2 research outputs found
Reconfiguration of Polygonal Subdivisions via Recombination
Motivated by the problem of redistricting, we study area-preserving reconfigurations of connected subdivisions of a simple polygon. A connected subdivision of a polygon ?, called a district map, is a set of interior disjoint connected polygons called districts whose union equals ?. We consider the recombination as the reconfiguration move which takes a subdivision and produces another by merging two adjacent districts, and by splitting them into two connected polygons of the same area as the original districts. The complexity of a map is the number of vertices in the boundaries of its districts. Given two maps with k districts, with complexity O(n), and a perfect matching between districts of the same area in the two maps, we show constructively that (log n)^O(log k) recombination moves are sufficient to reconfigure one into the other. We also show that ?(log n) recombination moves are sometimes necessary even when k = 3, thus providing a tight bound when k = 3
Characterizing Universal Reconfigurability of Modular Pivoting Robots
We give both efficient algorithms and hardness results for reconfiguring
between two connected configurations of modules in the hexagonal grid. The
reconfiguration moves that we consider are "pivots", where a hexagonal module
rotates around a vertex shared with another module. Following prior work on
modular robots, we define two natural sets of hexagon pivoting moves of
increasing power: restricted and monkey moves. When we allow both moves, we
present the first universal reconfiguration algorithm, which transforms between
any two connected configurations using monkey moves. This result
strongly contrasts the analogous problem for squares, where there are rigid
examples that do not have a single pivoting move preserving connectivity. On
the other hand, if we only allow restricted moves, we prove that the
reconfiguration problem becomes PSPACE-complete. Moreover, we show that, in
contrast to hexagons, the reconfiguration problem for pivoting squares is
PSPACE-complete regardless of the set of pivoting moves allowed. In the
process, we strengthen the reduction framework of Demaine et al. [FUN'18] that
we consider of independent interest