More efficient periodic traversal in anonymous undirected graphs

We consider the problem of periodic graph exploration in which a mobile entity with constant memory, an agent, has to visit all n nodes of an arbitrary undirected graph G in a periodic manner. Graphs are supposed to be anonymous, that is, nodes are unlabeled. However, while visiting a node, the robot has to distinguish between edges incident to it. For each node v the endpoints of the edges incident to v are uniquely identified by different integer labels called port numbers. We are interested in minimisation of the length of the exploration period. This problem is unsolvable if the local port numbers are set arbitrarily. However, surprisingly small periods can be achieved when assigning carefully the local port numbers. Dobrev et al. described an algorithm for assigning port numbers, and an oblivious agent (i.e. agent with no memory) using it, such that the agent explores all graphs of size n within period 10n. Providing the agent with a constant number of memory bits, the optimal length of the period was previously proved to be no more than 3.75n (using a different assignment of the port numbers). In this paper, we improve both these bounds. More precisely, we show a period of length at most 4 1/3 n for oblivious agents, and a period of length at most 3.5n for agents with constant memory. Moreover, we give the first non-trivial lower bound, 2.8n, on the period length for the oblivious case

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

Periods, partial words, and an extension of a result of Guibas and Odlyzko

"A well known and unexpected result of Guibas and Odlyzko states that the set of all periods of a word is independent of the alphabet size (alphabets with one symbol are excluded here). More specifically, for every word u, there exists a word v over the alphabet {0, 1} such that u and v have the same length and the same set of periods. Recently, Blanchet-Sadri and Chriscoe extended this fundamental result to words with one "do not know" symbol also called partial words with one hole. They showed that for every partial word u with one hole, there exists a partial word v with at most one hole over the alphabet {0, 1} such that u and v have the same length, the same set of periods, the same set of weak periods, and H(v) H(u)," where H(u) (respectively, H(v)) denotes the set of holes of u (respectively, v). In this paper, we extend this result further to a large class of partial words. Given a partial word u belonging to that class, our proof provides an algorithm to compute a partial word v over {0, 1} sharing the same length and same sets of periods and weak periods as u, and satisfying H(v) H(u)."--Abstract from author supplied metadata

Parameterized Algorithms for String Matching to DAGs: Funnels and Beyond

The problem of String Matching to Labeled Graphs (SMLG) asks to find all the paths in a labeled graph G = (V, E) whose spellings match that of an input string S ? ?^m. SMLG can be solved in quadratic O(m|E|) time [Amir et al., JALG 2000], which was proven to be optimal by a recent lower bound conditioned on SETH [Equi et al., ICALP 2019]. The lower bound states that no strongly subquadratic time algorithm exists, even if restricted to directed acyclic graphs (DAGs). In this work we present the first parameterized algorithms for SMLG on DAGs. Our parameters capture the topological structure of G. All our results are derived from a generalization of the Knuth-Morris-Pratt algorithm [Park and Kim, CPM 1995] optimized to work in time proportional to the number of prefix-incomparable matches. To obtain the parameterization in the topological structure of G, we first study a special class of DAGs called funnels [Millani et al., JCO 2020] and generalize them to k-funnels and the class ST_k. We present several novel characterizations and algorithmic contributions on both funnels and their generalizations

An updated annotated bibliography on arc routing problems

The number of arc routing publications has increased significantly in the last decade. Such an increase justifies a second annotated bibliography, a sequel to Corberán and Prins (Networks 56 (2010), 50–69), discussing arc routing studies from 2010 onwards. These studies are grouped into three main sections: single vehicle problems, multiple vehicle problems and applications. Each main section catalogs problems according to their specifics. Section 2 is therefore composed of four subsections, namely: the Chinese Postman Problem, the Rural Postman Problem, the General Routing Problem (GRP) and Arc Routing Problems (ARPs) with profits. Section 3, devoted to the multiple vehicle case, begins with three subsections on the Capacitated Arc Routing Problem (CARP) and then delves into several variants of multiple ARPs, ending with GRPs and problems with profits. Section 4 is devoted to applications, including distribution and collection routes, outdoor activities, post-disaster operations, road cleaning and marking. As new applications emerge and existing applications continue to be used and adapted, the future of arc routing research looks promising.info:eu-repo/semantics/publishedVersio

Collision-free network exploration

International audienc

Patrolling Grids with a Bit of Memory

We study the following problem in elementary robotics: can a mobile agent with $b$ bits of memory, which is able to sense only locations at Manhattan distance $V$ or less from itself, patrol a $d$-dimensional grid graph? We show that it is impossible to patrol some grid graphs with $0$ bits of memory, regardless of $V$, and give an exact characterization of those grid graphs that can be patrolled with $0$ bits of memory and visibility range $V$. On the other hand, we show that, surprisingly, an algorithm exists using $1$ bit of memory and $V=1$ that patrols any $d$-dimensional grid graph

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