Generalized distributed mutual exclusion algorithms allow processes to concurrently access a set of shared resources. However, they must ensure an exclusive access to each resource. In order to avoid deadlocks, many of them are based on the strong assumption of a prior knowledge about conflicts between processes' requests. Some other approaches, which do not require such a knowledge, exploit broadcast mechanisms or a global lock, degrading message complexity and synchronization cost. We propose in this paper a new solution for shared resources allocation which reduces the communication between non-conflicting processes without a prior knowledge of processes conflicts. Performance evaluation results show that our solution improves resource use rate by a factor up to 20 compared to a global lock based algorithm.