Low-power and Lossy Networks (LLNs) have been a key component in the Internet of Things (IoT) paradigm. Recently, a standardized algorithm, namely Trickle algorithm, is adopted for routing information maintenance in such networks. This algorithm is originally designed for disseminating code updates through a wireless sensor network. Thus, when it comes to routing maintenance in LLNs, Trickle suffers from some issues related to power, convergence time, network overhead and load-distribution. In this paper, a new algorithm for maintaining the network topology in LLNs is developed motivated by Trickle weaknesses, namely, Drizzle algorithm. Unlike Trickle, Drizzle uses an adaptive suppression mechanism that permits the nodes to have different transmission probabilities consistent with their transmission history. Another distinctive feature of Drizzle in comparison with Trickle, is the absence of the listen-only period from Drizzle's intervals, thus, leading to faster convergence time. Furthermore, a new policy for setting the redundancy coefficient has been used to mitigate the negative effect of the short-listen problem presented when removing the listen-only period and to further boost the fairness in the network. Our extensive simulation experiments confirm the superiority of the proposed algorithm over Trickle under different operating conditions.