A Cop and Drunken Robber Game on n-DimensionalInfinite-Grid Graphs

A Cop and Drunken Robber (CDR) game is one variation of a famous combinatorial game, called Cops and Robbers, which has been extensively studied and applied in the area of theoretical and computer science as demonstrated by several conferences and publications. In this paper, for a natural number n, we present two strategies for a single cop to chase a drunken robber on n-dimensional infinite-grid graphs. Both strategies show that if the initial distance between the cop and the drunken robber is s, then the expected capture time is s+o(s).

Circle formation by asynchronous opaque robots on infinite grid

This paper presents a distributed algorithm for circle formation problem under the infinite grid environment by asynchronous mobile opaque robots. Initially all the robots are acquiring distinct positions and they have to form a circle over the grid. Movements of the robots are restricted only along the grid lines. They do not share any global co-ordinate system. Robots are controlled by an asynchronous adversarial scheduler that operates in Look-Compute-Move cycles. The robots are indistinguishable by their nature, do not have any memory of their past configurations and previous actions. We consider the problem under luminous model, where robots communicate via lights, other than that they do not have any external communication system. Our protocol solves the circle formation problem using seven colors. A subroutine of our algorithm also solves the line formation problem using three colors.

Collectives of automata on infinite grid graph with deterministic vertex labeling

Automata walking on graphs are a mathematical formalization of autonomous mobile agents with limited memory operating in discrete environments. Under this model broad area of studies of the behaviour of automata in finite and infinite labyrinths (a labyrinth is an embedded directed graph of special form) arose and intensively developing. Research in this regard received a wide range of applications, for example, in the problems of image analysis and navigation of mobile robots. Automata operating in labyrinths can distinguish directions, that is, they have a compass. This paper examines vertex labellings of infinite square grid graph thanks to these labellings a finite automaton without a compass can walk along graph in any arbitrary direction. The automaton looking over neighbourhood of the current vertex and may move to some neighbouring vertex selected by its label. We propose a minimal deterministic traversable vertex labelling that satisfies the required property. A labelling is said to be deterministic if all vertices in closed neighbourhood of every vertex have different labels. It is shown that minimal deterministic traversable vertex labelling of square grid graph uses labels of five different types. Minimal deterministic traversable labelling of subgraphs of infinite square grid graph whose degrees are less than four are developed. The key problem for automata and labyrinths is the problem of constructing a finite automaton that traverse a given class of labyrinths. We say that automaton traverse infinite graph if it visits any randomly selected vertex of this graph in a finite time. It is proved that a collective of one automaton and three pebbles can traverse infinite square grid graph with deterministic labelling and any collective with fewer pebbles cannot. We consider pebbles as automata of the simplest form, whose positions are completely determined by the remaining automata of the collective. The results regarding to exploration of an infinite deterministic square grid graph coincide with the results of A.V. Andzhan (Andzans) regarding to traversal of an infinite mosaic labyrinth without holes. It follows from above that infinite grid graph after constructing a minimal traversable deterministic labelling on it and fixing two pairs of opposite directions on it becomes an analogue of an infinite mosaic labyrinth without holes.

Introduction

This introductory chapter provides an overview of the book, which deals with the plaid model. The plaid model is a rule for assigning a square tiling of the plane to each parameter A = p/q ɛ (0, 1) with pq even. Such parameters are called even rational. Based on the parameter A, even integers are assigned to the lines of the usual infinite grid of integer-spaced vertical and horizontal lines. These integers are called capacities. At the same time, a second grid of slanting lines is defined and odd integers are assigned to these lines. These odd integers are called masses. Then, a light point is placed at every intersection of the form σ‎ ∩ τ‎ where σ‎ is a slanting line, τ‎ is a horizontal or vertical line, and the mass of σ‎ has the same sign as the capacity of τ‎ and smaller absolute value.