Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Tilt Automata: Gathering Particles With Uniform External Control

Sándor P. Fekete, Jonas Friemel, Peter Kramer, Jan-Marc Reinhardt, Christian Rieck, Christian Scheffer

TL;DR

This work investigates the gathering of particles in the full tilt model of externally controlled motion planning and develops a polynomial-time algorithm for gathering in a completely filled polyomino as well as hardness reductions for approximating shortest gathering sequences.

Abstract

Motivated by targeted drug delivery, we investigate the gathering of particles in the full tilt model of externally controlled motion planning: A set of particles is located at the tiles of a polyomino with all particles reacting uniformly to an external force by moving as far as possible in one of the four axis-parallel directions until they hit the boundary. The goal is to choose a sequence of directions that moves all particles to a common position. Our results include a polynomial-time algorithm for gathering in a completely filled polyomino as well as hardness reductions for approximating shortest gathering sequences and for determining whether the particles in a partially filled polyomino can be gathered. We pay special attention to the impact of restricted geometry, particularly polyominoes without holes. As corollaries, we make progress on an open question from [Balanza-Martinez et al., SODA 2020] by showing that deciding whether a given position can be occupied remains NP-hard in polyominoes without holes and provide initial results on the parameterized complexity of tilt problems. Our results build on a connection we establish between tilt models and the theory of synchronizing automata.

Tilt Automata: Gathering Particles With Uniform External Control

TL;DR

This work investigates the gathering of particles in the full tilt model of externally controlled motion planning and develops a polynomial-time algorithm for gathering in a completely filled polyomino as well as hardness reductions for approximating shortest gathering sequences.

Abstract

Motivated by targeted drug delivery, we investigate the gathering of particles in the full tilt model of externally controlled motion planning: A set of particles is located at the tiles of a polyomino with all particles reacting uniformly to an external force by moving as far as possible in one of the four axis-parallel directions until they hit the boundary. The goal is to choose a sequence of directions that moves all particles to a common position. Our results include a polynomial-time algorithm for gathering in a completely filled polyomino as well as hardness reductions for approximating shortest gathering sequences and for determining whether the particles in a partially filled polyomino can be gathered. We pay special attention to the impact of restricted geometry, particularly polyominoes without holes. As corollaries, we make progress on an open question from [Balanza-Martinez et al., SODA 2020] by showing that deciding whether a given position can be occupied remains NP-hard in polyominoes without holes and provide initial results on the parameterized complexity of tilt problems. Our results build on a connection we establish between tilt models and the theory of synchronizing automata.
Paper Structure (4 sections, 2 figures)

This paper contains 4 sections, 2 figures.

Table of Contents

  1. Introduction
  2. Our contributions
  3. Related work
  4. Preliminaries

Figures (2)

  • Figure 1: Two examples where gathering a set of particles is impossible in the full tilt model. Both examples depict, from left to right, a set of particles (blue squares) inside a polyomino, a sequence of moves leading to all possible configurations, and a partition of reachable positions yielding a congruence. Example \ref{['sfig:gathering-ft-a']} prohibits gathering by keeping the particles in different congruence classes, whereas the particles in \ref{['sfig:gathering-ft-b']} always occupy distinct positions of the same class.
  • Figure 2: \ref{['sfig:polyomino-simple']} A simple polyomino with a darkly shaded pixel at the intersection of two lightly shaded segments and a corner pixel $p$. \ref{['sfig:polyomino-maze']} A non-simple maze and its dual graph. \ref{['sfig:moves']} Downwards moves in the blocking and merging variants of the full tilt model acting on a configuration.