Hardness of Reconfiguring Robot Swarms with Uniform External Control in Limited Directions

Caballero, David; Cantu, Angel A.; Gomez, Timothy; Luchsinger, Austin; Schweller, Robert T.; Wylie, Tim

doi:10.2197/ipsjjip.28.782

Cited by 8 publications

(6 citation statements)

References 10 publications

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“…A hierarchy of board geometries is described in [4]. It was shown that when limiting the number of available directions to 2 and with "monotone" board geometry the problem of relocation is NP-complete [9].…”

Section: Related Workmentioning

confidence: 99%

See 1 more Smart Citation

Uniform Robot Relocation Is Hard in only Two Directions Even Without Obstacles

Caballero

Cantu

Gomez

et al. 2023

Lecture Notes in Computer Science

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Given n robots contained within a square grid surrounded by four walls, we ask the question of whether it is possible to move a particular robot a to a specific grid location b by performing a sequence of global step operations in which all robots move one grid step in the same cardinal direction (if not blocked by a wall or other blocked robots). We show this problem is NP-complete when restricted to just two directions (south and west). This answers the simplest fundamental problem in uniform global unit tilt swarm robotics. We then consider a relaxed version of this problem in which the goal is to move a robot a to a specific row regardless of its horizontal placement. We show that if asking about the bottom-most row of the square grid, then this version of the problem is solvable in polynomial time. Finally, we discuss several areas for future research and open problems.

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Section: Related Workmentioning

confidence: 99%

“…This work is an extension of a conference version with the hardness proof [13], and of a 2-page abstract outlining the first row relocation problem [12]. The journal has extended these in several ways with full details and proofs of the positive result as well as additional future work and open problems.…”

Section: Contributionsmentioning

confidence: 99%

Uniform Robot Relocation Is Hard in only Two Directions Even Without Obstacles

Caballero

Cantu

Gomez

et al. 2023

Lecture Notes in Computer Science

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show abstract

“…Other papers considered some form of universality given a certain placement of blocked objects such as reconfiguration [11,26], particle computations [13], and building shapes when given edges of free objects stick together when in contact [9,10]. In the single-step model, for a given configuration of blocked objects, reconfiguration is NP-complete with single steps limited to two directions [15], and universality results also exist for a special configuration of blocked objects [16]. Note that the hardness of the trash compaction problem [5] implies the hardness of the problem of constructing a given shape in the single-step model.…”

Section: Related Workmentioning

confidence: 99%

Pushing Blocks by Sweeping Lines

Akitaya¹,

Löffler²,

Viglietta³

2022

Preprint

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We investigate the reconfiguration of n blocks, or "tokens", in the square grid using line pushes. A line push is performed from one of the four cardinal directions and pushes all tokens that are maximum in that direction to the opposite direction. Tokens that are in the way of other tokens are displaced in the same direction, as well. Since we consider configurations equivalent under translation, this model is equivalent to having a line barrier touching one of the sides of the bounding box of the configuration and trying to move all tokens toward the barrier by one unit.Similar models of manipulating objects using uniform external forces match the mechanics of existing games and puzzles, such as Mega Maze, 2048 and Labyrinth, and have also been investigated in the context of self-assembly, programmable matter and robotic motion planning [9][10][11]13]. The problem of obtaining a given shape from a starting configuration is know to be NP-complete [5].We show that, for every n, there are sparse initial configurations of n tokens (i.e., where no two tokens are in the same row or column) that can be compacted into any a × b box such that ab = n. However, only 1 × k, 2 × k and 3 × 3 boxes are obtainable from any arbitrary sparse configuration with a matching number of tokens. We also study the problem of rearranging labeled tokens into a configuration of the same shape, but with permuted tokens. For every initial configuration of the tokens, we provide a complete characterization of what other configurations can be obtained by means of line pushes.

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“…For a given set of particles within an obstacle environment, these problems ask for a sequence of tilts such that (i) any particle reaches a designated position, (ii) a specific particle reaches a designated position, and (iii) every particle reaches its respective target position. If particles are allowed to move a unit step on actuation, Caballero et al [16] show that (i) permits a linear-time algorithm, while the decision variants of (ii) and (iii) are NP-hard. More recently, Caballero et al [14] show PSPACE-completeness for (ii).…”

Section: Tilt Problemsmentioning

confidence: 99%

Particle-Based Assembly Using Precise Global Control

et al. 2022

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In micro- and nano-scale systems, particles can be moved by using an external force like gravity or a magnetic field. In the presence of adhesive particles that can attach to each other, the challenge is to decide whether a shape is constructible. Previous work provides a class of shapes for which constructibility can be decided efficiently when particles move maximally into the same direction induced by a global signal. In this paper we consider the single step model, i.e., a model in which each particle moves one unit step into the given direction. We restrict the assembly process such that at each single time step actually one particle is added to and moved within the workspace. We prove that deciding constructibility is NP-complete for three-dimensional shapes, and that a maximum constructible shape can be approximated. The same approximation algorithm applies for 2D. We further present linear-time algorithms to decide whether or not a tree-shape in 2D or 3D is constructible. Scaling a shape yields constructibility; in particular we show that the 2-scaled copy of every non-degenerate polyomino is constructible. In the three-dimensional setting we show that the 3-scaled copy of every non-degenerate polycube is constructible.

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Hardness of Reconfiguring Robot Swarms with Uniform External Control in Limited Directions

Cited by 8 publications

References 10 publications

Uniform Robot Relocation Is Hard in only Two Directions Even Without Obstacles

Uniform Robot Relocation Is Hard in only Two Directions Even Without Obstacles

Pushing Blocks by Sweeping Lines

Particle-Based Assembly Using Precise Global Control

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