We propose ZAP, an algorithm for the distributed channel assignment in cognitive radio (CR) networks. CRs are capable of identifying underutilized licensed bands of the spectrum, allowing their reuse by secondary users without interfering with primary users. In this context, efficient channel assignment is challenging as ideally it must be simple, incur acceptable communication overhead, provide timely response, and be adaptive to accommodate frequent changes in the network. Another challenge is the optimization of network capacity through interference minimization. In contrast to related work, ZAP addresses these challenges with a fully distributed approach based only on local (neighborhood) knowledge, while significantly reducing computational costs and the number of messages required for channel assignment. Simulations confirm the efficiency of ZAP in terms of (i) the performance tradeoff between different metrics and (ii) the fast achievement of a suitable assignment solution regardless of network size and density.