Coordinating Team Movements in Dangerous Territory

Suppose a two-member team is required to navigate through dangerous territory (a plane with polygonal obstacles) along two preselected polygonal paths. The team could consist of police officers, soldiers, or even robots on the surface of Mars. Team members are allowed to coordinate their movements, and it is imperative to keep the team close together during the mission so that if an emergency occurs, one team member can quickly run to the other member and offer assistance. We show how the team can navigate through a plane with polygonal obstacles such that the worst-case separation between the two team members is minimized during the mission. As part of this safety exercise, we develop geodesic and min-link algorithms for shortest path maps and the Fréchet distance [2] in a plane with obstacles. We also utilize a correspondence between shortest path maps and the free space diagram to help compute the Fréchet distance. We present the first algorithms for three types of Fréchet distance: (1) geodesic Fréchet distance in a polygonal domain (a plane with polygonal obstacles), (2) min-link Fréchet distance in a simple polygon, and (3) min-link Fréchet distance in a polygonal domain. A novel sub-quadratic approximation algorithm for the min-link (and traditional) Fréchet distance is also given. We also develop shortest path maps including (4) the first min-link shortest path map in a polygonal domain that supports queries from a continuous set of source points and (5) a geodesic shortest path map between two line segments in a polygonal domain. The latter structure is related to work by Chiang and Mitchell [8] but uses a new approach based on shortest path structures that we call dynamic and static spotlights. This work has been supported by the National Science Foundation grant NSF CAREER CCF-0643597. †Department of Computer Science, University of Texas at San Antonio, One UTSA Circle, San Antonio, TX 78249-0667. acook,[email protected]