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 k 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 G is an n-vertex r-regular graph whose transition matrix has second eigenvalue λ, then the COBRA cover time of G is O(log n), if 1 − λ is greater than a positive constant, and O((log n)/(1 − λ)3 )), if 1 − λ ≫ log(n)/n. These bounds are independent of r and hold for 3 ≤ r ≤ n − 1. They improve the previous bound of O(log 2 n) for expander graphs [Dutta et al., SPAA 2013].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 v is the source of an infection and remains permanently infected. At each step each vertex u other than v selects k neighbours, independently and uniformly, and u 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 process.
We consider an asynchronous voting process on graphs called discordant voting, which can be described as follows. Initially each vertex holds one of two opinions, red or blue. Neighbouring vertices with different opinions interact pairwise along an edge. After an interaction both vertices have the same colour. The quantity of interest is the time to reach consensus, i.e. the number of steps needed for all vertices have the same colour. We show that for a given initial colouring of the vertices, the expected time to reach consensus, depends strongly on the underlying graph and the update rule (push, pull, oblivious).
Grid pathfinding, an old AI problem, is central for the development of navigation systems for autonomous agents. A surprising fact about the vast literature on this problem is that very limited neighborhoods have been studied. Indeed, only the 4- and 8-neighborhoods are usually considered, and rarely the 16-neighborhood. This paper describes three contributions that enable the construction of effective grid path planners for extended 2k-neighborhoods. First, we provide a simple recursive definition of the 2k-neighborhood in terms of the 2k–1-neighborhood. Second, we derive distance functions, for any k >1, which allow us to propose admissible heurisitics which are perfect for obstacle-free grids. Third, we describe a canonical ordering which allows us to implement a version of A* whose performance scales well when increasing k. Our empirical evaluation shows that the heuristics we propose are superior to the Euclidean distance (ED) when regular A* is used. For grids beyond 64 the overhead of computing the heuristic yields decreased time performance compared to the ED. We found also that a configuration of our A*-based implementation, without canonical orders, is competitive with the "any-angle" path planner Theta$^*$ both in terms of solution quality and runtime.
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