Memory Optimal Dispersion by Anonymous Mobile Robots

Consider a team of $$k \le n$$ autonomous mobile robots initially placed at node an arbitrary graph G with n nodes. The dispersion problem asks for distributed algorithm that allows the to reach configuration in which each robot is distinct graph. If are anonymous, i.e., they do not have any unique identifiers, then solvable by deterministic algorithm. However, can be solved even anonymous if given access fair coin use generate random bits. In this setting, it known require $$\varOmega (\log {\varDelta })$$ bits memory achieve dispersion, where $$\varDelta $$ maximum degree G. On other hand, best upper bound $$min \{\varDelta , max\{\log }, \log {D}\}\}$$ (D = diameter G), $$\omega depending on values and D. paper, we close gap presenting optimal requiring $$O(\log memory.