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The resource discovery problem arises in the context of peer to peer (P2P) networks, where at any point of time a peer may be placed at or removed from any location over a general purpose network (e.g., an Internet site). A vertex (peer) can communicate with another vertex directly if and only if it knows a certain routing information to that other vertex. Hence, it is critical for peers to convey this routing information to each other. The problem was formalized by Harchol-Balter et al. (1999). The routing information needed for a vertex to reach another peer is that peer's identifier (e.g., IP address). A logical directed edge represents the fact that the peer at the tail of the edge knows the IP address of the one at its head. A number of algorithms were developed by Harchol-Balter et al. for this problem in the model of a synchronous network over a weakly connected directed graph. The best of these algorithms was randomized. Subsequently, a deterministic algorithm for the problem on synchronous networks with improved complexity was presented by Kutten et al. (2001). The current paper extends this deterministic algorithm to the environment of asynchronous networks, maintaining similar complexities (translated to the asynchronous model). These are lower than the complexities that would be needed to synchronize the system. The main technical difficulty in a directed, weakly connected system is to ensure that vertices take consistent steps, even if their knowledge about each other is not symmetric, and even if there is no timeout mechanism (which does exist in synchronous systems) to assist in that.