Problems on one way road networks
Journal of Graph Algorithms and Applications
A One-Way Road Network is an ordered pair OW RN = (Wx, Wy) comprising of a set Wx of m directed horizontal roads along with another set Wy of n directed vertical roads. An OW RN can also be viewed as a directed grid graph GG = (V, E), where V corresponds to intersections between every pair of horizontal and vertical roads, and there is a directed edge between every pair of consecutive vertices in V in the same direction corresponding to that road. A vehicle c is defined as a 3-tuple (t, s, P), where c starts moving at time t and moves with a constant speed s from its start vertex to destination vertex along pre-specified directed path P, unless a collision occurs. A collision between a pair of vehicles ci and cj(i ≠ j) occurs if they reach a vertex (Formula Presented) (a junction in OW RN) orthogonally at the same time. A traffic configuration on an OW RN is a 2-tuple T C = (GG, C), where C is a set of vehicles, each travelling on a pre-specified path on GG. A collision-free T C is a traffic configuration without any collision. We prove that finding a maximum cardinality subset (Formula Presented), such that T C = (GG, Cmax) is collision-free, is NP-hard. We also show that GG can be preprocessed into a data-structure in O(n + m) time and space, such that the length of the shortest path between any pair of vertices in GG can be computed in O(1) time and the shortest path can be computed in O(p) time, where p is the number of vertices in the path.
Ajay, Jammigumpula; Das, Avinandan; Dutta, Binayak; Karmakar, Arindam; Roy, Sasanka; and Saikia, Navaneeta, "Problems on one way road networks" (2020). Journal Articles. 466.