We consider replication-based distributed storage systems in which each node stores the same quantum of data and each data bit stored has the same replication factor across the nodes. Such systems are referred to as balanced distributed databases. When existing nodes leave or new nodes are added to this system, the balanced nature of the database is lost, either due to the reduction in the replication factor, or the non-uniformity of the storage at the nodes. This triggers a rebalancing algorithm, that exchanges data between the nodes so that the balance of the database is reinstated. The goal is then to design rebalancing schemes with minimal communication load. In a recent work by Krishnan \textit{et al.}, coded transmissions were used to rebalance a carefully designed distributed database from a node removal or addition. These coded rebalancing schemes have optimal communication load, however, require the file-size to be at least exponential in the system parameters. In this work, we consider a cyclic balanced database (where data is cyclically placed in the system nodes) and present coded rebalancing schemes for node removal and addition in such a database. Our algorithms have time-complexity polynomial in the system parameters. These databases (and the associated rebalancing schemes) require the file-size to be only cubic in the number of nodes in the system. In the node addition scenario, the rebalancing scheme presented is a simple uncoded scheme, which we show has optimal load. We show an information-theoretic lower bound for single node-removal rebalancing under the requirement of achieving a target cyclic database, and show that our achievable rebalancing loads are order-optimal (in the number of nodes), more specifically within a multiplicative gap of $2$ from the lower bound obtained.

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