2011/04/27 by Henry Lam, Zhenming Liu, Lam, Henry +7 · 1 citation
Computer Science · Mathematics · #Algorithms and Data Compression #Bounded function #Combinatorics #Computer science #Discrete Mathematics (cs.DM) #Discrete mathematics #Euclidean space #FOS: Computer and information sciences #FOS: Mathematics #Mathematical analysis #Mathematics #Optimization and Search Problems #Probability (math.PR) #Random walk #Space (punctuation) #Statistics #Stochastic processes and statistical mechanics #cs.DM #math.PR
paper · pdf · doi:10.48550/arxiv.1104.5268
58 pages, 1 figure
openalex publication_date 2011/04/27 · arxiv created 2011/10/02 · arxiv updated 2011/10/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
We study a natural information dissemination problem for multiple mobile agents in a bounded Euclidean space. Agents are placed uniformly at random in the d-dimensional space \-n, ..., n\d at time zero, and one of the agents holds a piece of information to be disseminated. All the agents then perform independent random walks over the space, and the information is transmitted from one agent to another if the two agents are sufficiently close. We wish to bound the total time before all agents receive the information (with high probability). Our work extends Pettarin et al.'s work (Infectious random walks, arXiv:1007.1604v2, 2011), which solved the problem for d ≤ 2. We present tight bounds up to polylogarithmic factors for the case d = 3. (While our results extend to higher dimensions, for space and readability considerations we provide only the case d=3 here.) Our results show the behavior when d ≥ 3 is qualitatively different from the case d ≤ 2. In particular, as the ratio between the volume of the space and the number of agents varies, we show an interesting phase transition for three dimensions that does not occur in one or two dimensions.