Information dissemination and search in networks are the most frequently performed operations in any large scale distributed system. Maximization of the number of distinctly visited nodes (coverage) under constraints is the prime target of any proposed algorithm for these operations. In order to quickly achieve high coverage, the existing algorithms tend to proliferate message packets at higher rate. Due to the nondeterministic nature of the algorithms, the blind proliferations create many redundant visits in the network which reduces the utilization of available resource. For the first time, we formally capture this existing tradeoff between the available time and available resource to achieve a certain amount of coverage in the network. Using methods from statistical mechanics, we showed that coverage can be optimized, without using any memory, for a certain combinations of resource and time. We also provide the algorithm which can achieve that optimal coverage. For the rest of the combinations of resource and time, we provide a notion of optimality employing a very little amount memory. We test our hypothesis through extensive computer simulations in various kinds of network topologies.