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).

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A tight lower bound for the capture time of the Cops and Robbers game

Theoretical Computer Science ◽

10.1016/j.tcs.2020.06.004 ◽

2020 ◽

Vol 839 ◽

pp. 143-163

Author(s):

Sebastian Brandt ◽

Yuval Emek ◽

Jara Uitto ◽

Roger Wattenhofer

Keyword(s):

Lower Bound ◽

Capture Time ◽

Cops And Robbers

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The Capture Time of the Hypercube

The Electronic Journal of Combinatorics ◽

10.37236/2921 ◽

2013 ◽

Vol 20 (2) ◽

Cited By ~ 5

Author(s):

Anthony Bonato ◽

Przemysław Gordinowicz ◽

Bill Kinnersley ◽

Paweł Prałat

Keyword(s):

Capture Time ◽

Coupon Collector ◽

Cops And Robbers ◽

Minimum Number

In the game of Cops and Robbers, the capture time of a graph is the minimum number of moves needed by the cops to capture the robber, assuming optimal play. We prove that the capture time of the $n$-dimensional hypercube is $\Theta (n\ln n)$. Our methods include a novel randomized strategy for the players, which involves the analysis of the coupon-collector problem.

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The Ihara Zeta Function of the Infinite Grid

The Electronic Journal of Combinatorics ◽

10.37236/3561 ◽

2014 ◽

Vol 21 (2) ◽

Author(s):

Bryan Clair

Keyword(s):

Functional Equation ◽

Zeta Function ◽

Cayley Graph ◽

Theta Functions ◽

Zeta Functions ◽

Multivalued Function ◽

Ihara Zeta Function ◽

Grid Graphs ◽

Infinite Grid ◽

Limiting Value

The infinite grid is the Cayley graph of $\mathbb{Z} \times \mathbb{Z}$ with the usual generators. In this paper, the Ihara zeta function for the infinite grid is computed using elliptic integrals and theta functions. The zeta function of the grid extends to an analytic, multivalued function which satisfies a functional equation. The set of singularities in its domain is finite.The grid zeta function is the first computed example which is non-elementary, and which takes infinitely many values at each point of its domain. It is also the limiting value of the normalized sequence of Ihara zeta functions for square grid graphs and torus graphs.

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On the Capture Time of Cops and Robbers Game on a Planar Graph

Combinatorial Optimization and Applications - Lecture Notes in Computer Science ◽

10.1007/978-3-319-48749-6_1 ◽

2016 ◽

pp. 3-17 ◽

Cited By ~ 2

Author(s):

Photchchara Pisantechakool ◽

Xuehou Tan

Keyword(s):

Planar Graph ◽

Capture Time ◽

Cops And Robbers

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Functions and functional on finite systems

Journal of Symbolic Logic ◽

10.2307/2275180 ◽

1992 ◽

Vol 57 (1) ◽

pp. 118-130 ◽

Cited By ~ 1

Author(s):

Libo Lo

Keyword(s):

Computer Science ◽

Natural Number ◽

Number System ◽

Input Function ◽

Finite System ◽

Global Function ◽

Finite Systems ◽

Recursive Functions ◽

The Continuum

The global function on finite systems is a new concept defined by Gurevich in [1] and discussed in [2] and [3]. In the last ten years this concept has become more and more useful in computer science and logic. Gurevich also pointed out the importance of global functionals on finite systems. In this paper we will give a brief introduction to the concepts of global functions and global functionals on finite systems.In studying the natural number system N = 〈N, +,0〉 we often refer to its functions and functionals. There are a lot of books and papers in this area. Kleene in [4] gave a detailed introduction to the recursive functions of N. The functionals of N are normally very difficult to compute because here we need to tell the machine what the input function is, which is not very easy to do. In developing the theory of finite systems the functions and functionals are also very useful. For computing the functionals in finite systems we can take the entire graph of a function as the input, which is not possible in N. We will discuss recursive functions and functionals for finite systems. The definitions of recursive functions are very similar to the case in N, but we will have a very different situation. In N the number of elements is infinite. The number of all possible functions from N to N is the continuum. In a finite system the number of all possible functions is finite. It seems that there is no necessity to define the global functions.

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