We seek to minimize the continuous diameter of a network by introducing shortcuts. In the discrete abstract setting, we consider an abstract graph G with unit weights and ask whether we can decrease the discrete diameter of G to at most D by adding at most k edges. For any fixed D ≥ 2, this problem is NP-hard [1, 6, 8] , has parametric complexity W[2]-hard [3, 4] , and remains NP-hard even if G is a tree [1] . For an overview of the approximation algorithms in terms of both D and k refer, for instance, to Frati et al. [3]. In the discrete geometric setting, we consider a geometric graph, where a shortcut connects two vertices. Große et al. [5] are the first to consider diameter minimization in this setting. They determine a shortcut that minimizes the discrete diameter of a path with n vertices in O(n log 3 n) time. The spanning ratio of a geometric network, i.e., the largest ratio between the network distance and the Euclidean distance of any two points, has been considered as target function for edge augmentation, as well.  Our results concern networks that are paths, cycles, and convex cycles. Figures 1 and 2 illustrate examples of optimal shortcuts for paths and cycles. In Section 2, we develop an algorithm that produces an optimal shortcut for a path with n vertices in O(n) time. In Section 3, we show that for cycles a single shortcut never suffices to reduce the diameter and that two shortcuts always suffice. We characterize pairs of optimal shortcuts for convex and non-convex cycles. Based on this characterization, we develop an algorithm in Section 4 that determines an optimal pair of shortcuts for a convex cycle with n vertices in O(n) time.  Consider a polygonal path P in the plane. We seek a shortcut pq for a path P that minimizes the continuous diameter of the augmented path P + pq, i.e., The following notation is illustrated in Figure 3 . Let s and e be the endpoints of P and let p be closer to s than q along P , i.e., d(s, p) < d (s, q). For a, b ∈ P , let P [a, b] denote the sub-path from a to b along P , and let C(p, q) be the simple cycle in P + pq. Augmenting a path P with a shortcut pq. The shortcut creates a cycle C(p, q) with the sub-path from p to q along P . The farthest point from p on this cycle isp andq is farthest from q on C(p, q). The distance d(q,p) betweenq andp along P matches the Euclidean distance between p and q, because of the following. When we move a point g from p to q along the shortcut pq, then the farthest pointḡ form g along C(p, q) moves fromp toq traveling the same distance as g, i.e., |pq|. Lemma 2.1. Let pq be a shortcut for P . Every continuous diametral path in P + pq contains an endpoint of P , except when the shortcut connects the endpoints of P . Proof. Let P be a polygonal path with endpoints s and e, and let pq be a shortcut for P . Let π be a path along P + pq. When an endpoint of π lies on P [s, p] or on P [e, q], we can create a longer path by extending π to s or to e. Suppose both endpoints of π lie on the simple cycle C(p, q). As illustrated in Figure 4 , we can move the endpoints of π at the same speed counter-clockwise along C(p, q) until one them coincides with p or with q. The resulting path π has the same length as π and can be extended by P [s, p] or P [e, q] creating a longer path than π, unless s = p and e = q. Therefore, for every shortcut pq other than se, every continuous diametral path in P + pq must contain and endpoint of P . According to Lemma 2.1, we have the following three candidates for continuous diametral paths in the augmented network P + pq, two of which are illustrated in Figure 5 . 1. The path U (p, q) from s to e via the shortcut pq, 2. the path S(p, q) from s to the farthest point from s on C(p, q), and 3. the path E(p, q) from e to the farthest point from e on C(p, q). Letp be the farthest point from p on C(p, q), and letq be the farthest point from q on C(p, q). denote the slack between p andq (and symmetrically betweenp and q) along C (p, q). With this notation, we have and we can express the lengths of U (p, q), S(p, q), and E(p, q) as follows. J.-L. De Carufel, C. Grimm, A. Maheshwari, and M. Smid The following lemma characterizes which of the paths U (p, q), S(p, q), and E(p, q) determine the diameter of P + pq. Notice that these cases overlap, for instance, E (p, q) and S(p, q) Proof. The claim follows, since the following relations are preserved for ∼ ∈ {<, =, >}. Lemma 2.3. For every path P , there is an optimal shortcut pq such that S(p, q) and Proof. Suppose we have a shortcut pq for a path P such that U (p, q) is continuous diametral in P + pq. In the notation of Lemma 2.2, this means z ≤ x and z ≤ y. By the triangle inequality, the length of U (p, q)-and, thus, the continuous diameterdecreases or remains the same as we move p closer to s (decreasing x) or q closer to e (decreasing y). Moreover, decreasing x or y increases z. Therefore, we can move the shortcut closer to s and to e until we have x = z or y = z, i.e., min(x, y, z) = min(x, y) while maintaining or decreasing the continuous diameter.
De Carufel et al. REF improved the running time to O(n) and also considered the continuous version of the problem for cycles that are embedded in the plane.
Minimizing the Continuous Diameter when Augmenting Paths and Cycles with Shortcuts
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