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Dispersion processes

Research output: Contribution to journalArticle

Original languageEnglish
Pages (from-to)561-585
JournalRANDOM STRUCTURES AND ALGORITHMS
Volume53
Issue number4
Early online date29 Oct 2018
DOIs
StatePublished - Dec 2018

King's Authors

Abstract

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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