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 $D$ 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 $2m$ 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 $O(m^4 D^2 + mD \log k)$ and $O(m^5 k^2)$ 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 $k = 2$ walks

Robustness of the Rotor-Router Mechanism

International audienceThe rotor-router model, also called the Propp machine, was first considered as a deter-ministic alternative to the random walk. The edges adjacent to each node v (or equivalently, the exit ports at v) are arranged in a fixed cyclic order, which does not change during the exploration. Each node v maintains a port pointer π(v) which indicates the exit port to be adopted by an agent on the conclusion of the next visit to this node (the "next exit port"). The rotor-router mechanism guarantees that after each consecutive visit at the same node, the pointer at this node is moved to the next port in the cyclic order. It is known that, in an undirected graph G with m edges, the route adopted by an agent controlled by the rotor-router mechanism forms eventually an Euler tour based on arcs obtained via replacing each edge in G by two arcs with opposite direction. The process of ushering the agent to an Euler tour is referred to as the lock-in problem. In [Yanovski et al., Algorithmica 37(3), 165–186 (2003)], it was proved that, independently of the initial configuration of the rotor-router mechanism in G, the agent locks-in in time bounded by 2mD, where D is the diameter of G. In this paper we examine the dependence of the lock-in time on the initial configuration of the rotor-router mechanism. Our analysis is performed in the form of a game between a player P intending to lock-in the agent in an Euler tour as quickly as possible and its adversary A with the counter objective. We consider all cases of who decides the initial cyclic orders and the initial values π(v). We show, for example, that if A provides its own port numbering after the initial setup of pointers by P, the complexity of the lock-in problem is O(m·min{log m, D}). We also investigate the robustness of the rotor-router graph exploration in presence of faults in the pointers π(v) or dynamic changes in the graph. We show, for example, that after the exploration establishes an Eulerian cycle, if k edges are added to the graph, then a new Eulerian cycle is established within O(km) steps

О направленном перемещении графоходного автомата без компаса на бесконечной цепи

Решена задача организации направленного перемещения графоходного автомата без компаса на бесконечной цепи (т. е. бесконечном связном 2-регулярном графе). Получены необходимые и достаточные условия в виде ограничений на свойства автомата и разметку цепи, при которых автомат сохраняет направление перемещения на цепи. Предложены два типа вершинной разметки цепи, допускающие направленное перемещение автомата: так называемые детерминированная и слабо детерминированная разметки. Разработаны методы и алгоритмы обхода автоматом конечных и бесконечных помеченных цепей. Для обоих типов разметки разработаны алгоритмы разметки цепей, все вершины которых не помечены или помечены одной и той же меткой. Полученные результаты закладывают основы для изучения навигации автоматов без компаса и их коллективов в стационарных однородных дискретных средах.Розв’язано задачу органiзацiї спрямованого перемiщення графохiдного автомату без компаса на нескiнченному ланцюзi (тобто нескiнченному зв’язному 2-регулярному графi). Отриманi необхiднi та достатнi умови у виглядi обмежень на властивостi автомата i розмiтку ланцюга, за яких автомат зберiгає напрямок перемiщення на ланцюзi. Запропоновано два типи вершинної розмiтки ланцюгу, що допускають спрямоване перемiщення автомата: так званi детермiнована i слабо детермiнована розмiтки. Розроблено методи та алгоритми обходу автоматом скiнченних i нескiнченних помiчених ланцюгiв. Для обох типiв розмiтки розроблено алгоритми розмiтки ланцюгiв, усi вершини яких не позначенi або позначенi однiєю i тiєю ж позначкою. Отриманi результати закладають основи для вивчення навiгацiї автоматiв без компасу та їх колективiв у стацiонарних однорiдних дискретних середовищах.This paper deals with the problem of organizing a directional movement of a graph-walking automaton on infinite path graph (i.e. infinite connected two-regular graph)

Exploiting graph structure in Active SLAM

Aplicando análisis provenientes de la teoría de grafos, la teoría espectral de grafos, la exploración de grafos en línea, generamos un sistema de SLAM activo que incluye la planificación de rutas bajo incertidumbre, extracción de grafos topológicos de entornos y SLAM activo \'optimo.En la planificación de trayectorias bajo incertidumbre, incluimos el análisis de la probabilidad de asociación correcta de datos. Reconociendo la naturaleza estocástica de la incertidumbre, demostramos que planificar para minimizar su valor esperado es más fiable que los actuales algoritmos de planificación de trayectorias con incertidumbre.Considerando el entorno como un conjunto de regiones convexas conectadas podemos tratar la exploración robótica como una exploración de grafos en línea. Se garantiza una cobertura total si el robot visita cada región. La mayoría de los métodos para segmentar el entorno están basados en píxeles y no garantizan que las regiones resultantes sean convexas, además pocos son algoritmos incrementales. En base a esto, modificamos un algoritmo basado en contornos en el que el entorno se representa como un conjunto de polígonos que debe segmentarse en un conjunto de polígonos pseudo convexos. El resultado es un algoritmo de segmentación que produjo regiones pseudo-convexas, robustas al ruido, estables y que obtienen un gran rendimiento en los conjuntos de datos de pruebas.La calidad de un algoritmo se puede medir en términos de cuan cercano al óptimo está su rendimiento. Con esta motivación definimos la esencia de la tarea de exploración en SLAM activo donde las únicas variables son la distancia recorrida y la calidad de la reconstrucción. Restringiendo el dominio al grafo que representa el entorno y probando la relación entre la matriz asociada a la exploración y la asociada al grafo subyacente, podemos calcular la ruta de exploración óptima.A diferencia de la mayoría de la literatura en SLAM activo, proponemos que la heurística para la exploración de grafos consiste en atravesar cada arco una vez. Demostramos que el tipo de grafos resultantes tiene un gran rendimiento con respecto a la trayectoria \'optima, con resultados superiores al 97 \% del \'optimo en algunas medidas de calidad.El algoritmo de SLAM activo TIGRE integra el algoritmo de extracción de grafos propuesto con nuestra versión del algoritmo de exploración incremental que atraviesa cada arco una vez. Nuestro algoritmo se basa en una modificación del algoritmo clásico de Tarry para la búsqueda en laberintos que logra el l\'imite inferior en la aproximación para un algoritmo incremental. Probamos nuestro sistema incremental en un escenario de exploración típico y demostramos que logra un rendimiento similar a los métodos fuera de línea y también demostramos que incluso el método \'optimo que visita todos los nodos calculado fuera de línea tiene un peor rendimiento que el nuestro.<br /

The power of a pebble : exploring and mapping directed graphs

Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1998.Includes bibliographical references (p. 36-39).by Amit Sahai.M.S

The multi-agent rotor-router on the ring: a deterministic alternative to parallel random walks

International audienceThe rotor-router mechanism was introduced as a deterministic alternative to the random walk in undirected graphs. In this model, an agent is initially placed at one of the nodes of the graph. Each node maintains a cyclic ordering of its outgoing arcs, and during successive visits of the agent, propagates it along arcs chosen according to this ordering in round-robin fashion. The behavior of the rotor-router is fully deterministic but its performance characteristics (cover time, return time) closely resemble the expected values of the corresponding parameters of the random walk. In this work Research partially supported by the ANR Project DISPLEXITY (ANR-11-BS02-014). This study has been carried out in the frame of the Investments for the future Programme IdEx Bordeaux-CP

Exploring Topological Environments

Simultaneous localization and mapping (SLAM) addresses the task of incrementally building a map of the environment with a robot while simultaneously localizing the robot relative to that map. SLAM is generally regarded as one of the most important problems in the pursuit of building truly autonomous mobile robots. This thesis considers the SLAM problem within a topological framework, in which the world and its representation are modelled as a graph. A topological framework provides a useful model within which to explore fundamental limits to exploration and mapping. Given a topological world, it is not, in general, possible to map the world deterministically without resorting to some type of marking aids. Early work demonstrated that a single movable marker was sufficient but is this necessary? This thesis shows that deterministic mapping is possible if both explicit place and back-link information exist in one vertex. Such 'directional lighthouse' information can be established in a number of ways including through the addition of a simple directional immovable marker to the environment. This thesis also explores non-deterministic approaches that map the world with less marking information. The algorithms are evaluated through performance analysis and experimental validation. Furthermore, the basic sensing and locomotion assumptions that underlie these algorithms are evaluated using a differential drive robot and an autonomous visual sensor

Exploring Topological Environments

Simultaneous localization and mapping (SLAM) addresses the task of incrementally building a map of the environment with a robot while simultaneously localizing the robot relative to that map. SLAM is generally regarded as one of the most important problems in the pursuit of building truly autonomous mobile robots. This thesis considers the SLAM problem within a topological framework, in which the world and its representation are modelled as a graph. A topological framework provides a useful model within which to explore fundamental limits to exploration and mapping. Given a topological world, it is not, in general, possible to map the world deterministically without resorting to some type of marking aids. Early work demonstrated that a single movable marker was sufficient but is this necessary? This thesis shows that deterministic mapping is possible if both explicit place and back-link information exist in one vertex. Such 'directional lighthouse' information can be established in a number of ways including through the addition of a simple directional immovable marker to the environment. This thesis also explores non-deterministic approaches that map the world with less marking information. The algorithms are evaluated through performance analysis and experimental validation. Furthermore, the basic sensing and locomotion assumptions that underlie these algorithms are evaluated using a differential drive robot and an autonomous visual sensor

Viscous streaming-enhanced inertial particle transport

Fluidic devices operating at the micro- and milli-meter scales employ several fundamental tasks involving pumping, mixing, separation, sorting, storing and transport of different fluids (or) species. An attractive fluid mechanism that can be leveraged to fulfill these wide range of tasks is viscous streaming, a non-linear effect characteristic of the scales above. In this thesis, we first show that numerical simulations based on the Remeshed Vortex Method (RVM) can accurately and efficiently capture viscous streaming dynamics. We test this algorithm on a wide variety of settings while simultaneously exhibiting the resultant streaming flow--structures, demonstrating both streaming's capability of effecting flow control and our solver's robustness in capturing these structures. We then consider the problem of an idealized two-dimensional inertial particle transport and prove that transport can be augmented by sensibly utilizing the streaming mechanism. We then successfully perform a forward--design study to devise shapes capable of enhanced transport using this mechanism, capitalizing on the insights gained from our demonstrations above. We envison such transport applications in the emergent technology of miniature robots, capable of traversing our blood stream to deliver payloads of therapeutical drugs