Optimal In-Place Compaction of Sliding Cubes

The sliding cubes model is a well-established theoretical framework that supports the analysis of reconfiguration algorithms for modular robots consisting of face-connected cubes. The best algorithm currently known for the reconfiguration problem, by Abel and Kominers [arXiv, 2011], uses O(n3) moves to transform any n-cube configuration into any other n-cube configuration. As is common in the literature, this algorithm reconfigures the input into an intermediate canonical shape. In this paper we present an in-place algorithm that reconfigures any n-cube configuration into a compact canonical shape using a number of moves proportional to the sum of coordinates of the input cubes. This result is asymptotically optimal. Furthermore, our algorithm directly extends to dimensions higher than three.

Compacting Squares

Edge-connected configurations of squares are a common model for modular robots in two dimensions. A well-established way to reconfigure such modular robots are so-called sliding moves. Dumitrescu and Pach [Graphs and Combinatorics, 2006] proved that it is always possible to reconfigure one edge-connected configuration of $n$ squares into any other using at most $O(n^2)$ sliding moves, while keeping the configuration connected at all times. For certain configurations $\Omega(n^2)$ sliding moves are necessary. However, significantly fewer moves may be sufficient. In this paper we present a novel input-sensitive in-place algorithm which requires only $O(\bar{P} n)$ sliding moves to transform one configuration into the other, where $\bar{P}$ is the maximum perimeter of the respective bounding boxes. Our Gather&Compact algorithm is built on the basic principle that well-connected components of modular robots can be transformed efficiently. Hence we iteratively increase the connectivity within a configuration, to finally arrive at a single solid $xy$-monotone component. We implemented Gather&Compact and compared it experimentally to the in-place modification by Moreno and Sacrist\'an [EuroCG 2020] of the Dumitrescu and Pach algorithm (MSDP). Our experiments show that Gather&Compact consistently outperforms MSDP by a significant margin, on all types of square configurations.

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 $O(n^3)$ 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.

Universal Reconfiguration of Facet-Connected Modular Robots by Pivots: The $O(1)$ Musketeers

We present the first universal reconfiguration algorithm for transforming a modular robot between any two facet-connected square-grid configurations using pivot moves. More precisely, we show that five extra "helper" modules ("musketeers") suffice to reconfigure the remaining $n$ modules between any two given configurations. Our algorithm uses $O(n^2)$ pivot moves, which is worst-case optimal. Previous reconfiguration algorithms either require less restrictive "sliding" moves, do not preserve facet-connectivity, or for the setting we consider, could only handle a small subset of configurations defined by a local forbidden pattern. Configurations with the forbidden pattern do have disconnected reconfiguration graphs (discrete configuration spaces), and indeed we show that they can have an exponential number of connected components. But forbidding the local pattern throughout the configuration is far from necessary, as we show that just a constant number of added modules (placed to be freely reconfigurable) suffice for universal reconfigurability. We also classify three different models of natural pivot moves that preserve facet-connectivity, and show separations between these models.