On maximum flows in polyhedral domains
Journal of Computer and System Sciences
An exact algorithm for kinodynamic planning in the plane
Discrete & Computational Geometry
Network flows: theory, algorithms, and applications
Network flows: theory, algorithms, and applications
Motion planning in the presence of moving obstacles
Journal of the ACM (JACM)
Handbook of discrete and computational geometry
An Optimal Algorithm for Euclidean Shortest Paths in the Plane
SIAM Journal on Computing
Robot Motion Planning
Time-minimal paths amidst moving obstacles in three dimensions
Theoretical Computer Science
Nonholonomic Motion Planning
Length-bounded disjoint paths in planar graphs
Discrete Applied Mathematics - Sixth Twente Workshop on Graphs and Combinatorial Optimization
Thick non-crossing paths and minimum-cost flows in polygonal domains
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Offline variants of the "lion and man" problem
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
New lower bound techniques for robot motion planning problems
SFCS '87 Proceedings of the 28th Annual Symposium on Foundations of Computer Science
Aircraft routing in the presence of hazardous weather
Aircraft routing in the presence of hazardous weather
Roadmap-based motion planning in dynamic environments
IEEE Transactions on Robotics
Simple wriggling is hard unless you are a fat hippo
FUN'10 Proceedings of the 5th international conference on Fun with algorithms
Routing multi-class traffic flows in the plane
Computational Geometry: Theory and Applications
Sweeping a terrain by collaborative aerial vehicles
Proceedings of the 21st ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems
Hi-index | 0.00 |
We consider the problem of finding a large number of disjoint paths for unit disks moving amidst static or dynamic obstacles. The problem is motivated by the capacity estimation problem in air traffic management, in which one must determine how many aircraft can safely move through a domain while avoiding each other and avoiding ''no-fly zones'' and predicted weather hazards. For the static case we give efficient exact algorithms, based on adapting the ''continuous uppermost path'' paradigm. As a by-product, we establish a continuous analogue of Menger's Theorem. Next we study the dynamic problem in which the obstacles may move, appear and disappear, and otherwise change with time in a known manner; in addition, the disks are required to enter/exit the domain during prescribed time intervals. Deciding the existence of just one path, even for a 0-radius disk, moving with bounded speed is NP-hard, as shown by Canny and Reif [J. Canny, J.H. Reif, New lower bound techniques for robot motion planning problems, in: Proc. 28th Annu. IEEE Sympos. Found. Comput. Sci., 1987, pp. 49-60]. Moreover, we observe that determining the existence of a given number of paths is hard even if the obstacles are static, and only the entry/exit time intervals are specified for the disks. This motivates studying ''dual'' approximations, compromising on the radius of the disks and on the maximum speed of motion. Our main result is a pseudopolynomial-time dual-approximation algorithm. If K unit disks, each moving with speed at most 1, can be routed through an environment, our algorithm finds (at least) K paths for disks of radius somewhat smaller than 1 moving with speed somewhat larger than 1.