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We address the problem of computing with mobile agents having small local maps. Several trade-offs concerning the radius of the local maps, the number of agents, the time complexity and the number of agent moves are proven. Our results are based on a generic simulation scheme allowing to transform any message passing algorithm into a mobile agent one. For instance, we show that using a near linear (resp. sublinear) number of agents having local maps of polylogarithmic (resp. sublinear) radius allows us to obtain a polylogarithmic (resp. sublinear) ratio between the time complexity of a message passing algorithm and its mobile agent counterpart. As a fundamental application, we show that there exists a universal algorithm that computes, from scratch, any global labeling function of any graph using nmobile agents knowing their o(n茂戮驴)-neighborhood (resp. without any neighborhood knowledge) in $\widetilde{O}(D)$ time (resp. $\widetilde{O}(\Delta+ D)$ expected time), where n,D,Δare respectively the size, the diameter, the maximum degree of the graph and 茂戮驴is an arbitrary small constant. For the leader election problem (resp. BFS tree construction), we obtain $\widetilde{O}(D)$ time algorithms under the additional restriction of using mobile agents having only logO(1)n(resp. $\widetilde{O}(n)$) memory bits.To the extent of our knowledge, the impact of local maps on mobile agent algorithms has not been studied in previous works. Our results prove that small local maps can have a strong global impact on the power of computing with mobile agents. Thus, we believe that the local map concept is likely to play an important role to a better understanding of the locality nature of mobile agent algorithms.