We prove better lower bounds on additive spanners and emulators, which are lossy compression schemes for undirected graphs, as well as lower bounds on shortcut sets, which reduce the diameter of directed graphs. We show that any O(n)-size shortcut set cannot bring the diameter below Ω(n1/6), and that any O(m)-size shortcut set cannot bring it below Ω(n1/11). These improve Hesse’s [15] lower bou...

This study introduces "shortcut routing," a novel routing mechanism in capsule networks that addresses computational inefficiencies by directly activating global capsules from local capsules, eliminating intermediate layers. An attention-based approach with fuzzy coefficients is also explored for improved efficiency. Experimental results on Mnist, smallnorb, and affNist datasets show comparable...

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, where 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...

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 shor...