4 research outputs found
Connected Coordinated Motion Planning with Bounded Stretch
We consider the problem of coordinated motion planning for a swarm of simple, identical robots: From a given start grid configuration of robots, we need to reach a desired target configuration via a sequence of parallel, continuous, collision-free robot motions, such that the set of robots induces a connected grid graph at all integer times. The objective is to minimize the makespan of the motion schedule, i.e., to reach the new configuration in a minimum amount of time. We show that this problem is NP-hard, even for deciding whether a makespan of 2 can be achieved, while it is possible to check in polynomial time whether a makespan of 1 can be achieved.
On the algorithmic side, we establish simultaneous constant-factor approximation for two fundamental parameters, by achieving constant stretch for constant scale. Scaled shapes (which arise by increasing all dimensions of a given object by the same multiplicative factor) have been considered in previous seminal work on self-assembly, often with unbounded or logarithmic scale factors; we provide methods for a generalized scale factor, bounded by a constant. Moreover, our algorithm achieves a constant stretch factor: If mapping the start configuration to the target configuration requires a maximum Manhattan distance of d, then the total duration of our overall schedule is ?(d), which is optimal up to constant factors
Connected Coordinated Motion Planning with Bounded Stretch
We consider the problem of connected coordinated motion planning for a large
collective of simple, identical robots: From a given start grid configuration
of robots, we need to reach a desired target configuration via a sequence of
parallel, collision-free robot motions, such that the set of robots induces a
connected grid graph at all integer times. The objective is to minimize the
makespan of the motion schedule, i.e., to reach the new configuration in a
minimum amount of time. We show that this problem is NP-complete, even for
deciding whether a makespan of 2 can be achieved, while it is possible to check
in polynomial time whether a makespan of 1 can be achieved. On the algorithmic
side, we establish simultaneous constant-factor approximation for two
fundamental parameters, by achieving constant stretch for constant scale.
Scaled shapes (which arise by increasing all dimensions of a given object by
the same multiplicative factor) have been considered in previous seminal work
on self-assembly, often with unbounded or logarithmic scale factors; we provide
methods for a generalized scale factor, bounded by a constant. Moreover, our
algorithm achieves a constant stretch factor: If mapping the start
configuration to the target configuration requires a maximum Manhattan distance
of , then the total duration of our overall schedule is ,
which is optimal up to constant factors.Comment: 28 pages, 18 figures, full version of an extended abstract that
appeared in the proceedings of the 32nd International Symposium on Algorithms
and Computation (ISAAC 2021); revised version (more details added, and typing
errors corrected
Space Ants: Episode II - Coordinating Connected Catoms (Media Exposition)
How can a set of identical mobile agents coordinate their motions to transform their arrangement from a given starting to a desired goal configuration? We consider this question in the context of actual physical devices called Catoms, which can perform reconfiguration, but need to maintain connectivity at all times to ensure communication and energy supply. We demonstrate and animate algorithmic results, in particular a proof of hardness, as well as an algorithm that guarantees constant stretch for certain classes of arrangements: If mapping the start configuration to the target configuration requires a maximum Manhattan distance of d, then the total duration of our overall schedule is in ?(d), which is optimal up to constant factors
Space Ants: Episode II – Coordinating Connected Catoms
International audienceHow can a set of identical mobile agents coordinate their motions to transform their arrangement from a given starting to a desired goal configuration? We consider this question in the context of actual physical devices called Catoms, which can perform reconfiguration, but need to maintain connectivity at all times to ensure communication and energy supply. We demonstrate and animate algorithmic results, in particular a proof of hardness, as well as an algorithm that guarantees constant stretch for certain classes of arrangements: If mapping the start configuration to the target configuration requires a maximum Manhattan distance of d, then the total duration of our overall schedule is in O(d), which is optimal up to constant factors