Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  seek to augment a geometric network in the Euclidean plane with shortcuts to minimize its continuous diameter, i.e., the largest network distance beten any two points on the augmented network. Unlike in the discrete setting where a shortcut connects two vertices and the diameter is measured beten vertices,  take all points along the edges of the network into account when placing a shortcut and when measuring distances in the augmented network. #R##N##R##N# study this network augmentation problem for paths and cycles. For paths,  determine an optimal shortcut in linear time. For cycles,  show that a single shortcut never decreases the continuous diameter and that two shortcuts always suffice to reduce the continuous diameter. Furthermore,  characterize optimal pairs of shortcuts for convex and non-convex cycles. Finally,  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. #R##N##R##N#Our work reveals some of the underlying challenges that must be overcome when addressing the discrete version of this network augmentation problem, where  minimize the discrete diameter of a network with shortcuts that connect only vertices.
['Jean-Lou De Carufel', 'Carsten Grimm', 'Anil Maheshwari', 'Michiel H. M. Smid']
I am writing a research paper on a idea Minimizing the Continuous Diameter when Augmenting Paths and Cycles with Shortcuts
[{"from": "human", "value": "I am writing a research paper on a idea Minimizing the Continuous Diameter when Augmenting Paths and Cycles with Shortcuts"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  seek to augment a geometric network in the Euclidean plane with shortcuts to minimize its continuous diameter, i.e., the largest network distance beten any two points on the augmented network. Unlike in the discrete setting where a shortcut connects two vertices and the diameter is measured beten vertices,  take all points along the edges of the network into account when placing a shortcut and when measuring distances in the augmented network. #R##N##R##N# study this network augmentation problem for paths and cycles. For paths,  determine an optimal shortcut in linear time. For cycles,  show that a single shortcut never decreases the continuous diameter and that two shortcuts always suffice to reduce the continuous diameter. Furthermore,  characterize optimal pairs of shortcuts for convex and non-convex cycles. Finally,  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. #R##N##R##N#Our work reveals some of the underlying challenges that must be overcome when addressing the discrete version of this network augmentation problem, where  minimize the discrete diameter of a network with shortcuts that connect only vertices."}]