523 research outputs found
Lock-in Problem for Parallel Rotor-router Walks
The rotor-router model, also called the Propp machine, was introduced as a
deterministic alternative to the random walk. In this model, a group of
identical tokens are initially placed at nodes of the graph. Each node
maintains a cyclic ordering of the outgoing arcs, and during consecutive turns
the tokens are propagated along arcs chosen according to this ordering in
round-robin fashion. The behavior of the model is fully deterministic. Yanovski
et al.(2003) proved that a single rotor-router walk on any graph with m edges
and diameter stabilizes to a traversal of an Eulerian circuit on the set of
all 2m directed arcs on the edge set of the graph, and that such periodic
behaviour of the system is achieved after an initial transient phase of at most
2mD steps. The case of multiple parallel rotor-routers was studied
experimentally, leading Yanovski et al. to the conjecture that a system of k
\textgreater{} 1 parallel walks also stabilizes with a period of length at
most steps. In this work we disprove this conjecture, showing that the
period of parallel rotor-router walks can in fact, be superpolynomial in the
size of graph. On the positive side, we provide a characterization of the
periodic behavior of parallel router walks, in terms of a structural property
of stable states called a subcycle decomposition. This property provides us the
tools to efficiently detect whether a given system configuration corresponds to
the transient or to the limit behavior of the system. Moreover, we provide
polynomial upper bounds of and on the
number of steps it takes for the system to stabilize. Thus, we are able to
predict any future behavior of the system using an algorithm that takes
polynomial time and space. In addition, we show that there exists a separation
between the stabilization time of the single-walk and multiple-walk
rotor-router systems, and that for some graphs the latter can be asymptotically
larger even for the case of walks
The Fagnano Triangle Patrolling Problem
We investigate a combinatorial optimization problem that involves patrolling
the edges of an acute triangle using a unit-speed agent. The goal is to
minimize the maximum (1-gap) idle time of any edge, which is defined as the
time gap between consecutive visits to that edge. This problem has roots in a
centuries-old optimization problem posed by Fagnano in 1775, who sought to
determine the inscribed triangle of an acute triangle with the minimum
perimeter. It is well-known that the orthic triangle, giving rise to a periodic
and cyclic trajectory obeying the laws of geometric optics, is the optimal
solution to Fagnano's problem. Such trajectories are known as Fagnano orbits,
or more generally as billiard trajectories. We demonstrate that the orthic
triangle is also an optimal solution to the patrolling problem.
Our main contributions pertain to new connections between billiard
trajectories and optimal patrolling schedules in combinatorial optimization. In
particular, as an artifact of our arguments, we introduce a novel 2-gap
patrolling problem that seeks to minimize the visitation time of objects every
three visits. We prove that there exist infinitely many well-structured
billiard-type optimal trajectories for this problem, including the orthic
trajectory, which has the special property of minimizing the visitation time
gap between any two consecutively visited edges. Complementary to that, we also
examine the cost of dynamic, sub-optimal trajectories to the 1-gap patrolling
optimization problem. These trajectories result from a greedy algorithm and can
be implemented by a computationally primitive mobile agent
When Patrolmen Become Corrupted: Monitoring a Graph Using Faulty Mobile Robots
A team of k mobile robots is deployed on a weighted graph whose edge weights represent distances. The robots move perpetually along the domain, represented by all points belonging to the graph edges, without exceeding their maximum speed. The robots need to patrol the graph by regularly visiting all points of the domain. In this paper, we consider a team of robots (patrolmen), at most f of which may be unreliable, i.e., they fail to comply with their patrolling duties. What algorithm should be followed so as to minimize the maximum time between successive visits of every edge point by a reliable patrolman? The corresponding measure of efficiency of patrolling called idleness has been widely accepted in the robotics literature. We extend it to the case of untrusted patrolmen; we denote by Ifk(G) the maximum time that a point of the domain may remain unvisited by reliable patrolmen. The objective is to find patrolling strategies minimizing Ifk(G). We investigate this problem for various classes of graphs. We design optimal algorithms for line segments, which turn out to be surprisingly different from strategies for related patrolling problems proposed in the literature. We then use these results to study general graphs. For Eulerian graphs G, we give an optimal patrolling strategy with idleness Ifk(G)=(f+1)|E|/k, where |E| is the sum of the lengths of the edges of G. Further, we show the hardness of the problem of computing the idle time for three robots, at most one of which is faulty, by reduction from 3-edge-coloring of cubic graphs—a known NP-hard problem. A byproduct of our proof is the investigation of classes of graphs minimizing idle time (with respect to the total length of edges); an example of such a class is known in the literature under the name of Kotzig graphs
CPC: Crime, Policing and Citizenship - Intelligent Policing and Big Data
Crime, Policing and Citizenship (CPC) – Space-Time Interactions of Dynamic Networks has been a major UK EPSRC-funded research project. It has been a multidisciplinary collaboration of geoinformatics, crime science, computer science and geography within University College London (UCL), in partnership with the Metropolitan Police Service (MPS). The aim of the project has been to develop new methods and applications in space-time analytics and emergent network complexity, in order to uncover patterning and interactions in crime, policing and citizen perceptions. The work carried out throughout the project will help inform policing at a range of scales, from the local to the city-wide, with the goal of reducing both crime and the fear of crime. The CPC project is timely given the tremendous challenging facing policing in big cities nationally and globally, as consequences of changes in society, population structure and economic well-being. It addresses these issues through an intelligent approach to data-driven policing, using daily reported crime statistics, GPS traces of foot and vehicular patrols, surveys of public attitudes and geo-temporal demographic data of changing community structure. The analytic focus takes a spatio-temporal perspective, reflecting the strong spatial and temporal integration of criminal, policing and citizen activities. Street networks are used throughout as a basis for analysis, reflecting their role as a key determinant of urban structure and the substrate on which crime and policing take place. The project has presented a manifesto for ‘intelligent policing’ which embodies the key issues arising in the transition from Big Data into actionable insights. Police intelligence should go beyond current practice, incorporating not only the prediction of events, but also how to respond to them, and how to evaluate the actions taken. Cutting-edge network-based crime prediction methods have been developed to accurately predict crime risks at street segment level, helping police forces to focus resources in the right places at the right times. Methods and tools have been implemented to support senior offices in strategic planning, and to provide guidance to frontline officers in daily patrolling. To evaluate police performance, models and tools have been developed to aid identification of areas requiring greater attention, and to analyse the patrolling behaviours of officers. Methods to understand and model confidence in policing have also been explored, suggesting strategies by which confidence in the police can be improved in different population segments and neighbourhood areas. A number of tools have been developed during the course of the project include data-driven methods for crime prediction and for performance evaluation. We anticipate that these will ultimately be adopted in daily policing practice and will play an important role in the modernisation of policing. Furthermore, we believe that the approaches to the building of public trust and confidence that we suggest will contribute to the transformation and improvement of the relationship between the public and police
Fast Two-Robot Disk Evacuation with Wireless Communication
In the fast evacuation problem, we study the path planning problem for two
robots who want to minimize the worst-case evacuation time on the unit disk.
The robots are initially placed at the center of the disk. In order to
evacuate, they need to reach an unknown point, the exit, on the boundary of the
disk. Once one of the robots finds the exit, it will instantaneously notify the
other agent, who will make a beeline to it.
The problem has been studied for robots with the same speed~\cite{s1}. We
study a more general case where one robot has speed and the other has speed
. We provide optimal evacuation strategies in the case that by showing matching upper and lower bounds on the
worst-case evacuation time. For , we show (non-matching)
upper and lower bounds on the evacuation time with a ratio less than .
Moreover, we demonstrate that a generalization of the two-robot search strategy
from~\cite{s1} is outperformed by our proposed strategies for any .Comment: 18 pages, 10 figure
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