This technical note introduces a new generalized consensus problem where nodes in a network are guaranteed to reach the final consensus on a common vector whose elements are exactly the initial values arbitrarily chosen by nodes. We propose a fully decentralized algorithm able to solve the above problem and derive conditions to guarantee that consensus is reached in a finite number of steps.
In particular, we prove that in a network composed by n nodes, the proposed consensus problem is solvable in at most 2n steps. Moreover, we introduce a finite-field solution based on the Chinese Remainder Theorem (CRT) able to reduce the complexity of the proposed approach in the case of capacity constraints, and we discuss an illustrative case study.