  Unlike the classical distributed consensus protocols enabling the group of
agents as a whole to reach an agreement regarding a certain quantity of
interest in a distributed fashion, the distributed biased min-consensus
protocol (DBMC) has been proven to generate advanced complexity pertaining to
solving the shortest path problem. As such a protocol is commonly incorporated
as the first step of a hierarchical architecture in real applications, e.g.,
robots path planning, management of dispersed computing services, an impedance
limiting the application potential of DBMC lies in, the lack of results
regarding to its convergence within a user-assigned time. In this paper, we
first propose two control strategies ensuring the state error of DBMC decrease
exactly to zero or a desired level manipulated by the user, respectively. To
compensate the high feedback gains incurred by these two control strategies,
this paper further investigates the nominal DBMC itself. By leveraging small
gain based stability tools, this paper also proves the global exponential
input-to-state stability of DBMC, outperforming its current stability results.
Simulations have been provided to validate the efficacy of our theoretical
result.
