scholarly journals The Capture Time of the Hypercube

10.37236/2921 ◽  
2013 ◽  
Vol 20 (2) ◽  
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.

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

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

scholarly journals Cops, robbers, and barricades

2021 ◽  
Author(s):  
Erin Kathleen McKenna Meger
Keyword(s):  
Complete Graphs ◽  
New Variant ◽  
Cops And Robbers ◽  
Minimum Number ◽  
Compare And Contrast

Cops, Robbers, and Barricades is a new variant of the game on graphs, Cops and Robbers. In this variant, the robber may build barricades that restrict the movements of the cops. The minimum number of cops required to capture the robber on a graph G is called the barricade-cop number, denoted cB(G). If cB(G) = 1, then G is called barricade-cop-win. The game can be generalized so that the robber may build b(k)-many barricades on vertices during her kth turn, in accordance with barricade rules that dictate the permissible positions of these barricades. The barricade-cop number is determined exactly for complete graphs, cycles, and paths, and we provide bounds on trees and locally-path-like graphs. We compare and contrast variants on the barricade rules, and give an algorithmic characterization of barricade-cop-win graphs with any set of barricade rules.

scholarly journals Cops, robbers, and barricades

2021 ◽  
Author(s):  
Erin Kathleen McKenna Meger
Keyword(s):  
Complete Graphs ◽  
New Variant ◽  
Cops And Robbers ◽  
Minimum Number ◽  
Compare And Contrast

Cops, Robbers, and Barricades is a new variant of the game on graphs, Cops and Robbers. In this variant, the robber may build barricades that restrict the movements of the cops. The minimum number of cops required to capture the robber on a graph G is called the barricade-cop number, denoted cB(G). If cB(G) = 1, then G is called barricade-cop-win. The game can be generalized so that the robber may build b(k)-many barricades on vertices during her kth turn, in accordance with barricade rules that dictate the permissible positions of these barricades. The barricade-cop number is determined exactly for complete graphs, cycles, and paths, and we provide bounds on trees and locally-path-like graphs. We compare and contrast variants on the barricade rules, and give an algorithmic characterization of barricade-cop-win graphs with any set of barricade rules.

scholarly journals Cops and Robbers on Geometric Graphs

2012 ◽  
Vol 21 (6) ◽  
pp. 816-834 ◽  
Author(s):  
ANDREW BEVERIDGE ◽  
ANDRZEJ DUDEK ◽  
ALAN FRIEZE ◽  
TOBIAS MÜLLER
Keyword(s):  
Lower Bound ◽  
Absolute Constant ◽  
High Probability ◽  
Probability Space ◽  
Geometric Graph ◽  
Geometric Graphs ◽  
Random Geometric Graphs ◽  
Cops And Robbers ◽  
Minimum Number ◽  
Vertex Set

Cops and robbers is a turn-based pursuit game played on a graph G. One robber is pursued by a set of cops. In each round, these agents move between vertices along the edges of the graph. The cop number c(G) denotes the minimum number of cops required to catch the robber in finite time. We study the cop number of geometric graphs. For points x1, . . ., xn ∈ ℝ2, and r ∈ ℝ+, the vertex set of the geometric graph G(x1, . . ., xn; r) is the graph on these n points, with xi, xj adjacent when ∥xi − xj∥ ≤ r. We prove that c(G) ≤ 9 for any connected geometric graph G in ℝ2 and we give an example of a connected geometric graph with c(G) = 3. We improve on our upper bound for random geometric graphs that are sufficiently dense. Let (n,r) denote the probability space of geometric graphs with n vertices chosen uniformly and independently from [0,1]2. For G ∈ (n,r), we show that with high probability (w.h.p.), if r ≥ K1 (log n/n)1/4 then c(G) ≤ 2, and if r ≥ K2(log n/n)1/5 then c(G) = 1, where K1, K2 > 0 are absolute constants. Finally, we provide a lower bound near the connectivity regime of (n,r): if r ≤ K3 log n/ then c(G) > 1 w.h.p., where K3 > 0 is an absolute constant.

scholarly journals Finding a Princess in a Palace: a Pursuit-Evasion Problem

10.37236/2296 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
John R Britnell ◽  
Mark Wildon
Keyword(s):  
Winning Strategy ◽  
Finite Graph ◽  
Pursuit Evasion ◽  
Single Room ◽  
Cops And Robbers ◽  
Minimum Number ◽  
Evasion Problem

This paper solves a pursuit-evasion problem in which a prince must find a princess who is constrained to move on each day from one vertex of a finite graph to another. Unlike the related and much studied `Cops and Robbers Game', the prince has no knowledge of the position of the princess; he may, however, visit any single room he wishes on each day. We characterize the graphs for which the prince has a winning strategy, and determine, for each such graph, the minimum number of days the prince requires to guarantee to find the princess.

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

2016 ◽  
pp. 3-17 ◽  
Author(s):  
Photchchara Pisantechakool ◽  
Xuehou Tan
Keyword(s):  
Planar Graph ◽  
Capture Time ◽  
Cops And Robbers

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

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2107
Author(s):  
Nuttanon Songsuwan ◽  
Thiradet Jiarasuksakun ◽  
Anuwat Tangthanawatsakul ◽  
Pawaton Kaemawichanurat
Keyword(s):  
Computer Science ◽  
Natural Number ◽  
Combinatorial Game ◽  
Grid Graphs ◽  
Initial Distance ◽  
Capture Time ◽  
Cops And Robbers ◽  
Infinite Grid

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).

Information capacity and bandwidth limitations in the SEM

1989 ◽  
Vol 47 ◽  
pp. 84-85
Author(s):  
D. C. Joy ◽  
R. D. Bunn
Keyword(s):  
Signal To Noise Ratio ◽  
Critical Dimension ◽  
Information Capacity ◽  
Acquisition Time ◽  
Signal To Noise ◽  
Sem Image ◽  
Probe Size ◽  
Minimum Number ◽  
Noise Ratio ◽  
Signal Channel

The information available from an SEM image is limited both by the inherent signal to noise ratio that characterizes the image and as a result of the transformations that it may undergo as it is passed through the amplifying circuits of the instrument. In applications such as Critical Dimension Metrology it is necessary to be able to quantify these limitations in order to be able to assess the likely precision of any measurement made with the microscope.The information capacity of an SEM signal, defined as the minimum number of bits needed to encode the output signal, depends on the signal to noise ratio of the image - which in turn depends on the probe size and source brightness and acquisition time per pixel - and on the efficiency of the specimen in producing the signal that is being observed. A detailed analysis of the secondary electron case shows that the information capacity C (bits/pixel) of the SEM signal channel could be written as :

Applying Generalizability Theory to Optimize Analysis of Spontaneous Teacher Talk in Elementary Classrooms

2020 ◽  
Vol 63 (6) ◽  
pp. 1947-1957
Author(s):  
Alexandra Hollo ◽  
Johanna L. Staubitz ◽  
Jason C. Chow
Keyword(s):  
Special Education Teachers ◽  
Generalizability Theory ◽  
Child Outcomes ◽  
Elementary Classrooms ◽  
Teacher Talk ◽  
Group Instruction ◽  
Data Set ◽  
School Year ◽  
Minimum Number ◽  
Representative Samples

Purpose Although sampling teachers' child-directed speech in school settings is needed to understand the influence of linguistic input on child outcomes, empirical guidance for measurement procedures needed to obtain representative samples is lacking. To optimize resources needed to transcribe, code, and analyze classroom samples, this exploratory study assessed the minimum number and duration of samples needed for a reliable analysis of conventional and researcher-developed measures of teacher talk in elementary classrooms. Method This study applied fully crossed, Person (teacher) × Session (samples obtained on 3 separate occasions) generalizability studies to analyze an extant data set of three 10-min language samples provided by 28 general and special education teachers recorded during large-group instruction across the school year. Subsequently, a series of decision studies estimated of the number and duration of sessions needed to obtain the criterion g coefficient ( g > .70). Results The most stable variables were total number of words and mazes, requiring only a single 10-min sample, two 6-min samples, or three 3-min samples to reach criterion. No measured variables related to content or complexity were adequately stable regardless of number and duration of samples. Conclusions Generalizability studies confirmed that a large proportion of variance was attributable to individuals rather than the sampling occasion when analyzing the amount and fluency of spontaneous teacher talk. In general, conventionally reported outcomes were more stable than researcher-developed codes, which suggests some categories of teacher talk are more context dependent than others and thus require more intensive data collection to measure reliably.

RGB Frequency Based Image Steganography

2020 ◽  
pp. 549-553
Author(s):  
Himanshu Kumar ◽  
Nitesh Kumar
Keyword(s):  
Image Steganography ◽  
Minimum Number

In this paper, we introduced a new RGB technique for image steganography. In this technique we introduced the idea of storing a different number of bits per channel (R, G or B) of a pixel based on the frequency of color values of pixel. The higher color frequency retains the maximum number of bits and lower color frequency stores the minimum number of bits.

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