In building failure diagnosis systems for discrete-event systems (DES), two performance indexes must be specified: first, the maximum time the diagnosis system takes to detect the failure occurrence, usually referred to as T-diagnosability, and second, if no time information is available, how many events must occur after the failure in order for the diagnosis system to detect and isolate its occurrence, usually referred to as K-diagnosability. In this technical note, we consider the problem of verifying if these specifications can be met for DES modeled by finite-weighted automaton: we regard the ordinary finite automaton as a particular case of weighted automaton with all weights equal to one. The resulting algorithms have polynomial time complexity. being the lowest worst case computational complexity among those already proposed in the literature for T- and K-codiagnosability verification of finite-state automata.