Research output: Contribution to journal › Article

Colin Desmond Cooper, Andrew Jabez McDowell, Tomasz Radzik, Nicolas Andres Rivera Aburto, Takeharu Shiraga

Original language | English |
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Journal | RANDOM STRUCTURES AND ALGORITHMS |

State | Accepted/In press - 28 Sep 2018 |

We study a synchronous process called dispersion. Initially M particles are placed at a distinguished origin vertex of a graph G. At each time step, at each vertex v occupied by more than one particle at the beginning of this step, each of these particles moves to a neighbour of v chosen independently and uniformly at random. The process ends at the first step when no vertex is occupied by more than one particle.
For the complete graph K_n, for any constant d > 1, with high probability, if M <= n/2(1 - d), the dispersion process finishes in O(log n) steps, whereas if M >= n/2(1 + d), the process needs e^Omega(n) steps to complete, if ever. A lazy variant of the process exhibits analogous behaviour but at a higher threshold, thus allowing faster dispersion of more particles.
For paths, trees, grids, hypercubes and Abelian Cayley graphs of large enough size, we give bounds on the time to finish and the maximum distance traveled from the origin as a function of M.

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