We consider the SFC embedding (SFCE) problem in the Slice as a Service (SlaaS) model. In this model, a slice provider leases resources from multiple cloud and network providers in order to instantiate the Service Function Chain (SFC) requested by a slice tenant. As the slice provider has no visibility on the infrastructures of the resource providers, in which resources may be purchased and released quite rapidly, it has to query them to determine what are the possible allocations and their costs. We show that when there are many resource providers and many VNFs composing the SFC, the number of queries to be made for discovering a minimum cost SFC embedding grows quickly, leading to excessively long deployment times. In order to reduce the latter quantity, we propose to query resource providers strategically, rather than collecting the information on all possible allocations at once. We provide bounds on the number of queries to be made in this approach, and propose to exploit a Shortest Path Discovery algorithm in order to reduce this number of queries and thus the SFC deployment time. Our numerical results suggest that this algorithm is fairly efficient, and that the deployment times can be significantly shortened, in particular when initial estimates of allocation costs can be provided by the slice provider.