Pushing lines helps: Efficient universal centralised transformations for programmable matter

Almethen, Abdullah; Michail, Othon; Potapov, Igor

doi:10.1016/j.tcs.2020.04.026

Cited by 15 publications

(17 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We then investigate the transformation of nice shapes. A Nice Shape (defined in [20]) is a shape S which has a central line L where for all nodes u in S either u ∈ L or u is connected to L by a line of nodes perpendicular to L. We provide a lower bound of Omega(n 2 ) for transforming a line of n nodes into a nice shape. We show that it is possible to transform such a line into a nice shape of n − 1 nodes using a 3-node seed in O(n 2 ) time.…”

Section: Contributionmentioning

confidence: 99%

“…Recent progress in this direction has been made in a previous paper [20], covering questions related to a specific model of programmable matter where nodes exist in the form of a shape on a 2D grid and are capable of performing two specific movements: rotation around each other and sliding a node across two other nodes. They presented 3 problems: transformations with only rotations (Rot-Transformability), transformations with rotations with the restriction that shapes must always remain connected (RotC-Transformability) and transformations with both rotation and sliding movements (RS-Transformability).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Centralised Connectivity-Preserving Transformations for Programmable Matter: A Minimal Seed Approach

Connor¹,

Michail²,

Потапов³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

We study a model of programmable matter systems consisting of n devices lying on a 2-dimensional square grid which are able to perform the minimal mechanical operation of rotating around each other. The goal is to transform an initial shape A into a target shape B. We investigate the class of shapes which can be constructed in such a scenario under the additional constraint of maintaining global connectivity at all times. We focus on the scenario of transforming nice shapes, a class of shapes consisting of a central line L where for all nodes u in S either u ∈ L or u is connected to L by a line of nodes perpendicular to L. We prove that by introducing a minimal 3-node seed it is possible for the canonical shape of a line of n nodes to be transformed into a nice shape of n − 1 nodes. We use this to show that a 4-node seed enables the transformation of nice shapes of size n into any other nice shape of size n in O(n 2 ) time. We leave as an open problem the expansion of the class of shapes which can be constructed using such a seed to include those derived from nice shapes.

show abstract

Section: Contributionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Centralised Connectivity-Preserving Transformations for Programmable Matter: A Minimal Seed Approach

Connor¹,

Michail²,

Потапов³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…A more recent study along this direction is shown in [4], and introduces the line-pushing model. In this model, an individual entity can push the whole line of consecutive entities one position in a given direction in a single time-step.…”

Section: Related Workmentioning

confidence: 99%

“…a path starting from some p ∈ S, visiting every agent in S and ending at some p ∈ S, where p = p . In this work, each agent is equipped with the linear-strength mechanism introduced in [4], called the line pushing mechanism. A line L consists of a sequence of k agents occupying consecutive cells on the grid, say w.l.o.g, L = (x, y), (x + 1, y), .…”

Section: Modelmentioning

confidence: 99%

Distributed Transformations of Hamiltonian Shapes based on Line Moves

Almethen¹,

Michail²,

Потапов³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

We consider a discrete system of n simple indistinguishable devices, called agents, forming a connected shape SI on a two-dimensional square grid. Agents are equipped with a linear-strength mechanism, called a line move, by which an agent can push a whole line of consecutive agents in one of the four directions in a single time-step. We study the problem of transforming an initial shape SI into a given target shape SF via a finite sequence of line moves in a distributed model, where each agent can observe the states of nearby agents in a Moore neighbourhood. Our main contribution is the first distributed connectivity-preserving transformation that exploits line moves within a total of O(n log 2 n) moves, which is asymptotically equivalent to that of the best-known centralised transformations. The algorithm solves the line formation problem that allows agents to form a final straight line SL, starting from any shape SI , whose associated graph contains a Hamiltonian path.

show abstract

“…They also presented O(n 2 )-time reconfiguration algorithm by rotations and slidings. Almethen et al considered reconfiguration by line-pushing, where each module is equipped with the ability of pushing a line of modules [1]. They presented O(n log n)-time universal reconfiguration algorithm that does not promise connectivity of intermediate shapes and O(n √ n)-time reconfiguration algorithm that transforms a diagonal line into a straight chain with preserving connectivity.…”

Section: Introductionmentioning

confidence: 99%

Search by a Metamorphic Robotic System in a Finite 3D Cubic Grid

Yamada

Yamauchi²

2021

Preprint

View full text Add to dashboard Cite

We consider search in a finite 3D cubic grid by a metamorphic robotic system (MRS), that consists of anonymous modules. A module can perform a sliding and rotation while the whole modules keep connectivity. As the number of modules increases, the variety of actions that the MRS can perform increases. The search problem requires the MRS to find a target in a given finite field. Doi et al. (SSS 2018) demonstrate a necessary and sufficient number of modules for search in a finite 2D square grid. We consider search in a finite 3D cubic grid and investigate the effect of common knowledge. We consider three different settings. First, we show that three modules are necessary and sufficient when all modules are equipped with a common compass, i.e., they agree on the direction and orientation of the x, y, and z axes. Second, we show that four modules are necessary and sufficient when all modules agree on the direction and orientation of the vertical axis. Finally, we show that five modules are necessary and sufficient when all modules are not equipped with a common compass. Our results show that the shapes of the MRS in the 3D cubic grid have richer structure than those in the 2D square grid.

show abstract

Pushing lines helps: Efficient universal centralised transformations for programmable matter

Cited by 15 publications

References 25 publications

Centralised Connectivity-Preserving Transformations for Programmable Matter: A Minimal Seed Approach

Centralised Connectivity-Preserving Transformations for Programmable Matter: A Minimal Seed Approach

Distributed Transformations of Hamiltonian Shapes based on Line Moves

Search by a Metamorphic Robotic System in a Finite 3D Cubic Grid

Contact Info

Product

Resources

About