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For the task of moving a group of indistinguishable agents on a connectedgraph with unit edge lengths into an arbitrary goal formation, it waspreviously shown that distance optimal paths can be scheduled to complete witha tight convergence time guarantee, using a fully centralized algorithm. Inthis study, we show that the problem formulation in fact induces a morefundamental ordering of the vertices on the underlying graph network, whichdirectly leads to a more intuitive scheduling algorithm that assures the sameconvergence time and runs faster. More importantly, this structure enables adistributed scheduling algorithm once individual paths are assigned to theagents, which was not possible before. The vertex ordering also readily extendsto more general graphs - those with non-unit capacities and edge lengths - forwhich we again guarantee the convergence time until the desired formation isachieved.

Shortest Path Set Induced Vertex Ordering and its Application to Distributed Distance Optimal Multi-agent Formation Path Planning