In this paper, we formulate and investigate a generalized consensus algorithm which makes an attempt to unify distributed averaging and maximizing algorithms considered in the literature. Each node iteratively updates its state as a time-varying weighted average of its own state, the minimal state, and the maximal state of its neighbors. This part of the paper focuses on time-dependent communication graphs. We prove that finite-time consensus is almost impossible for averaging under this uniform model. Then various necessary and/or sufficient conditions are presented on the consensus convergence. The results characterize some similarities and differences between distributed averaging and maximizing algorithms.