Pushing Blocks by Sweeping Lines

by   Hugo A. Akitaya, et al.

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. 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. The problem of obtaining a given shape from a starting configuration is know to be NP-complete. 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.


page 4

page 17

page 19

page 20


Cyclic Shift Problems on Graphs

We study a new reconfiguration problem inspired by classic mechanical pu...

Dependency-Based Neural Representations for Classifying Lines of Programs

We investigate the problem of classifying a line of program as containin...

Theoretical Foundation of Colored Petri Net through an Analysis of their Markings as Multi-classification

Barwise and Seligman stated the first principle of information flow: "In...

Parameterized complexity of reconfiguration of atoms

Our work is motivated by the challenges presented in preparing arrays of...

Full Tilt: Universal Constructors for General Shapes with Uniform External Forces

We investigate the problem of assembling general shapes and patterns in ...

Exploration of Finite 2D Square Grid by a Metamorphic Robotic System

We consider exploration of finite 2D square grid by a metamorphic roboti...

Directing Chemotaxis-Based Spatial Self-Organization via Biased, Random Initial Conditions

Inspired by the chemotaxis interaction of living cells, we have develope...

Please sign up or login with your details

Forgot password? Click here to reset