13 research outputs found
The coalescing-branching random walk on expanders and the dual epidemic process
Information propagation on graphs is a fundamental topic in distributed
computing. One of the simplest models of information propagation is the push
protocol in which at each round each agent independently pushes the current
knowledge to a random neighbour. In this paper we study the so-called
coalescing-branching random walk (COBRA), in which each vertex pushes the
information to randomly selected neighbours and then stops passing
information until it receives the information again. The aim of COBRA is to
propagate information fast but with a limited number of transmissions per
vertex per step. In this paper we study the cover time of the COBRA process
defined as the minimum time until each vertex has received the information at
least once. Our main result says that if is an -vertex -regular graph
whose transition matrix has second eigenvalue , then the COBRA cover
time of is , if is greater than a positive
constant, and , if . These bounds are independent of and hold for . They improve the previous bound of for expander graphs.
Our main tool in analysing the COBRA process is a novel duality relation
between this process and a discrete epidemic process, which we call a biased
infection with persistent source (BIPS). A fixed vertex is the source of an
infection and remains permanently infected. At each step each vertex other
than selects neighbours, independently and uniformly, and is
infected in this step if and only if at least one of the selected neighbours
has been infected in the previous step. We show the duality between COBRA and
BIPS which says that the time to infect the whole graph in the BIPS process is
of the same order as the cover time of the COBRA proces
Static and expanding grid coverage with ant robots: Complexity results
AbstractIn this paper we study the strengths and limitations of collaborative teams of simple agents. In particular, we discuss the efficient use of âant robotsâ for covering a connected region on the Z2 grid, whose area is unknown in advance, and which expands at a given rate, where n is the initial size of the connected region. We show that regardless of the algorithm used, and the robotsâ hardware and software specifications, the minimal number of robots required in order for such a coverage to be possible is Ω(n). In addition, we show that when the region expands at a sufficiently slow rate, a team of Î(n) robots could cover it in at most O(n2lnn) time. This completion time can even be achieved by myopic robots, with no ability to directly communicate with each other, and where each robot is equipped with a memory of size O(1) bits w.r.t. the size of the region (therefore, the robots cannot maintain maps of the terrain, nor plan complete paths). Regarding the coverage of non-expanding regions in the grid, we improve the current best known result of O(n2) by demonstrating an algorithm that guarantees such a coverage with completion time of O(1kn1.5+n) in the worst case, and faster for shapes of perimeter length which is shorter than O(n)
A Time-Space Trade-off for Undirected s-t Connectivity
Version 3 makes use of the Metropolis-Hastings walkInternational audienceIn this paper, we make use of the Metropolis-type walks due to Nonaka et al. (2010) to provide a faster solution to the --connectivity problem in undirected graphs (USTCON). As our main result, we propose a family of randomized algorithms for USTCON which achieves a time-space product of in graphs with nodes and edges (where the -notation disregards poly-logarithmic terms). This improves the previously best trade-off of , due to Feige (1995). Our algorithm consists in deploying several short Metropolis-type walks, starting from landmark nodes distributed using the scheme of Broder et al. (1994) on a modified input graph. In particular, we obtain an algorithm running in time which is, in general, more space-efficient than both BFS and DFS. We close the paper by showing how to fine-tune the Metropolis-type walk so as to match the performance parameters (e.g., average hitting time) of the unbiased random walk for any graph, while preserving a worst-case bound of on cover time
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