Minimizing the Continuous Diameter when Augmenting Paths and Cycles with Shortcuts

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

Shortcut sets for the locus of plane Euclidean networks

We study the problem of augmenting the locus N of a plane Euclidean network N by in- serting iteratively a finite set of segments, called shortcut set , while reducing the diameterof the locus of the resulting network. There are two main differences with the classicalaugmentation problems: the endpoints of the segments are allowed to be points of N as well as points of the previously inserted segments (instead of only vertices of N ), and the notion of diameter is adapted to the fact that we deal with N instead of N . This increases enormously the hardness of the problem but also its possible practical applications to net- work design. Among other results, we characterize the existence of shortcut sets, computethem in polynomial time, and analyze the role of the convex hull of N when inserting a shortcut set. Our main results prove that, while the problem of minimizing the size of ashortcut set is NP-hard, one can always determine in polynomial time whether insertingonly one segment suffices to reduce the diameter.Ministerio de Economía y Competitividad MTM2015-63791-

Analysing trajectory similarity and improving graph dilation

In this thesis, we focus on two topics in computational geometry. The first topic is analysing trajectory similarity. A trajectory tracks the movement of an object over time. A common way to analyse trajectories is by finding similarities. The Fr\'echet distance is a similarity measure that has gained popularity in the theory community, since it takes the continuity of the curves into account. One way to analyse trajectories using the Fr\'echet distance is to cluster trajectories into groups of similar trajectories. For vehicle trajectories, another way to analyse trajectories is to compute the path on the underlying road network that best represents the trajectory. The second topic is improving graph dilation. Dilation measures the quality of a network in applications such as transportation and communication networks. Spanners are low dilation graphs with not too many edges. Most of the literature on spanners focuses on building the graph from scratch. We instead focus on adding edges to improve the dilation of an existing graph